Functional Programming
and Input Output
Andrew Donald Gordon
King s College
A dissertation submitted to the University of Cambridge
towards the degree of Do ctor of Philosophy
August
Contents
Summary vii
Preface ix
Preface to the Bo ok Edition xi
Intro duction
A brief history of functional I O
Semantics of functional languages
Hyp othesis
Synopsis
Results
How to read the dissertation
Mathematical preliminaries
A calculus of recursive typ es
Typ es of
Terms of
Reduction in
Combinators to simulate Elim and Intro
Two examples
Strong normalisation for
A metalanguage for semantics
Syntax of M
Typ e assignmentinM
Op erational semantics of M i
ii CONTENTS
Convergence
Op erational precongruence
Contexts
Exp eriments and contextual exp eriments
Ground and con ned relations in M
Contextual and exp erimental order
Applicative and compatible similarity
Contextual order and applicative similarity
Theory of the metalanguage
Laws of op erational equivalence in M
Empty and one p ointtyp es in M
Iterated pro ducts and sums in M
Bo oleans in M
Natural numbers in M
An op erational theory of functional programming
H a small functional language
Ground and con ned relations in H
Op erational equivalence for H
Denotational equivalence for H
A theory of H programming
Derived laws of programming
HX an extension of H with exceptions
teletyp e I O Four mechanisms for
Side e ecting I O
Lab elled transition systems and bisimilarity
Stream transformers
Landin stream I O
Synchronised stream I O
Continuation passing I O
Maps b etween three of the mechanisms
Discussion
CONTENTS iii
Monadic I O
Programming I O in a monadic style
Aweak monad for teletyp e I O
Aweak monad for the MRC computer
Pro of of the motivating example
Conclusion
Summary
A calculus of recursivetyp es
A metalanguage for semantics
Op erational precongruence
Theory of the metalanguage
An op erational theory of functional programming
Four mechanisms for telet yp e I O
Monadic I O
Bibliography
Notation Index
Index
iv CONTENTS
List of Tables
Terms of M
Typ e assignment rules for M
Evaluation rules for M
A translation of M typ es and terms into
b
De nition of R
Rules concerning con ned relations
Laws of equivalence in M
Syntax of H
Typ e assignment rules for H
Some standard functions
Interpretations of Haskell functions in H
Op erational semantics of H
Denotational semantics of H
b
De nition of R and rules concerning con ned relations
Laws of equivalence in H
HX an extension of H with exceptions
Three mechanisms for functional I O
Translations in HX between three styles of I O
Aweak monad for teletyp e I O
Deriving the MRC programming mo del
The motivating example
A monadic style of I O for Standard ML v
vi LIST OF TABLES
Summary
A common attraction to functional programming is the ease with which pro ofs can be
given of program prop erties A common disapp ointment with functional programming
is the di culty of expressing input output I O while at the same time b eing able to
verify programs In this dissertation we show how a theory of functional programming
can b e smo othly extended to admit b oth an op erational semantics for functional I O and
veri cation of programs engaged in I O
The rst half develops the op erational theory of a semantic metalanguage used in the sec
ond half The metalanguage M is a simply typ ed calculus with pro duct sum function
lifted and recursive typ es We study two de nitions of op erational equivalence Morris
We style contextual equivalence and a typ ed form of Abramsky s applicative bisimulation
prove op erational extensionality for M that these two de nitions give rise to the same
op erational equivalence We prove equational laws that are analogous to the axiomatic
domain theory of LCF and derive a co induction principle
The second half de nes a small functional language H and shows how the semantics of
H can b e extended to accommo date I O H is essentially a fragmentofHaskell Wegive
b oth op erational and denotational semantics for H The denotational semantics uses M
in a case study of Moggi s prop osal to use monads to parameterise semantic descriptions
We de ne op erational and denotational equivalences on H and show that denotational
implies op erational equivalence We develop a theory of H based on equational laws and
a co induction principle
We study simpli ed forms of four widely implemented I O mechanisms side e ecting
Landin stream synchronised stream and continuation passing I O We give reasons why
side e ecting I O is unsuitable for lazy languages We extend the semantics of H to
include the other three mechanisms and prove that the three are equivalenttoeach other
in expressivepower
We investigate monadic I O a high level mo del for functional I O based on Wadler s
suggestion that monads can express interaction with state in a functional language We
describ e a simple monadic programming mo del and give its semantics as a particular form
of state transformer Using the semantics weverify a simple programming example vii
viii SUMMARY
Prayer
Let the light of truth and the help of grace be vital principles of action in us
that we may in the time of life attain the ends for which we live
and that our religion which begins in know ledge
may proceed in action settle in temper and end in happiness
Preface
This dissertation is the result of myown work and includes nothing which is the outcome
of work done in collab oration This dissertation is not substantially the same as any that I
have submitted or am currently submitting for a degree diploma or any other quali cation
at any other university
Iacknowledge here many p eople who have help ed me complete this work whether named
here or not I thank all the p eople who have beengoodtomeover mytimeinCambridge
y sup ervisor Larry Paulson who has read commented and Iam extremely grateful to m
discussed many drafts of my work Perhaps the most imp ortant thing I ve learnt from
Larry is to keep computer science relevant to practical computational concerns I am
grateful also to Alan Mycroft who acted as joint sup ervisor for fruitful discussions ab out
many drafts and ideas
Ihave had many useful conversations with my friend Roy Crole His help over the years
with understanding many mathematical ideas has been invaluable I am particularly
grateful to him for explaining Moggi s approach to denotational semantics to me in simple
terms and for commenting on an early draft of this dissertation in detail Robin Milner s
ideas have b een an enormous in uence on this dissertation I am grateful to have b een in
his undergraduate concurrency class at Edinburgh and to have had many helpful conver
with him during his sabbatical in Cambridge Jon Fairbairn enlivened my early sations
years in Cambridge with his idiosyncratic views he graciously passed on to me debates
ab out I O in the Haskell committee During his absence from the Lab he is greatly missed
I thank Andy Pitts for reading and discussing this work on many o ccasions and for his
encouragement Mike Gordon provided funding for me to try out my semantic ideas on
a data ow language for hardware description a go o d way to work out how semantics ix
x PREFACE
can serve practical needs Ken Mo o dy o ered me students to teach many encouraging
conversations and the o dd b ottle of wine I thank Roger Needham head of department
for many things not least that he suggest I apply for an internship at DEC SRCinPalo
Alto I learnt a great deal from my host Jim Horning and others at DEC for which I
am very grateful
I thank Ian Poole of the Medical Research Council at the Western General Hospital in
Edinburgh for the time he sp ent explaining their system to me asking searching questions
ab out functional I O and suggesting the example that motivates Chapter
This work was supp orted partially by an SERC studentship This dissertation was pre
pared on a Fire y computer donated by DEC Mart yn Johnson keeps the Fire ies going
one of his many services to the Lab The exp erimental side of this work owes much to
Dave Matthews and Mark Jones Dave Matthews made available his p ort of Poly ML to
the Fire y and Mark Jones his Gofer interpreter for Haskell
Juanito Camilleri Jo e Dixon Lee Fedder and Mads Rosendahl started with me they are
good friends Thanks to the following who read and discussed drafts of this work Nick
Benton Gavin Bierman Paul Curzon Eike Ritter and Stuart Wray My o cemates in
and TM have b een go o d company Mike Burrows was a go o d friend in my rst year
and during my time at DEC Mike Gordon s Hardware Veri cation Group is a stimulating
and friendly environment I v e enjoyed conversation with Tom Melham in all sorts of
places I thank the following for helpful conversations ab out functional I O and semantics
Peter Breuer John Cupitt Andrew Dwelly Nicholas Graham John Hughes Eugenio
Moggi Luke Ong Nigel Perry Gordon Plotkin Raman Sundaresh Allen Stoughton
Simon Thompson David N Turner John Williams and Ed Wimmers
I do not knowhow I could have sustained this work without the supp ort of p eople in two
good veggie shared houses Bryant s Bend in Palo Alto and Glob House in Cambridge
Evensong at King s signi es beyond what I can say My op ening prayer used often
is by Benjamin Whichcote provost Stephen Cherry the present at King s
chaplain has given me much My friends in the Student Christian Movement at Holy
Trinity St Columba s and the Romsey Lab our Club have b een go o d to know Thanks
esp ecially to Shelagh Phil Sue Cathy and Julie Yvonne is an Irish blessing
Thanks to my brother Dave for haggis and thuds amongst other gifts My rst and
greatest debt is to my Mum and Dad who never fail to love and supp ort me
Andrew D Gordon August
Preface to the Bo ok Edition
This b o ok is a slight revision of the submitted dissertation Ihave simpli ed the language
M intro duced in Chapter incorp orated some changes made to Chapter when it was
published in FPCA and simpli ed Chapter somewhat I have not however
attempted to take account in the main text of the publications that have app eared or
come to my attention since submission These include the following
Semantics of Functional Languages Recent textb o oks cover domain
theoretic semantics of recursively typ ed languages such as M Pitts gives a
simple pro of of adequacy in the context of recursively de ned domains a
result derived from his theory of relational prop erties of domains A domain
theoretic semantics for a fragmentofM app ears in a pap er with Crole Mason
and Talcott and Felleisen et al have written series of pap ers on entirely
op erational theories of functional programming Their work go es b eyond this disser
tation in considering side e ects It do es not however takeadvantage of applicative
bisimulation and co induction in the way Chapter do es Applicative bisimulation
is studied by Pitts and Stark in the presence of dynamically allo cated names
and by Sands as a measure of p erformance Je rey and Launchbury
haverecently develop ed theories of call by need evaluation
Graphical User Interfaces Several recent implementations including Concurrent
Clean Fudgets and Budgets express graphical user interfaces in a
functional style Noble is an extensivesurvey
Monadic I O Peyton Jones and Wadler s work has been further develop ed by
Launchbury and Peyton Jones The MRC pro ject using monadic I O is still in
progress Programs and sp eci cations are based on the monadic mo del suggested in
Chapter although formal pro ofs of program prop erties are impractical at present
This is a good opp ortunityto thank my examiners Simon Jones and Andrew Pitts Fi
company is nally I want to thank Lindy for all her supp ort and encouragement Her
good
Andrew D Gordon
May xi
xii PREFACE TO THE BOOK EDITION
Chapter
Intro duction
One of the main arguments in favour of functional programming is that it is easier to
prove prop erties of functional programs than of imp erative programs Pro ofs of functional
programs are claimed to be easier to construct than pro ofs of imp erative programs On
the other hand input output I O has long b een viewed as problematic for functional
languages I O mechanisms for functional languages have either been impractically in
expressiveornotbeenintegrated into the language semantics The working programmer
can argue that there is no published evidence such as realistic examples of how prop
erties can be proved of functional programs that p erform realistic I O Hence a ma jor
theoretical advantage of functional programming that programs are easy to understand
and verify do es not carry over to practical programs engaged in I O
This dissertation is a study of how to give semantics to I O mechanisms for functional
languages and how to use such semantics to prove prop erties of programs engaged in
I O It is meant as a step towards convincing the working programmer that functional
programming can b e practical though much remains to b e done
The purp ose of this chapter is to intro duce the problem of functional I O survey previ
ous work and outline the contribution of this dissertation x de nes terminology used
here concerning functional programming x discusses previous work on I O in func
tional languages and identi es four widely implemented mechanisms Any semantics of
functional I O has to build on a semantics of functional languages x reviews semantic
metho ds suitable for functional languages x states the hyp othesis of this dissertation
x outlines each chapter x states the original results of this dissertation and x
o ers advice to the reader Finally x intro duces some of the mathematical material
needed here
Functional programming
Many functional or applicative languages have been put forward since the pioneering
of this work on LISP ISWIM and POP in the s For the purp ose
dissertation we distinguish two classes of functional languages dep ending on the seman
tics of function application Recall the terms call by value and call by name from ALGOL
CHAPTER INTRODUCTION
When a function is applied to an actual parameter under call by value semantics
the function s formal parameter is b ound to the value obtained by evaluating the actual
parameter Under call by name semantics the formal parameter is bound to the un
evaluated actual parameter and each time the value of the formal is required the actual
parameter is evaluated An eager language is one in which function application has call
by value semantics a lazy language is one in which function application has call by name
semantics By this de nition LISP Scheme and ML are eager whereas Miranda Lazy
ML and Haskell are lazy For the sake of e ciency application in lazy languages is usually
implemented using call by need as in graph reduction for instance Call by
need is the same as call by name except that after the rst evaluation of the actual
parameter its value is retained and used whenever the formal parameter is subsequently
used
The primary fo cus of this dissertation is I O for lazy languages such as Miranda or Haskell
although we discuss eager languages brie y Wemake no claim to have considered all the
varieties of functional language for instance the work here is not immediately applicable
to data ow languages like Lucid or Silage in which every expression stands
for an in nite stream
Much has b een written in praise of functional programming The curious reader is referred
to the pap er by Hughes or any of the many textb o oks on functional programming
b eginning this research was the thought My own motivation when
that unlike imp erative programs lazy functional programs are easy to manipulate when
proving program prop erties but it was not clear howto reason ab out programs engaged
in I O
A brief history of functional I O
Many mechanisms have b een implemented and prop osed for functional I O We identify
four classes of I O mechanism which together cover most of the prop osed schemes
Side e ecting I O
Like functional programming functional I O b egins with McCarthy LISP had a
side e ecting I O mechanism The core of LISP can b e explained as applications of
functions to arguments but the LISP Programming System needed other op erations such
as commands to e ect an action such as the op eration of input output whichwere called
pseudo functions The pseudo function print wrote its S expression argument to
the printer The pseudo function read to ok no arguments but returned an S expression
from the input device This side e ecting style of I O p ersists in LISP and is also used in
other eager languages suchasScheme or ML Many language theorists hav e viewed side
e ecting I O with suspicion b ecause unlike pure LISP the evaluation of programs using
side e ects cannot simply b e explained as the applications of functions to arguments To
paraphrase Stoy there is more to the meaning of an expression than just its value
the side e ects and order of evaluation of sub expressions b ecome signi cant The same
suspicions are aroused by LISP or ML programs that use the assignment statement
A BRIEF HISTORY OF FUNCTIONAL I O
Although suspicious to some side e ecting I O is by far the most widely used I O mech
anism in eager languages In an eager language it is fairly easy to predict the order in
which expressions are evaluated so programs using side e ecting I O can b e fairly simple
to write and debug if not to reason ab out formally To the b est of the author s knowledge
Williams and Wimmers pap er is the only work to consider howtoprove prop erties
of programs using side e ecting I O in an eager language They develop an algebra for
FL a descendantofBackus FP I O is achieved in FL by op erations on histories
ob jects that enco de the status of all I O devices and which are implicitly passed to and
from every function One can view this as a form of side e ecting I O
On the other hand it is not usually easy to predict the order in which a program in a lazy
language will call such side e ecting pseudo functions Evaluation order is determined
ance Side e ects mixed with by data dep endencies which can be hard to predict in adv
lazy evaluation make programs hard to understand Another reason why side e ecting
I O is hard to use with a lazy language is that call by need can cease to be a correct
implementation of call by name as weshow in Chapter
Landin stream I O
A stream is a potentially endless list of values generated as need arises Streams were
used by Landin in his calculus semantics of ALGOL to represent the values of lo op
variables he remarked that streams could have been used to represent I O in AL
GOL Streams were b eing used ab out the same time by Strachey in his imp erative
GPM language to represent I O Streams can b e represented as elements of certain
recursively de ned domains In an in uential pap er Kahn applied domain theory
to concurrency using collections of stream pro cessing functions to mo del the semantics
of certain kinds of pro cess network With MacQueen he showed that these pro cess
networks could b e implemented in POP extended with certain side e ecting op erations
on streams
Ab out the same time the pioneers of lazy languages argued that the list cons
op eration likeany other function should not evaluate its arguments In an eager language
like LISP every list is nite On the other hand if the cons op eration do es not evaluate
its arguments in nite lists can b e represented whose elements are computed on demand
Notionally in nite lists are an imp ortant to ol for the programmer in a lazy language
emerged that the input and output streams used in giving semantics to I O The idea
could be implemented within a lazy language itself In what we call Landin stream
I O interaction with a teletyp e is sp eci ed by a functional program that maps a lazy list
of input characters to a lazy list of output characters The reduction mechanism needs
to be extended so that demand for values in the input stream is met by obtaining fresh
input from the keyb oard Jones and Sinclair credit Henderson as b eing the rst
to prop ose the use of a stream pro cessing function to implement teletyp e I O By the
covered in intro ductory textb o oks mid s this was a standard technique
CHAPTER INTRODUCTION
Synchronised stream I O
Synchronised stream I O is a generalisation of Landin stream I O where the program
is a function mapping a stream of acknowledgements to a stream of requests In Landin
stream I O inputs and outputs need not be synchronised inputs occur when demand
arises for the value in the input stream outputs o ccur when the value of the next item in
the output stream has b een determined In synchronised stream I O input and output is
synchronised the functional program must pro duce an output request before examining
the corresp onding input acknowledgement The p ower of synchronised streams is that the
typ e of requests can enco de any kind of imp erative command Synchronised streams were
rst rep orted as the underlying implementation technique for Karlsson s Nebula op erating
system The same essential idea was indep endently discovered by Stoye in his
op erating system for SKIM I I and also by O Donnell Synchronised streams were
c hosen as the basic I O mechanism in Haskell
Continuation passing I O
Karlsson derived continuation passing I O op erations from the underlying
synchronised stream mechanism of Nebula In the context of teletyp e I O
continuation passing I O is based on a typ e CPS short for continuation passing style
with three op erations INPUT Char CPS CPS OUTPUT Char CPS CPS and
DONE CPS The typ e CPS can b e implemented as an algebraic typ e within the functional
language There is no change to the language s evaluation mechanism but a top level
program of typ e CPS can be interpreted or executed as follows To execute INPUT k
input a character v from the keyb oard and then execute k v To execute OUTPUT v q
output character v to the printer and then execute program q To execute DONE simply
terminate The style is called continuation passing b ecause the argument k to the INPUT
op eration is reminiscent of continuations in denotational semantics Holm
str om used a continuation passing style in PFL an eager dialect of ML extended
primitives Perry and McLoughlin and Hayes implemented with concurrency
continuation passing I O mechanisms in lazy dialects of Hop e Reb elsky s recent prop osal
of I O trees is essentially a form of continuation passing I O Unlike side e ecting
and either kind of stream based I O the continuation passing style is suitable for either
lazy or eager languages
The Haskell I O system is based on synchronised stream I O based on a typ e called
Dialogue but there is a standard set of continuation passing op erations These op erations
are programmed in terms of the underlying synchronised stream mechanism in the same
spirit as Nebula Hudak and Sundaresh discuss translations b etween the two mechanisms
that were discovered by the Haskell committee One fruit of the formal semantics for
functional I O develop ed in Chapter is a pro of of correctness of translations between
Landin stream synchronised stream and continuation passing I O
A BRIEF HISTORY OF FUNCTIONAL I O
Combinators for I O
Programmers using eager languages nd that programs using side e ecting I O are fairly
easy to understand if not to reason ab out formally The order of evaluation and hence of
side e ects is fairly easy to control On the other hand programs using stream based I O
can b e hard to develop for two reasons explicit plumbing of streams around a program
is easy to get wrong the order in which input and output is interleaved can be hard to
predict b ecause of lazy evaluation Wray and Dwelly rep ort problems of this
sort
Several authors have derived combinators to abstract op erations that are commonly needed
with stream based programming Karlsson programmed continuation passing op er
ations using a synchronised stream mechanism Wray suggested combinators for
sequential comp osition and iteration In his seminal work on the semantics and pragmat
binators ics of Landin stream I O in lazy languages Thompson suggested a range of com
with which to construct Landin stream programs The combinators construct programs of
typ e interact a b a program of this typ e is intended to representaninteractive compu
tation with state of typ e a that when executed will return a value of typ e b Thompson s
combinators include op erations such as sequential comp osition and iteration He devel
op ed a trace theory to verify their correctness the rst work on semantics of I O for lazy
languages
In developing the Kent Applicative Op erating System KAOS a line
Miranda program John Cupitt re ned Thompson s combinators He worked with a typ e
interact a which represented interactive computations that return values of typ e a He
used two basic combinators return and comp
return a interact a
interact a interact b comp a interact b
A program return v is the trivial computation that immediately returns the value v a
program comp f p is a sequential comp osition rst execute p to return a value v and
then execute program f v Stream based programs such as KAOS written using these
and other combinators have neither of the disadvantages mentioned earlier There is no
explicit plumbing of streams The order of input and output is controlled by sequential
comp osition
Moggi has shown that structures of the form interact return comp occur
often in the denotational semantics of programming languages The semantics of such
a structure can be given as a computational mo del in the sense of Moggi a categorical
structure based on a strong monad Wadler showed that such structures are a
versatile to ol for functional programming particularly when writing programs to interact
with state
In uenced bythework of Cupitt Moggi and Wadler Chapter of this dissertation advo
cates what wecallmonadic I O in whichcombinators like Cupitt s are used to structure
programs Monadic I O is a high level construct that can be implemented using any of
the four low level I O mechanisms Monadic programs are easier to understand than pro
grams written in the three low level styles suitable for lazy languages there are no explicit
streams or continuations to tangle a program
CHAPTER INTRODUCTION
Summary of previous work on functional I O
Wehave discussed four classes of I O mechanism These will b e covered in greater length
in Chapter in the context of teletyp e I O The semantic to ols develop ed earlier in the
dissertation will allowustogive semanticstoeach of the four I O mechanisms
To summarise we considered side e ecting Landin stream synchronised stream and
continuation passing mechanisms of I O Side e ecting I O is not suitable for lazy lan
guages b ecause of the di culty of predicting the order in which side e ects o ccur The
semantics of b oth eager and lazy languages are complicated by the presence of side e ects
making program prop erties harder to prove Although the input stream needs to be im
plemen ted sp ecially the semantics of a lazy language need not b e a ected by the presence
of stream based I O input and output streams are simply lazy lists I O mechanisms in
certain eager languages have b een based on streams but the typ e of streams
is kept distinct from lists and used only for I O Continuation passing I O has b een used
with both lazy and eager languages Evaluation of expressions remains unchanged but
some kind of interpreter needs to b e added to the implementation to execute continuation
passing programs Various sets of combinators have b een prop osed for programming at a
higher level than the basic I O mechanism
The four classes cover the most widely implemented mechanisms for functional I O To the
b est of the author s knowledge the only mechanism suitable for teletyp e I O not covered
here is the extension of Landin stream I O with hiatons as implemented in Lazy ML
The problem is for a program to b e able to p oll the keyb oard Landin stream I O has
blo cking input in the sense that once demand arises for the next input value computation
is halted until a key is typ ed The solution is that a sp ecial value a hiaton app ears
arisen for a character but none is available in the input stream whenever demand has
from the keyb oard Hiatons have not b een widely implemented Another solution to the
p olling problem is to add a command to poll the keyb oard to synchronised stream
or continuation passing I O but we do not pursue this idea here
There are go o d literature surveys on functional I O by Hudak and Sundaresh Jones
and Perry Historically many ideas ab out functional I O have arisen from adding
nondeterminism or concurrency to functional languages We do not study such mecha
nisms in this dissertation We refer the interested reader to pap ers containing surveys on
the following topics functional programming and op erating systems nondeterminism
and stream based semantics real time functional programming and concur
rent extensions of ML Kelly s b o ok cites manyworks on parallel systems based
on functional languages
Although there has b een a great deal of work on functional I O there has b een very little
work on semantics The primary goal of this dissertation is to explain the semantics of
functional I O and hence make prop erties provable of functional programs engaged in
I O To the b est of the author s knowledge Thompson s pap er on the semantics
of Landin streams in Miranda is the only prior work on the semantics of I O in lazy
languages In the context of eager languages there is Williams and Wimmers
work on semantics for what is essentially side e ecting I O and several pap ers giving
op erational semantics for concurrent constructs but with no development
of a theory for program pro ofs Dyb jer and Sander rep ort work on the related problem
SEMANTICS OF FUNCTIONAL LANGUAGES
of expressing concurrency using streams They verify a communications proto col expressed
as a concurrentnetwork of stream based functions
Semantics of functional languages
To obtain a theory of functional I O wemust b egin with a theory of functional program
ming itself In the context of this dissertation such a theory has two purp oses to sp ecify
precisely the computational behaviour of functional programs so that implementations
could b e veri ed and to enable program prop erties to b e stated and proved
Abramsky p oints out that although the untyp ed calculus has often b een viewed as the
prototypical functional language actual implementations of lazy languages do not conform
to the standard theory Abramsky considers two functional programs x and
where is a lo oping or divergent program He p oints out that according to the standard
theory the two are equal but in the implementation of lazy languages such as Miranda
of the second diverges or Lazy ML evaluation of the rst converges whereas evaluation
Motivated by this example Abramsky develops his lazy calculus as a step towards a
theory of lazy functional programming Following Plotkin s study of PCF Abramsky
equips the lazy calculus with a structural op erational semantics and a domain
theoretic denotational semantics He then proves an adequacy theorem to relate the two
semantics
As far as this dissertation is concerned Abramsky s key mano euvre is to view his lazy
calculus as a pro cess calculus Led by Milner a great many op erationally based metho ds
have b een develop ed for the CCS theory of concurrency Bisimilarity found by
taking the greatest xp oint of a certain functional is a cornerstone of this theory
Since it is a greatest xp ointit admits co inductive pro ofs Abramsky builds
a bridge between CCS and the calculus by prop osing applicative bisimulation as
the notion of op erational equivalence in the lazy calculus Applicativebisimulation is a
reworking of CCS bisimulation for the calculus
We follow Abramsky and construct a theory of functional programming based on struc
tural op erational semantics and applicative bisimulation Veri cation of an implemen
tation could be based on the op erational semantics but this is beyond the scop e of the
dissertation Pro ofs of program prop erties are based on a theory of applicative bisimulation
that parallels that of CCS It is imp ortant that applicative bisimulation is a congruence
relation that is a substitutive equivalence relation Abramsky s original pro of that ap
plication bisimulation is a congruence dep ended on domain theoretic results Stoughton
and Howe made two suggestions for how congruence could be proved directly from the
op erational semantics Stoughton suggested a variant of Milner s context lemma
Howe with an ingenious construction proved congruence for a broad range of lazy compu
tation systems In Chapter wewillinvestigate b oth the context lemma and Howe s
metho d In related work Milner and Sangiorgi link the lazy calculus with the
theory of calculus a development of CCS with mobile pro cesses Smith builds
on Howe s work to construct semantic domains from op erational semantics
A semantics of functional programs has often been based on domain theoretic
denotational seman tics Stoy s classic pap er shows how domain
CHAPTER INTRODUCTION
theoretic metho ds such as Scott induction can b e applied to prove prop erties of functional
programs Instead of domain theory we use op erational semantics to sp ecify the I O b e
haviour of functional programs Were we to app eal to domain theoretic principles in pro ofs
of functional programs wewould need to relate the op erational and domain theoretic se
mantics Lester Simpson and Burn have proved such a relation usually
known as adequacy in the context of a lazy functional language other related work is
more theoretical Weleave the relation b etween the op erational and
domain theoretic semantics of the functional languages studied here as future work For
the purp ose of proving program prop erties wehave not felt the lack of a domain theoretic
semantics as a loss examples arising here that might have required Scott induction in a
domain theoretic setting have b een proved using co induction
In summary asasteptowards a theory of functional I O wedevelop a theory of functional
programming in which the functional language is viewed as a kind of pro cess calculus The
theory is based on structural op erational semantics and applicative bisimulation
Hyp othesis
This dissertation aims to show the following
An op erational theory of functional programming is suitable for precisely sp ecifying
a functional language and proving prop erties of functional programs
Such an op erational theory can b e extended to sp ecify and prove prop erties of the
most widely implemented mechanisms for I O in lazy functional languages
A semantics for a simple form of monadic I O may be expressed within the func
tional language Hence programs using monadic I O maybeveri ed using standard
techniques
Synopsis
The rst half of the dissertation de nes a semantic metalanguage M which is used in
the second half for the investigation of functional I O
typ es In this chapter we provea technical result Chapter A calculus of recursive
needed in Chapter Mendler has proved con uence and strong normalisation for
the Girard Reynolds p olymorphic calculus extended with p ositive recursivetyp es
This chapter proves strong normalisation for an extension called of Mendler s
calculus
Chapter A metalanguage for semantics The metalanguage M is a simply typ ed
calculus with pro duct sum function lifted and recursive typ es This chapter
de nes its syntax typ e assignment relation and its lazy and deterministic op erational
semantics The main result of this chapter is a convergence theorem that apart
from terms of lifted typ es evaluation of every term converges
Chapter Op erational precongruence We investigate two op erationally de ned
preorders on the terms of M contextual order after Morris and Plotkin and ap
SYNOPSIS
plicative similarity after Milner and Abramsky We de ne a notion of op erational
adequacy to mean that a preorder resp ects evaluation in a certain way We show
that each preorder is an op erationally adequate precongruence The pro ofs use a
variantofMilner scontext lemma for contextual order and a typ ed reworking
of a metho d due to Howe for applicative similarity Given that applicativesim
ilarity is a precongruence it is routine to establish op erational extensionality
that applicative similarity coincides with contextual order
Chapter Theory of the metalanguage We adopt applicative bisimilarity the
equivalence corresp onding to contextual order and applicative similarit yas equiva
lence on terms of M We prove equational laws that are analogues of the axiomatic
domain theory of LCF We derive a principle of co induction from the
de nition of applicative bisimilarity We investigate prop erties of empty one p oint
iterated sum iterated pro duct b o olean and natural number typ es
The second half investigates a range of ways in which functional languages can b e extended
to express I O
Chapter An op erational theory of functional programming We de ne a
functional language H which is essentially a subset of Haskell H has lazy al
gebraic typ es and both call by name and call by value function applications We
give a deterministic op erational semantics and a denotational semantics using M
The denotational semantics is a case study of Moggi s prop osal to use monads to
parameterise semantic descriptions We prove a close corresp ondence between the
op erational and denotational semantics We de ne op erational and denotational
equivalences as ob ject level applicative bisimilarity and equivalence in the metalan
guage resp ectively We show that a theory of programming which consists of a
set of equational laws together with a co induction principle holds for op erational
but we equivalence The equational laws are valid for denotational equivalence
leave op en whether the co induction principle holds We conclude the chapter by
considering the semantics and theory of HX a language obtained from H by adding
a parameterless exception mechanism
Chapter Four mechanisms for teletyp e I O We take teletyp e I O interaction
with a keyb oard and printer as a simple I O mo del In this context we discuss
the formal semantics of four widely implemented mechanisms for functional I O
side e ecting Landin stream synchronised stream and continuation passing I O
We explain why side e ecting I O combines badly with call by name semantics of
function application The other three mechanisms are suitable for use with call
by name semantics We prove in a precise sense that they are of equal expressive
power
Chapter Monadic I O Wedevelop a monadic style of functional I O to supp ort an
application of functional I O at the Medical Research Council MRC in Edinburgh
We describ e a simple monadic programming mo del and express its semantics within
H as a particular form of state transformer Using the semantics weverify a simple
programming example
Chapter Conclusion Conclusions are drawn and further work suggested
CHAPTER INTRODUCTION
Results
The main contribution of this dissertation is to develop an op erational theory of lazy
functional programming to extend it to accommo date various I O constructs and to
show how it can be applied to prove prop erties of functional programs engaged in I O
Here are sp eci c original results
A theory of a non trivial functional language H based on equational reasoning and
co induction and develop ed op erationally from rst principles
A case study of the monadic approach to denotational semantics based on an
op erationally de ned metalanguage M Pro of that the denotational semantics cor
resp onds closely to the op erational semantics
A formal semantics for side e ecting Landin stream synchronised stream and
continuation passing I O
uation passing I O are A pro of that Landin stream synchronised stream and contin
equally expressiveinthecontext of teletyp e I O
A case study of the monadic approach to I O motivated by an application of func
tional programming to medical electronics
An investigation of the relationship b etween the context lemma and Howe s metho d
of proving precongruence for op erationally de ned preorders
How to read the dissertation
A bird s eye view can b e had by reading this chapter reading the unnumbered intro ductory
sections of Chapters to and then the whole of the last chapter The intro ductory
sections are intended to motivate and sketch the results of eachchapter without mentioning
any mathematical details
As mentioned ab ove the rst half develops a semantic metalanguage M for use in the
the second half to give denotational semantics for a functional ob ject language H H is
basis for the study of functional I O in the second half The argument in the dissertation
is linear in that almost every chapter dep ends on all its predecessors to some degree but
none on its successors That said a great deal of the second half can be understo o d
without knowing the development of M in detail After obtaining a bird s eye view the
reader primarily interested in functional I O mightskimx to x to get an impression
of the de nition of M browse Chapter on the theory of M and b egin reading more
thoroughly at Chapter
The extension of Mendler s calculus develop ed in Chapter is only used explicitly in x
The reader wishing to understand M in detail is advised to b egin at Chapter and skip
x on rst reading and then to study Chapter b efore reading x
MATHEMATICAL PRELIMINARIES
Mathematical preliminaries
Syntax alpha conversion and the variable convention
We deal with three formal languages in this dissertation intro duced in Chapter
M intro duced in Chapter and H intro duced in Chapter Chapter also intro duces
a variant of H called HX Here we state general syntactic conventions that apply to all
these languages except that term variables are treated sp ecially in
We assume two countably in nite sets of typ e variables and term variables ranged
over by the metavariables X Y Z and f g u v w x y z resp ectively We almost
always refer to term variables simply as variables We will p oint out binding o ccurrences
of variables when de ning each formal language If t is a phrase of syntax we write ftv t
and fv t for the sets of typ e and term variables resp ectively that occur free in t The
iterated notation fv t t is short for fv t fv t we use a similarly iterated
n n
t t
X andt x for the outcomes of and t are phrases of syntax we write t form of ftv If t
substituting t for each free o ccurrence of X or x respectively int withchange of b ound
variables in t to avoid variable capture We refer the reader to Hindley and Seldin s
textb o ok for a clear treatment of substitution and alpha conversion We make free
use of the prop erties of substitution and alpha conversion develop ed in Section B of their
b o ok
We follow the standard practice of identifying phrases of syntax up to alpha conversion
that is treating a syntactic phrase as if it were its alpha equivalence class We use the
symbol for alpha conversion We adopt Barendregt s variable convention and as
sume that all the b ound variables in a term are distinct from each other and from any free
variables This is legitimate when phrases of syntax are identi ed up to alpha conversion
We will make it clear whenever we are not treating a phrase of syntax as its alpha
equivalence class When we are not we can write bv t for the set of b ound term variables
in t A context C or D is a term p ossibly containing one or more holes written as
We write C t which we call an instantiation for the term obtained by lling in each
hole in C with the term t Contexts are not identi ed up to alpha conversion we write
C D to mean that contexts C and D are literally the same The only signi cant use of
contexts is in Chapter contexts are covered in greater detail in x
Typ es and p olymorphic de nitions
Each of the formal languages M and H is a typ ed calculus in which terms are
information Such typ e information can often be inferred from the tagged with typ e
narrative text and then we omit it Occasionally we omit typ e information from the
de ning equation of an M or H term and sp ecify a typ e scheme For instance we might
def
make the de nition id x x and comment that term id has typ e scheme The
idea of typ e schemes or p olymorphic typ es is widely used in functional languages like
ML or Haskell Rather than burden the reader with a formal mechanism of p olymorphic
def
typ es we trust that the equation id x x and others like it can be understo o d as
the de nition of a whole family of terms id x x for eac htyp e
CHAPTER INTRODUCTION
Relations
If S and T are sets we treat a binary relation b etween S and T as a subset of the pro duct
S T If R S T we write sRt and s t R interchangeably Relational comp osition
is written as juxtap osition that is if R S S and R S S then R R S S
is the comp osition of R and R If R S S then R and R are its transitive closure
and re exive transitive closure resp ectively We write Id for the identity relation If R is
a relation R is its inverse
Bisimulation and co induction
We make extensive use of preorders and equivalence relations de ned to be the greatest
xp oints of certain functionals The prototypical use of this technique in computer science
is bisimilarity in CCS suggested byPark and develop ed by Milner
We will intro duce such relations by rst de ning the functional denoted parenthetically
by or h i Then we de ne notions of simulation and similarity as intro duced in the
following theorem which states general prop erties of the relations de ned in this way
Prop osition Supp ose the following
Metavariable t ranges over a set of terms Term
Metavariable S ranges over subsets of Term Term
Functional is a monotone function over subsets of Term Term
that is if S S then S S
A simulation isarelationS such that S S
is de ned to b e the union of all simulations Similarity
Wehave
Similarity is the greatest xp ointof
A xp oint of is a relation S suchthat S S
Similarity is the greatest simulation
t t i there is a simulation S such that tS t
If the identity relation on Term is a simulation then is re exive
If S S is a simulation whenever b oth S and S are then is transitive
If S is simulation whenever S is then is symmetric
Pro of Part is a sp ecial case of the Knaster Tarski theorem in xp oint theory see
Davey and Priestley pages
That similarityisa simulation follows from It is the largest since by de nition it
contains any other
simulation S to be itself For the backwards For the forwards direction take the
direction wehave S so t t S implies t t
For any t pair t t is in a simulation the identity relation so bypart we have
MATHEMATICAL PRELIMINARIES
t t
t Since similarityisa simu t and t For arbitrary t t and t supp ose that t
t by de nition of relational comp osition lation so is the comp osition Wehave t
t as required so pair t t iscontained in a simulation Hence by wehave t
t as required Supp ose that t t Since is a simulation wehave t
Prop erty of this prop osition is an imp ortant pro of technique to prove t t itsu ces
to nd some S containing pair t t and prove that S S This technique has been
called Park s rule or greatest xp oint induction or co induction
If the relation de ned by a functional is a preorder a re exive and transitive relation
we call a similarity and any S such that S S we call a simulation as we did
ab ove On the other hand if relation is an equivalence we call a bisimilarity and
any S such that S S wecallabisimulation
CHAPTER INTRODUCTION
Chapter
A calculus of recursive typ es
This chapter prepares the way for Chapter where we will de ne a simply typ ed metalan
guage for programming language semantics M We wish to prove in Chapter that for
every term in a certain class of M typ es evaluation cannot diverge This result is used in
Chapter where wewilldevelop a theory of M We obtain this result by translating such
terms of M into a p olymorphic calculus that p ossesses a strong normalisation prop erty
that no in nite reduction sequence starts from anyterm
The typ es of M we wish to enco de are sums pro ducts functions and a certain class of
recursive typ es called p ositive recursive typ es A recursive typ e X is p ositive
within typ e is to the left of an just when each o ccurrence of the b ound typ e variable X
even numb er of function arrows For instance the typ es X X of natural numb ers
or X X of nite lists of typ e are p ositive recursivetyp es As an example
of a non p ositivetyp e consider X X X which can enco de any term of the untyp ed
calculus Evidently there are terms of this typ e whose evaluation diverges
We seek a calculus with a strong normalisation result whose typ e structure is expressive
enough to enco de the typ es of M and hence to be the basis of a normalisation pro of
for M One candidate is the p olymorphic calculus of Girard and Reynolds which
has a rich typ e structure able to enco de a wide range of typ es Functions are primitive
in and there are standard enco dings of sums and pro ducts As for recursive typ es
Girard outlines a scheme for enco ding the class of recursivetyp es which are sums of
pro ducts and equipping each with primitive recursive functions Wraith describ es
a general scheme for enco ding any p ositive recursivetyp e and primitive recursion within
A second candidate is Mendler s extension of which adds new typ e
constructors for p ositive recursive typ es together with constants and reduction rules to
recursive typ es and compute primitive recursive functions Mendler construct terms of
proves that his extension of isChurch Rosser and strongly normalising
We use Mendler s calculus b ecause it allows a direct representation of recursivetyp es In
this chapter we augment Mendler s calculus with two new families of constants that are
needed to simulate certain op erators in M The purp ose of the chapter is to prove that
even with the additional constants and their reduction rules all the terms of the calculus
Paul Francis Mendler is also known as Nax Paul Mendler
CHAPTER A CALCULUS OF RECURSIVE TYPES
are strongly normalisable
x de nes the typ e structure of the calculus develop ed in this chapter to b e the
same as in Mendler s calculus x de nes the term structure of to be the same as
in Mendler s calculus but for the addition of two new families of constants x de nes
the reduction relation for terms of and states Mendler s theorem that anytermof
is strongly normalisable provided none of the new constants o ccurs in it x de nes
combinators in Mendler s calculus that corresp ond to the new constants x shows in two
examples that the combinators can simulate the new constants x proves that the new
constants can always be simulated by combinators contained within Mendler s original
calculus
Typ es of
Calculus is a generalisation of the Girard Reynolds p olymorphic calculus We
typ e variables ranged over bymetavariables X Y Z assume a countably in nite set of
The set of typ es with metavariables and isgiven by the grammar
X typ e variable
j function typ e
j X p olymorphic typ e X b ound in
j recursivetyp e
j
X initial typ e X b ound in
X terminal typ e X b ound in
Conventional notions of free typ e variables and substitution apply as stated in x Typ es
are identi ed up to alpha conversion
W esaythatatyp e variable X o ccurs p ositively in a typ e i each o ccurrence of X in
is to the left of an even number of s Dually atyp e variable X o ccurs negatively
in a typ e i each o ccurrence of X in is to the left of an o dd number of s
Recursivetyp es take one of the forms X or X Beware that the symbol or
is used b oth as a metavariable for initial typ es or terminal typ es and as part of the
syntax as the typ e constructor itself For an M typ e to b e well formed we require two
any recursive typ e X or X that o ccurs in First we require prop erties of
that eachsuch recursivetyp e is p ositive whichistosay that the b ound variable X o ccurs
positively in Second we require that each recursivetyp e is closed or equivalently that
ftv fX g For instance typ e X X X is closed but not p ositive typ e X Y X
is closed and p ositive but is not well formed b ecause it contains a typ e Y X whichis
positive but not closed If we make the standard de nitions of sums and pro ducts
def
Z ftv Z Z Z Z
def
Z Z Z Z ftv
Mendler p oints out that X o ccurs p ositively in suchtyp es i X o ccurs p ositively in typ es
and Let typ e b e X X a one p ointtyp e whichweinvestigate in the context of
M in x If is a well formed closed typ e we can de ne well formed typ es X X
TERMS OF
X X and X X of natural numb ers nite lists and notionally in nite
streams resp ectively
We insist that recursivetyp es b e closed and p ositive for the following reasons Recursive
typ es that are not p ositive are problematic because they can typ e terms that are not
normalisable For instance the typ e X X X can typ e a version of the combinator
from the untyp ed calculus Positive recursivetyp es that are not closed complicate some
of the syntactic constructions used in this chapter We rule them out as a convenience
just for this reason Mendler requires recursive typ es to be positive but not necessarily
closed We will assume implicitly that any typ e we deal with is well formed Note that
the set of well formed typ es is closed under substitution
Parentheses will often b e omitted when the intended grouping is implied by asso ciativity
and scop e rules These rules are that the function arrow asso ciates to the right
and the scop e of bound variables continues as far to the right as p ossible For example
the typ e is short for
Terms of
As mentioned in x typ e variables are treated sp ecially in We assume given a
countably in nite set of names ranged over by metavariables f g x y z This set
of names is the same as the set of term variables used in M and H In a term
variable called simply a variable in the following is of the form x where x is a name
and is a typ e The raw terms of with metavariables L M and N are pro duced
by the grammar
M x typ ed variable
j x M abstraction variable x bound in M
j MN application
j X M typ e abstraction X b ound in M
X not free in anytyp e o ccurring in fv M
j M typ e application
j k constant
where constants k are drawn from the set of constants Con given b elow
def
Con fIntro Elim R Intro Elim S g
Let ftv M andfv M b e the sets of all typ e and term variables that o ccur free in M
M ftv M fv M
x ftv fx g
x M ftv ftv M fv M fx g
MN ftv M ftv N fv M fv N
ftv M fX g fv M X M
M ftv M ftv fv M
k
N
Typ e and term substitution is written as M X and M x Alpha conversion is up to
renaming of the names of b ound term variables but not their typ e lab els x M
CHAPTER A CALCULUS OF RECURSIVE TYPES
y
y M x if y fv M We identify terms up to alpha conversion The restriction
X
on variables in typ e abstractions comes from the without it fv X x is not well
de ned up to alpha conversion of terms We adopt a name convention for
that all the names of all the b ound variables in a term are distinct from each other and
from the names of anyfreevariables we can always achieve this by alpha con version
The only di erence between the calculus given here and Mendler s calculus is the
presence of the constants Elim and Intro To obtain his calculus Mendler added the
initial and terminal typ es to the Girard Reynolds calculus together with the constants
Intro R Elim and S We use a notation slightly di erent from Mendler s Elim instead
of In Intro instead of out X instead of X and di erent metavariables We need
the extra constants so as to mo del corresp onding Intro and Elim op erators in M
The typ e assignment relation of a predicate of the form M where M is a raw
term and is a typ e is de ned inductively by the rules
Intro X
Elim X
R Y X X X Y Y Y
Intro X
Elim X
S Y X X Y X Y Y
x
M M
M N M X
x M X M X
MN M X
where X and X If M can be inferred we say that M is a term and
that it has typ e Henceforth the metavariables L M and N will range over just the
terms rather than all the rawterms
For the sake of legibilitytyp e sup erscripts will often be omitted Parentheses will often
b e omitted when the intended grouping is implied by asso ciativity and scop e rules These
rules are that typ e and term applications asso ciate to the left and the scop e of b ound
variables continues as far to the right as p ossible For example the term LM N is short
for LM N and the term x y x y y is short for x y xy y
Reduction in
Let us say that a binary relation R on terms of iscompatible just when it is closed
under the following rules
M R N M RN M RN
x M R x N ML R NL LM R LN
M RN M RN
X M R X N M R N
COMBINATORS TO SIMULATE ELIM AND INTRO
Let Id stand for the identity function x x The reduction relation on terms is
the least compatible relation closed under the following eight axiom schemes
N
x M N M x
x Mx M if x fv M
X M M X
X MX M if X ftv M
Elim Intro M M
R M Intro N M Id R M N
Elim Intro M M
Elim S M N M Id S M N
In this dissertation wetakeadvantage of the following theorem
Theorem Mendler If no constant Elim or Intro o ccurs in a term M then M
is strongly normalisable which is to say that there is no in nite sequence of reductions
starting from M
Pro of The pro of is contained in an article by Mendler His calculus do es not contain
the new families of constants Elim or Intro The reduction rules applicable to terms of
that do not contain the new constants are exactly the same as the rules in Mendler s
article If M M is a reduction and M do es not contain one of the new constants
then neither do es M Therefore any sequence of reductions starting from a term not
containing the new constants can be exactly simulated by a sequence of reductions in
Mendler s calculus Therefore his result tells us that any term M in is strongly
normalisable provided that none of the new constants o ccurs in M
In a private communication June Mendler said that he omitted the Elim and Intro
constants from his calculus to simplify the con uence pro of whichwe make no use of in
this dissertation and was not concerned at the loss of expressiveness b ecause they can b e
mimicked using the recursion constants R or S resp ectively This presentwork app ears
to be the rst to work through a construction of combinators to mimic the constants
omitted by Mendler
The rest of this chapter is devoted to a pro of that anytermof is strongly normalis
able
Combinators to simulate Elim and Intro
We prove that every term of is strongly normalisable by simulating the new con
stants Elim and Intro with combinators elim and intro de nable in Mendler s original
calculus from constants R and S resp ectively Before de ning these combinators in
De nition we de ne two auxiliary functions Spec and Gen
De nition We de ne the partial functions not terms of the calculus Spec and
Gen that map quadruples of the form X M to a term where is a recursivetyp e
X is a typ e variable is a typ e and M is a term First cho ose some name f Then de ne
CHAPTER A CALCULUS OF RECURSIVE TYPES
the two functions simultaneously by structural induction on the third argument
def
X
Spec X X M f M
def
Spec X Y M M if X Y
def
Spec X M M
def
Spec X Y M Y Spec X M Y where Y ftv X M
def
Spec X M x Spec X M Gen X x
where X
and name x f do es not o ccur in fv M
def
X
Gen X X M f M
def
Gen X Y M M if X Y
def
Gen X M M
def
Gen X Y M Y Gen X M Y where Y ftv X M
def
M x Gen X M Spec X x Gen X
where name x f do es not o ccur in fv M
The functions are partial only so far as Spec X M andGen X M are unde ned
when is the typ e variable X this is unproblematic b ecause we only apply Spec to a
terminal typ e X orGen to an initial typ e X when X o ccurs negatively in
De nition For each initial typ e X let be the typ e X and de ne
the combinator family elim as follows
def
X X
elim R X f g x Spec X x
Dually for each terminal typ e X let be X and de ne the combinator
family intro as follows
def
X X
g x Gen X x intro S X f
The name f is the same as the one chosen in the de nition of Spec and Gen
In x we will prove that these combinators p ossess the following prop erties
elim Intro M M
Elim intro M M
for any suitably typ ed term M Given these prop erties it is not hard to show that every
term in is strongly normalising First we showtwo examples
Two examples
First recall the typ e X X We can calculate the combinator intro as follows
X X
S o X f g x Gen X X x intr
X X X
S X f g x f x
STRONG NORMALISATION FOR
Nowwe can checkthatElim intro M M for any term M of typ e
X X X
Elim intro M Elim S X f g x f x M
X X X
X f g x f x Id intro M
f x Id intro M f g x
g x Id x intro M
x Id x M
Id M
M
Second let be the typ e X and initial typ e be X We b egin by
calculating Spec X x which has typ e
y Spec X x Gen X X y Spec X x
y Spec X x
X X
z Gen X y Spec X X z
y Spec X x
X X X
z Gen X y f z
X X X
ec X x z y f z y Sp
X X X
y x z y f z
The combinator elim is de ned as follows where is
X X
elim R X f g x Spec X x
Wecancheck that elim Intro M M for any term M of typ e
X X
elim Intro M R X f g x Spec X x
Intro M
X X
X f g x Spec X x
Id elim M
f g x
f z y x z y
Id elim M
g x y x z y Id z
elim M
x y x z y Id z M
z y M z y Id
z y M z y
y M y
M
Notice that the reductions make use of the b eta and eta rules and dep end on the variable
named f b eing replaced by the identity function
Strong normalisation for
Weprove a series of prop ositions that culminates in a strong normalisation result for
The rst prop osition states all the prop erties that we rely on in the functions Spec and
CHAPTER A CALCULUS OF RECURSIVE TYPES
Gen parts a and d are of particular interest
Prop osition Let be any initial typ e X and let s be the substitution
Id
X f Then for anytyp e and term M
If M and X occurs positively in let N Spec X M and then
a N has typ e X
b fv N fv M and name f do es not o ccur in fv N
ftv ftv M fX g and c ftv N
d Ns Ms
If M X and X o ccurs negatively in letN Gen X M and then
a N has typ e and
b fv N fv M and name f do es not o ccur in fv N
c ftv N ftv ftv M fX g and
d Ns Ms
Pro of We prove all parts simultaneously by induction on the size of the typ e and
pro ceed by analysis of its structure For the sake of brevity we omit all the details of
parts b and c They are no harder to prove than the other parts
Case X
X
a Here N f M whichhastyp e which equals X
d Ns Id Ms Ms
Gen X X M is unde ned but this case cannot arise b ecause X o ccurs p os
itively in X
Case Y X
a N M Y and Y X
d Ns Ms Ms
a N M Y and Y
d Ns Ms Ms
Case This case is trivial because of the restriction in that all recursive
typ es b e closed If we had not made this restriction the de nitions of Spec and Gen
would b e rather more complicated
a Here N M which has typ e All recursivetyp es are closed so X
d Ns Ms Ms
N M which has typ e X since all recursivetyp es are closed a
d Ns Ms Ms
Case Y We may assume Y ftv X M since typ es are identi ed up to
alpha conversion
a Here N Y Spec X MY By IH we have that Spec X MY is
of typ e X Therefore N is of typ e Y X X
d Ns Y Spec X MY s
Y MY s IH
Y Ms Y
Ms
STRONG NORMALISATION FOR
a N Y Gen X MY By IH Gen X MY isoftyp e Therefore
N is of typ e Y
d Ns Y Gen X MY s
Y MY s IH
Y Ms Y
Ms
Case
a N x Spec X M Gen X x where x has t yp e X
Since X o ccurs negatively in by IH we have Gen X x Then
M Gen X x has typ e Since X o ccurs p ositively in by IH we
have Spec X M Gen X x has typ e X Therefore N has typ e
X X which equals X
d Ns x Spec X M Gen X x s
x M Gen X x s IH
x Ms Gen X x s
x Ms xs IH
x Ms x
Ms
a Dually N x Gen X M Spec X x where x has typ e Since
X o ccurs p ositively in by IH we have Spec X x has typ e X Then
M Spec X x has typ e X Since X o ccurs negatively in by IH we
N has typ e have Gen X M Spec X x has typ e Finally
which equals
d Ns x Gen X M Spec X x s
x M Spec X x s IH
x Ms Spec X x s
x Ms xs IH
x Ms x
Ms
The following is analogous to Prop osition but for terminal typ es
Prop osition Let be any terminal typ e X and let s be the substitution
Id
X f Then for anytyp e and term M
If M and X o ccurs negatively in letN Spec X M and then
a N has typ e X and
b fv N fv M and name f do es not o ccur in fv N
ftv M fX g and c ftv N ftv
d Ns Ms
If M X and X o ccurs p ositively in letN Gen X M and then
a N has typ e and
b fv N fv M and name f do es not o ccur in fv N
c ftv N ftv ftv M fX g and
d Ns Ms
CHAPTER A CALCULUS OF RECURSIVE TYPES
Pro of Omitted but dual to the previous pro of
Nowweprove prop erties of the combinators elim and intro
Prop osition For any recursivetyp es X and X
There are no typ e or term variables free in elim or intro that is they are combi
nators
The typ e assignments elim X and intro X are derivable that
is the combinators are well typ ed
For anytermM
a if M X then elim Intro M M
b if M X then Elim intro M M
Pro of Consider the term elim Let be the typ e X Since
fn Spec X x fx f g we can calculate fn elim fn Spec X x
ff x gg Therefore fv elim Similarly ftv Spec X x ftv fX g
fX g since ftv fX g Wehave
ftv elim ftv X X ftv Spec X x fX g
Similarly wecancheck that the term intro has no free typ e or term variables
Combinator elim will b e well typ ed if Spec X x isoftyp e X and Prop o
X
sition a says so Similarly for intro to b e well typ ed the term Gen X x
needs to havetyp e and according to Prop osition a it do es
Again let be X By app eal to Prop osition d let us calculate
elim Intro M
X X
R X f g x Spec X x Intro M
X X
X f g x Spec X x
Id elim M
f g x Spec X x X
Id elim M
Id
g x Spec X x X f
elim M
elim Id
g M x Spec X x X f
Id
x Spec X x X f M
Id
x x X f M
Id M
M
By a similar calculation we can compute the other part of the prop osition
Nowthatweknowhowtosimulate the new constants that app ear in our extended calculus
in terms of the original calculus we can simulate each term N of the extended calculus
by a term M with each of the new constants replaced by one of the combinators wehave just de ned
STRONG NORMALISATION FOR
De nition De ne the simulation relation M N on terms of as the least
compatible relation closed under the following axiom schemes
elim Elim
intro Intro
k k for any k Con fElim Intro g
Later we will rely on the fact that simulation is preserved by substitution
Lemma For any terms M and N such that M N
If M N then fv M fv N and ftv M ftv N
M X N X for any and X and
L L
M x N x for any L L and x with L L
Pro of Part is an easy induction on the depth of inference of M N Part is also
by induction on the depth of inference of M N We showseveral cases
Case M elim Elim N Since neither side contains any free typ e variable
free term M X M N N X Similarly since neither side contains a
L L
variable M x N x
Case M intro Elim N The result follows by the same argument as the previous
case that neither side containsafreetyp e or term variable
Case M k k N Same again
Case M y y N Wehave M X N X so M X N X For either
L L L L
x y or not If so M x L L N x If not M x y y N x
Case M y M y N N We may pick the b ound variable y such that
L L
y x By induction hyp othesis wehave M X N X and M x N x
X X
Then we can derive that M X y M X x M X
L L L L
N X and M x y M x x M x N x as required
Case M M M N N N By induction hyp othesis M X N X and
i i
L L
M x N x for i Then wehave
i i
M X M X M X N X N X N X
L L L L L L
x N x N x as required x M x N and M x M
The case for abstractions is similar Part follows by a similar argument
Now we come to the key prop erty of the simulation relation that if M N then any
reduction of N can b e mimicked by a non empty sequence of reductions starting from M
Prop osition If M N and N N then there exists M such that M M and
M N
Pro of The pro of is by induction on the depth of inference of N N pro ceeding by
o cases where an analysis of how the inference was derived We show here only the tw
combinators do the work of constants and one of the inductive cases
CHAPTER A CALCULUS OF RECURSIVE TYPES
Case N Elim Intro N N N
Since M N wehave M elim Intro M withM N
Set M M and by Prop osition a wehave M M and weknow M N
Case N Elim Intro N N N
Since M N wehave M Elim intro N withM N
Set M M and by Prop osition b wehave M M and weknow M N
Case N x N x N N
where N N Since M N wehave M x M withM N
By induction hyp othesis there is M such that M M and M N
x M and wehave M M and M N Set M
The other cases of the pro of follow the same pattern
A simple consequence of Prop osition is strong normalisation
Theorem Each term N of is strongly normalisable
Pro of Supp ose to the contrary that there is a well typ ed term N that admits an in nite
chain of reductions Construct a term M such that M N by replacing each constant
of form Elim or Intro that o ccurs in N by the corresp onding combinator elim and
intro resp ectively From Prop osition it follows that since there is an in nite chain of
reductions starting from N there is also one starting at M But M is a term of Mendler s
published calculus whichis known to be strongly normalising Theorem so we have
reached a contradiction Therefore eachtermof is strongly normalisable
We have extended Mendler s second order calculus with recursive typ es to have two
new families of constants Elim and Intro This provides a basis for the normalisation
pro of of the semantic metalanguage M develop ed in later chapters of this dissertation
Chapter
A metalanguage for semantics
The purp ose of this chapter is to de ne a metalanguage for programming language se
mantics called M In Chapter we de ne a small functional language called H Wewill
give a denotational semantics for H in terms of M and also show how the denotational
semantics can b e extended to account for an exception mechanism and side e ecting I O
Hence each of the four classes of functional I O discussed in Chapter can be de ned
from the denotational semantics for H extended as necessary
We adopt a variant of Plotkin s threefold metho dology for denotational semantics
First he develop ed a theory of partial functions and so called b ottomless domains Second
calculus as a semantic metalanguage his calculus had pro duct he prop osed a typ ed
sum function lifted and recursive typ es His calculus had both domain theoretic and
op erational semantics He proved adequacy results relating the op erational and domain
theoretic semantics of his metalanguage Third to study a particular ob ject language he
gave a direct op erational semantics and an indirect domain theoretic semantics induced by
mapping the ob ject language into the metalanguage Pro of of the corresp ondence b etween
the op erational and domain theoretic semantics of the ob ject language can takeadvantage
of the adequacy results for the metalanguage This approach is general in that for each
ob ject language studied the pro of of adequacy factors into two a general result proved
once and for all for the metalanguage and a comparatively simpler pro of relating the
semantics of the ob ject and metalanguages Plotkin s work has b een develop ed by Moggi
and Jones
We develop in this dissertation a simply typ ed calculus called M for the denotational
semantics of H and its variants The most imp ortant di erence b etween the op erational
semantics of Plotkin s calculus and M is that in M the evaluation of any term of a certain
class of typ es cannot diverge In Plotkin s calculus there is a divergent term at every typ e
We partition the typ es of M in two the p ossibly divergent typ es are the lifted typ es
the certainly convergent typ es are all the others One reason for this is to convey
information ab out op erational b ehaviour in the typ e system for instance anynumeral in
M term of typ e Num which is a certainly convergenttyp e representing H is mapp ed to an
the natural numbers The typ e of the translation of the numeral conveys that it cannot
diverge
In this and the following two chapters we investigate M b efore applying it to the study
CHAPTER A METALANGUAGE FOR SEMANTICS
of functional programming and I O in Chapters and In this chapter we de ne M
and assign it an op erationally based rather than domain theoretic semantics M has a
typ e system similar to Plotkin s calculus pro duct sum function lifted and recursive
typ es Using a metho dology similar to Plotkin s we use the metalanguage to decomp ose
a more complex ob ject language H In Chapter we investigate two op erationally
de ned preorders on the terms of M Morris style contextual order and a typ ed form
of Abramsky s applicative bisim ulation We will prove them to be equal and take the
equivalence they generate as equivalence on terms of M In Chapter we develop a
theory of M based on equational laws and a co induction principle We study certain
derived M typ es that are needed for the semantics of H b o oleans natural numb ers and
iterated sums and pro ducts
x de nes the syntax of M The typ e system is de ned in x and the op erational
semantics in x x proves the ma jor result of this c hapter that only terms of lifted
typ e can diverge
Syntax of M
We assume countably in nite sets of typ e variables and term variables ranged over by
letters X Y Z and f g x y z resp ectively The syntactic conventions of x apply to
M The typ es are given by the following grammar
X j j j j j X
and the terms by the grammar in Table
Atyp e is p ossibly divergent i it is a lifted typ e that is has the form Any other
typ e is certainly convergent The main result of this chapter is a convergence theorem
Theorem which says that evaluation of any closed term of a certainly convergent
typ e must converge A corollary is that only terms of a p ossibly divergent typ e may
diverge The pro duct disjoint sum and function typ es and
resp ectively are certainly convergenttyp es
They take the form X Just as in Chapter Recursivetyp es are certainly convergent
the symbol is used b oth as a metavariable for recursivetyp es and as part of the syntax
as the typ e constructor itself There are two syntactic restrictions on these typ es
each free o ccurrence of the typ e variable X in must b e to the left of an even number of
s and the only typ e variable free in can b e X Restriction rules out mutually
recursive typ es These conditions are required so that the convergence theorem can be
proved by mapping any certainly convergenttyp e in M to a corresp onding typ e in
ell formed typ es is We will assume implicitly that typ es are well formed The set of w
closed under substitution
Typ e assignment in M
An environment is a nite mapping from variables to closed typ es written as
x x where the variables in the list are pairwise distinct The domain of
n n
TYPE ASSIGNMENT IN M
Syntax
U V hM N i pair
j Inl M j Inr M injection
j x M abstraction x b ound in M
j Lift M Lift term
j Intro M Intro term
L M N x variable
j V canonical term
j Split M x y N Split term x y bound in N
j Case M x N x N Case term x bound in N i
i i
j MN application
j Seq M x N Seq term x bound in N
j Fix x M Fix term x bound in M
j Elim M Elim term
Table Terms of M
x x
M N
M x y N
hM N i
Split M x y N
M M
Inl M Inr M
M x N x N
Case M x N x N
x M
M N
x M
MN
M M x N x M
Lift M Seq M x N Fix x M
M X M X X
Intro M Elim M X
Table Typ e assignment rules for M
CHAPTER A METALANGUAGE FOR SEMANTICS
an environment is denoted by Dom We write for the concatenation of two
environments this notation is well formed only when Dom Dom
De nition The M typ e assignment relation M is inductively de ned by
the rules in Table Each rule has the implicit side condition that any environments
appearingintherulearewell formed
Weshow that the typ e assignment relation p ossesses some standard prop erties
Prop osition
If M and M then
N
If x M and N then M x
If M then fv M Dom
If M and then M
Pro of By structural induction on M The pro of is straightforward b ecause terms are
lab elled with typ e information where necessary to ensure unique typ e assignment
By induction on the depth of inference of the typ e assignment x M
By simple inductions on the depth of inference of M
We de ne classes of programs and con ned terms
De nition
A program is a term M such that M and hence ftv M for some
necessarily unique typ e The typ e is called the typ e of M which itself is
called a program
A con ned term is a pair M such that there is a necessarily unique typ e
with M The typ e is called the typ e of M and is called the
Occasionally we represent a con ned term M of environment of M
typ e with the typ e assignmentsentence M itself
Op erational semantics of M
We de ne a deterministic op erational semantics for M programs We use an auxiliary
notion of an exp eriment E which is a function on programs such that E M is obtained
by wrapping a selector around program M Exp eriments are discussed at greater length
in x They are an alternativetoFelleisen s evaluation contexts
De nition Exp eriments E are de ned by the grammar at the top of Table
Write E M for the term obtained by replacing the o ccurrence of in E by the term
M
The reduction and evaluation relations for M are the binary relations on M programs
and resp ectively de ned inductively by the rules in Table
The canonical programs are the outcomes of evaluation The op erational semantics is
lazy in the sense that subterms of canonical programs may be non canonical Witness
OPERATIONAL SEMANTICS OF M
Reduction Semantics
E Split x y M j Case x N x N j M j Seq x N j Elim
M N
E M E N
M M
Split hM M i x x N N x x
M
Case Inl M x N x N N x
M
Case Inr M x N x N N x
N
x M N M x
M
M x N N x Seq Lift
Fix x M
Fix x M M x
Elim Intro M M
Evaluation Semantics
V V
M M
L hM M i N x x V
Split L x x N V
M M
L Inl M N x V L Inr M N x V
Case L x N x N V Case L x N x N V
N
L x M M x V
LN V
M
L Lift M N x V
Seq L x N V
Fix x M
M x V
Fix x M V
M Intro N N V
Elim M V
Table Evaluation rules for M
CHAPTER A METALANGUAGE FOR SEMANTICS
Inr Fix x x which is canonical but has non canonical Fix x x as a subterm
Prop osition
If M and M N then N
If M and M V then V
If L M and L N then M N
If L U and L V then U V
The canonical terms are the normal forms of one step reduction where a normal
M such that for no term form is a program that cannot b e reduced that is a term
N do es M N
Supp ose M N Then for any V N V implies M V
M V just when M V
Pro of Parts and followby straightforward inductions on the depth of inference of
M N and M V resp ectively Parts and follow similarly by inductions on the
depth of inference of L M and L U resp ectively
For part we must show that each canonical program has no reductions and that
if a term has no reductions then it is canonical First no canonical program has any
reductions b ecause none of the reduction rules in Table is applicable to any canonical
program Second each program is either canonical a Fix term or of the form E M for
some exp eriment E and program M There are rules in Table to reduce a Fix term
and each p ossible well typ ed term E M so any program with no reductions must be
canonical
Part is by induction on the depth of inference of the reduction M N The forwards
direction of is by induction on the depth of inference of evaluation M V We prove
the backwards direction of via a metho d used in Crole s dissertation Supp ose that
M M M M M V We show that M V by lling in the following
n n
diagram from right to left starting with the fact that M V V and establishing for
n
each i that M V from M V and part of the prop osition
i i
M M M M M V
n n
V V V V
Both directions established part is proved
We de ne terminology for termination of evaluation
De nition Supp ose that M is a program Say that M converges and write M
i there is a necessarily unique canonical program V such that M V Conversely say
that M diverges and write M i M do es not converge
Convergence
The argument that certainly convergent typ es deserve the name pro ceeds as follows
CONVERGENCE
Typ es
def
X X
def
X X X where X ftv
def
X X X X where X ftv
def
def
X X X
def
X X
Terms
def
x
x x
def
X
hM N i X f f M N
def
Split M x x N M
x x x x N
where M
def
X X
Inl M X f g f M
def
X X
Inr M X f g g M
def
Case M x N x N M
x x N
x x N
where M
def
x M x x M
def
MN M N
def
X X
Lift M X x x
def
X X
Seq M x N X x x
def
X X
Fix x M X x x
def
Intro M Intro M
def
Elim M Elim M where M
Table A translation of M typ es and terms into
In Table we de ne a translation of M typ es and terms into Prop osition
proves that well typ ed terms of M are translated into well typ ed terms of
Lemma is a substitution lemma saying that the translation of an M term into
is preserved under typ e and term substitution
We prove a simulation theorem Theorem if an M program M of certainly
convergenttyp e reduces to program N then the translation of M into reduces
to the translation of N
The Convergence Theorem for certainly convergent programs Theorem then
follows easily from the simulation theorem and the fact Theorem that no
in nite chains of reductions exists in
De nition Each M typ e is translated to a typ e according to the mapping
inductively de ned by the rules in Table
CHAPTER A METALANGUAGE FOR SEMANTICS
Each M con ned term M is translated to a term M according to the
mapping inductively de ned by the rules in Table Any b ound names that app ear in
the right hand side of translations but not in the left hand side are assumed to b e new
The purp ose of this construction is to show that each reduction of a certainly convergent
term in M is simulated by one or more reduction steps in so as to prove by con
tradiction that no certainly convergent term diverges The translations of pro ducts and
sums are the standard ones mentioned in x For the present purp ose there is no need
for reductions of p ossibly divergent terms to b e simulated by their translations indeed it
would b e imp ossible Hence we simply map each p ossibly divergenttyp e to X X X
X X
and can map any p ossibly divergent term to X x x
Prop osition
For any M typ e ftv ftv
For any M con ned term M of typ e ftv M and the set of
names free in M equals the set of M variables free in M
If M con ned term M has typ e term M has typ e
Pro of Parts and are by induction on the structure of and the depth of inference
of M resp ectively
By induction on the depth of inference of the typ e assignment M
We need some substitution prop erties of the translation
Prop osition
X X
If x M and N
N N
then M x x M x
Pro of By induction on the structure of Weshowseveral cases Supp ose Y If
X Y then b oth sides equal If not then b oth sides equal Supp ose
Then lhs Y X X Y Y where we may assume that b ound
Y Y rhs The variable Y ftv X By IH lhs Y X X
other cases are similar
By induction on the depth of inference of x M
Here is the key theorem of the argument
Prop osition For any two M programs of the same certainly convergent typ e M
and N ifM N then M N
Pro of The pro of is by induction on the depth of inference of M N As one might
exp ect as the translations are standard the reductions of pro ducts sums and functions
go through straightforwardly We show the case of pro ducts as an example and also the
inductive case
CONVERGENCE
M M
x N We may assume typ e x Case M Split hM M i x x M M
assignments hM M i N and x x M Wehave
hM M i x x x x M M
and
X
hM M i X f f M M
So
M x x x x M M M
M M
x x M x x
and hence by the substitution lemma Prop osition wehave
M N
as required
Case M E M E N N where M N First note from the typ e assignment
rules in Table that since M is a program of certainly convergenttyp e then so to o
are M and N Therefore the induction hyp othesis applies to the reduction M N
and wehave M N Then the conclusion that M N
follows by insp ection of the translation in Table of any p ossible E M and E N
Reductions involving recursive typ es also go through straightforwardly Reductions con
cerning lifted typ es do not arise b ecause only terms of certainly convergent typ e are
considered
Theorem If is a certainly convergent typ e and M is a program then there is
a canonical program V for which M V
Pro of By contradiction assume that there is an in nite chain of M terms b eginning
M M M By the previous prop osition there is a chain of terms
b eginning M M M But this is a contradiction b ecause
by Theorem is strongly normalising so no suchchain can exist Therefore the
theorem follows
CHAPTER A METALANGUAGE FOR SEMANTICS
Chapter
Op erational precongruence
We will develop in the next chapter an entirely op erational theory of the semantic meta
language M The purp ose of the present chapter is to investigate how an op erational
equivalence can b e de ned on terms of M For the sake of simplicitywe consider in this
chapter op erational preorders Each such preorder induces an equivalence in the usual
way
The theory of M in Chapter dep ends on M having an op erationally adequate precon
gruence An op erationally adequate preorder is one that resp ects M evaluation in certain
ways A preorder is a precongruence i whenever M N then C M C N for all
contexts C Roughly sp eaking a context C is a term containing holes the term C M is
obtained by lling in eachholeincontext C with the term M
C
and applicative similarity We consider two candidate preorders contextual order
A
x investigates contextual order attributed by Abramsky to Morris and used
by Plotkin and Milner If M and N are two M programs of the same typ e
C
It is not hard to N i for all contexts C ifC M converges then so do es C N then M
showthatcontextual order is a precongruence We show that it is op erationally adequate
E
which is evidently op erationally by de ning an auxiliary preorder exp erimental order
adequate and showing that exp erimental and contextual order are the same To do so
weuseavariant of Milner s context lemma
A
The other candidate preorder is applicative similarity investigated in x This
is a typ ed formulation of Abramsky s applicative bisimulation Applicative similarity
is the greatest xp oint of a certain functional Roughly sp eaking if M and N are two M
A
programs of the same typ e then M N i whenever M V there is U with N U such
that U and V have the same outermost syntactic constructor and their corresp onding
subterms are applicatively similar It is not hard to prove that applicative similarity is
op erationally adequate We show that it is a precongruence via a typ ed reworking of an
ingenious metho d due to Howe
In x we proveanop erational extensionality result that contextual order equals
applicative similarity Hence wehave two indep endentcharacterisations of the same pre
order This preorder generates an equivalence in the usual way which we refer to as
or applicative bisimilarity in the remainder of the disserta op erational equivalence
CHAPTER OPERATIONAL PRECONGRUENCE
tion
Mathematical material needed to do the work of the chapter is intro duced at the b eginning
Contexts are de ned in x Exp eriments were used to de ne the reduction semantics of
M in Chapter and they are needed here to de ne exp erimental order In x werecall
the de nition of an exp eriment and de ne contextual exp eriments which are exp eriments
that act on contexts and terms The detailed syntactic op erations in x and x are
used here only in the study of contextual and exp erimental orders The theory of M in
Chapter is based on programs and con ned terms whichwere de ned in Chapter In
x we formulate certain prop erties of relations on con ned terms or programs that are
needed later
Contexts
p ossibly containing holes written A context is pro Intuitively a context is a term
duced from the grammar for M terms with metavariables C and D instead of M and
N augmented with an additional rule C Contexts are not identi ed up to alpha
conversion unlike phrases of abstract syntax such as terms which are Write C C to
mean that contexts C and C are literally the same The sets of free and b ound variables
is a of a context C are written fv C and bv C resp ectively A canonical context V
context generated from the grammar for M canonical terms with metavariable V instead
of V augmented with an additional rule V
For any term M and context C de ne the instantiation of M in C C M to b e the term
obtained by lling each hole in C with the term M Variables free in M may b ecome
b ound in C M suchas x when M hx y i and C x If V is a canonical context
then term V M is canonical for anytermM
Supp ose C and D are contexts and x is a v ariable If bv C fv D x de ne
D
the substitution C x to be the context obtained by replacing each o ccurrence of the
variable x in C with the context D This is a literal substitution there is no renaming of
the variables b ound in C whichhave b een assumed to b e distinct from the variables free
in D
Weshow that in certain circumstances substitution and instantiation commute
Lemma Supp ose C and D are contexts x is a variable and M is a term If bv C
D D M
fv D x and x fv M then C x M C M x
Pro of By structural induction on context C We consider the following cases in detail
D M
Case C lhs M M rhs M x M since x fv M
D M
Case C x lhs D M rhs x x D M
D M
y x y Case C y x lhs y M y rhs
EXPERIMENTS AND CONTEXTUAL EXPERIMENTS
Case C C C
D
lhs C C x M
D D
C x M C x M
D M D M
C M x C M x IH
D M
C M C M x
M D
x C C M
rhs
Case C y C Wehave y x since bv C fxg
D
lhs y C x M
D
y C x M
D M
y C M x IH
D M
y C M x
D M
y C M x
rhs
The other cases follow similarly to the last two and are omitted
Exp eriments and contextual exp eriments
Exp eriments were de ned in De nition and used in Table as an economical notation
for de ning the reduction relation on programs of M Recall the de ning grammar
E Split x y M j Case x N x N j M j Seq x N j Elim
We use exp eriments in this chapter to de ne exp erimental order on programs When we
come to prove that exp erimental order coincides with contextual order we will use a variant
of Milner s context lemma In this variant we need to extend the idea of an exp eriment
to contexts A contextual exp eriment E is given by the following grammar
x y C j Case x C x C j C j Seq x C j Elim E Split
The symbol o ccurs once in each exp erimentorcontextual exp eriment The term E M
and the context E C are obtained by replacing the o ccurrence of in E or E by the
term M or context C resp ectively Intuitively an exp eriment wraps a destructor term
t wraps a destructor context around a around a term similarly a contextual exp erimen
context Exp eriments are identi ed up to alpha conversion contextual exp eriments are
not Wede nefv E and fv E to the sets of variables free in exp eriment E and contextual
exp eriment E resp ectively
If E is a p ossibly empty list of exp eriments E E then write E M to mean the term
n
E E M Similarly if E is a p ossibly empty list of contextual exp eriments
n
E E write E C tomeanthecontext E E C
n n
Lemma
If M is a program then there is a unique list of exp eriments E such that either
M E V for some canonical program V or
M E Fix x N for some program Fix x N
CHAPTER OPERATIONAL PRECONGRUENCE
If C is a context then there is a unique list of contextual exp eriments E such that
either
C E or
C E x for some variable x or
C E V for some canonical context V or
C E Fix x C for some context Fix x C
Pro of By structural inductions on program M and context C
Exp eriments relate to reduction as follows
Lemma For any exp eriment E list of exp eriments E and programs M and N such
that terms E M E N E M and E N are programs
E M E N if M N
E M E N if M N
whenever E M V there is U such that M U and E U V
whenever E M V there is U such that M U and E U V
Pro of In Table reduction is closed under this inference rule so part holds
Part is a corollary of Part follows by an insp ection of the ve kinds of
exp eriment and the corresp onding evaluation rules in Table The pro of of is by
induction on the size of the list E The base case when the list is empty follows at once
Otherwise supp ose that E E M V By IH there is some U such that E M U and
E U V By there is some U such that M U and E U U From we have
E E M E E U E U V thatis E E M V
If E is an exp erimental context and M is a term then the instantiation of E with M
E M is the exp eriment obtained by lling in each o ccurrence of a hole in E with the term
M
Lemma For anycontextual exp eriment E context C term M and list of contextual
exp eriments E wehave
E C M E M C M
E C M E M C M
C x y C and we have Pro of Supp ose E Split x y C Then E C Split
E M Split x y C M so E C M Split C M x y C M E M C M The
other four cases are similar Part is a corollary of
GROUND AND CONFINED RELATIONS IN M
b
Rules of R
b
x R x
M R N x y M RN
M R N i
i i
b
b
Split M x y M R Split N x y N
hM M i RhN N i
M RN M RN
b b
Inl M R Inl N Inr M R Inr N
M R N x M RN i
i i i i
b
Case M x M x M R Case N x N x N
x M RN
M R N i
i i
b
b
x M R x N
M M R N N
M RN x M R N M RN
b b
Lift M R Lift N Seq M x M R Seq N x N
b
x M R N
Fix x M R Fix x N
M RN M RN
b b
Intro M R Intro N Elim M R Elim N
b
Table De nition of R
Ground and con ned relations in M
Recall that we de ned notions of programs and con ned terms in De nition
De nition
A ground relation R is a binary relation b etween programs of the same typ e
A con ned relation R is a binary relation between con ned terms of the same
typ e and environment Write M RN to mean that M N R
We can immediately make precise the notion of op erational adequacy which we seek to
of prove for contextual order and applicative similarity Let M program be Fix x x
typ e scheme for any
De nition A ground relation R is op erationally adequate i for all programs
M N and canonical programs V
If M N then N RM
If M V then V RM
M i M R
CHAPTER OPERATIONAL PRECONGRUENCE
Prop erties of Con ned Relations
M RN
Weak
M RN
M RN
Stren
M RN
x M RM N
Sp ec
N N
M x RM x
M RN
Precong
C M RC N
b
M R N
Comp
M RN
x M R N M RN
Sub
M N
M x RN x
Table Rules concerning con ned relations
M i for some canonical V V RM
Con ned relations are imp ortant for the theory of M in Chapter We formulate some
useful op erations and prop erties of con ned relations including precongruence
De nition
b
If R is a con ned relation let the con ned relation R be de ned by the rules in
Table
We de ne inference rules Weak Stren Sp ec Precong Comp and Sub in Ta
ble All these rules have the implicit side condition that any sentence denoting
ell formed i a pair of con ned terms is well formed A sentence M RN is w
pairs M and N are con ned terms of the same typ e
A con ned relation is natural i the rules Weak Stren and Sp ec are valid
A con ned relation is a precongruence i the rule Precong is valid A congru
ence is a con ned relation that is b oth a precongruence and an equivalence relation
Prop osition
If R is transitive and rules Sp ec and Precong are valid then rule Sub is valid to o
If R is a preorder then rule Precong is valid i rule Comp is valid
Pro of Supp ose x M RN and M RN By Precong we have
M N
M x R M x where we treat term M as a context By Sp ec we have
N M N N
x x RN x By transitivitywehave M x RN M
GROUND AND CONFINED RELATIONS IN M
Given that R is a preorder and Precong is valid we are to prove that
M RN implies M RN We pro ceed by an analysis of which rule from Table
derived M RN We examine the rule for Case terms as a typical example Given
the inference
M RN x M R N i
i i i i
b
Case M x M x M R Case N x N x N
we are to show that Case M x M x M R Case N x N x N From
Precong we can calculate
Case M x M x M R Case N x M x M
Case N x M x M R Case N x N x M
Case N x N x M R Case N x N x N
These together with the transitivityof R prove the result The other cases are similar
Given that rule Comp is valid for preorder R weproveby induction on the structure
of context C that for all M andN if M RN then C M RC N Consider any
C and cho ose any M and N that satisfy the assumption M RN We examine three
cases
Case C Goal C M RC N is precisely M RN
Case C x xRx follows from re exivityofR
Case C Case C x C x C By IH wehave C M RC N x C M RC N
and x C M RC N These together with the rule for Case terms in Table
b
give C M RC N and hence C M RC N as required
The other cases are similar
We will often need to induce a con ned relation from a ground relation and vice versa
De nition
Let be an environment x x Then a closure is an iterated substi
n n
L L
n
tution x x where each L is a program The order of substitution
n
i i
do es not matter b ecause the variables are disjoint and each L is closed
i
The con ned extension of a ground relation R is the con ned relation R such
G
L L L
x R N x that M RN i for all closures x M
G
If R is a con ned relation then its ground restriction is the ground relation
f M N j M RN g
If R is a con ned relation write M RN to mean that pair M N is in the ground
restriction of R
Wetypically use the same symb ol for a ground relation and its con ned extension If R
G
d b
is a ground relation and R is its con ned extension we write R to mean R
G
Prop osition The con ned extension of a ground relation is natural
rst two we Pro of We are to prove rule Weak Stren and Sp ec To begin with the
CHAPTER OPERATIONAL PRECONGRUENCE
must show the following inferences are valid
M RN
M RN
Weak Stren
M RN
M RN
where R is the con ned extension of a ground relation R for any given M and
G
N suchthat the sentences M RN and M RN are well formed By de nition
of R these inferences would b e valid if the following condition were true
L L L L L L
M x R N x i M x x R N x x
G G
L L
for all closures x and closures x But we know that fv M N Dom from
L L L
the well formedness of sentence M RN and therefore wehave M x x M x
L L L
N x These equations immediately imply condition as required and N x x
Finally we are to prove the validity of the Sp ec rule
x M RM N
N N
M x RM x
L L L L
The sentence x M RM means that M x x R N x x for any program
G
L
x whichwe knowto be a program L and suitably typ ed program list L Let L be N
L
L L N x N L L
x foreach x x M x x x M since fv N Dom x Since M
i i i
N L N L N N
i we have M x x R M x x which is to say that M x R M x as
G
required
Contextual and exp erimental order
We b egin with a notion of op erational ordering whichwecallcontextual order
C
such that De nition De ne contextual order to be the con ned relation
C
N i for all contexts C such that C M and C N are programs of the same M
typ e if C M converges so do es C N
Programs C M and C N can be of any typ e in Plotkin s formulation they must be of
ground integer typ e
Prop osition Contextual order is a precongruence
C C
Pro of It su ces to verify the Precong rule for Supp ose that M N For
C
some particular context D we are to show that D M D N This is to say that for
all contexts C suchthat C D M and C D N are programs of the same typ e if C D M
C
converges then so do es C D N This follows at once from M N by de nition
It is not so straightforward to prove that contextual order is op erationally adequate
We do so by de ning a second relation exp erimental order which is not hard to show
op erationally adequate and then showing that contextual and exp erimental order coincide
CONTEXTUAL AND EXPERIMENTAL ORDER
E
De nition Ground exp erimental order is the ground relation such that
E
M N i whenever E is a list of exp eriments with E M and E N programs of the
same typ e if E M converges so do es E N The con ned relation exp erimental order
is the con ned extension of ground exp erimental order
It is not hard to see that ground exp erimental order is re exive and transitive and hence
that exp erimental order is a preorder Weprove op erational adequacy
Prop osition Ground exp erimental order is op erationally adequate
Pro of We are to show for all programs M N and canonical programs V
E
N then N M If M
E
If M V then V M
E
M i M
E
M M i for some canonical V V
We are to show for all exp eriment lists E if E N then E M Since M N we
Corollary of have E M E N So if the latter converges so must the former
E
and the transitivityof
E
we need that for For the forwards direction supp ose that M To show M
all E if E M converges then so do es E But for no E do es E M converge For
E
So for all E if E M converges then so the backwards direction supp ose that M
do es E Pro ceed by con tradiction and supp ose that M converges Taking E to b e the
empty list wehavethat converges since M do es Contradiction
For the forwards direction M means that M V for some V Then bypart we
E E
M From the de nition M For the backwards direction supp ose that V have V
E
we have that if V converges then so do es M take the list of exp eriments to be of
empty But V converges b eing canonical
It is not hard to show the following
Prop osition Contextual order implies exp erimental order
C
N for some context x x and terms M Pro of Supp ose that M
n n
E
L L
and N It is necessary to show for any closure N whereM M x that M x
L
and N N x This is to say that for each suitably typ ed E ifE M converges then so
N Picksome list E and de ne context C E x x L L do es E
n n n
n n
E N Then if E M converges then so must C M E M andC N Wehave C M
C
If C M converges then so to o do es C N given that M N But then E N must
converge as required
The other direction is harder The next two lemmas are needed to prove Prop osition
that the ground restriction of contextual order coincides with ground exp erimental order
Lemma Supp ose context C E V for some contextual exp eriment E and canonical
context V Then there is a context D such that for all programs L such that C L is a
CHAPTER OPERATIONAL PRECONGRUENCE
program C L D L
Pro of We assume that all the b ound variables in context C are distinct if they are not
we can easily nd a context C in which all b ound variables are distinct such that for
any program L C L alpha converts to C L Since C L is a program fv C fv E
fv V Given sucha C the context D can b e de ned as follows
C C
y x V hC C i and E Split x y C then D C if
C
if V Inl C andE Case x C x C then D C x
C
if V Inr C andE Case x C x C then D C x
C
x if V x C andE C then D C
C
if V Lift C andE Seq x C then D C x
if V Intro C and E Elim then D C
Given the assumptions wehave made ab out context C ineach case we can check that the
conditions hold for substitution of a context into another to be well formed we omit the
C L is a program details From the typ e assignment rules in Table all the cases when
are covered ab ove and from the de nition of one step reduction it is not hard to see that
C L D L given Lemma
The pro of of the following lemma is based on a similar pro of due to Allen Stoughton for
the lazy calculus and based on Milner s context lemma
E
Lemma Context For anytwo programs M and N of the same typ e with M
N and any context C such that C M and C N are b oth programs of the same typ e if
C M converges then C N converges
E
N We Pro of Cho ose anytwo programs M and N of the same typ e and assume M
induction on n that for any context C prove the following hyp othesis by mathematical
suchthat C M and C N are programs of the same typ e
if C M converges in n steps then C N converges
We pro ceed by a case analysis of C According to Lemma there is a p ossibly empty
list of contextual exp eriments E such that one of four cases holds
Case C E V Either list E is empty in which case b oth C M and C N are canonical
and hence converge or the list takes the form E E and wehave C E E V Let
C E V and by Lemma there is a context D such that for any program L
if C L is a program then C L D L Let D E D and for any program L
wehave C L E E V L E L C L E L D L D L using Lemmas
and Since M and N are programs wehave C M D M andC N D N Now
D M must converge in n steps so by the induction hyp othesis D N converges
and therefore so to o do es C N as reduction is deterministic
Case C E Fix x C We may assume that all the b ound variables in context C are
by working with a con distinct if they are not we could prove the hyp othesis
text C in which all b ound variables are distinct and such that for any program
Fix x C
L C L alpha converts to C L Set D E C x For any program L
CONTEXTUAL AND EXPERIMENTAL ORDER
Fix x C L
we have C L E Fix x C L E L Fix x C L E L C L x
Fix x C
E L C x L D L using Lemmas and Hence we have
C M D M and C N D N Just as in the previous case since D M must
converge in n steps by the induction hyp othesis D N converges and therefore
so to o do es C N
Case C E Let context D E M and then D M E M M C M so D M
converges in n steps But context D must take one of the forms already considered
E
so we can conclude that D N E N M converges But we have M N so
E N N converges and E N N C N
Case C E x Trivial b ecause for no term L is C L a program
All cases considered the hyp othesis is proved for all n and the lemma follows
A corollary of this lemma and Prop osition is that ground exp erimental order and the
ground restriction of contextual order are the same
C E
Prop osition For all programs M and N of the same typ e M N i M N
Pro of The forwards direction is a sp ecial case of Prop osition For the backwards
E
N and that C M converges where C is a context Hence direction supp ose that M
by the context lemma C N converges to o
We can extend this result to op en terms via the next three lemmas
d
E E
N where programs M and N are as follows N then M Lemma If M
M hM M i and N hN N i
M Split M x x L and N Split N x x L
M Inl M and N Inl N
M Inr M and N Inr N
M Case M x L x L and N Case N x L x L
M Lift M and N Lift N
M Seq M x L and N Seq N x L
M Intro M and N Intro N
M Elim M and N Elim N
M M M and N N N
E C
Pro of Since each M and N is a program and M N ineachcasewehave M N
i i i i i i
C
by Prop osition Contextual order is a precongruence so M N in each case and
E
therefore M N by Prop osition as required
d
E E
Lemma If M N then M N where programs M and N are as follows
M x M and N x N
M Split L x x M and N Split L x x N
CHAPTER OPERATIONAL PRECONGRUENCE
M Case L x M x M and N Case L x N x N
M Seq L x M and N Seq L x N
Pro of For any list E supp ose that E M We are to showthat E N Wehave that
L L
M by Lemma Hence L hL L i and wehave M M x x and N
d
E E
L L L L L L
N x N we can deduce M x From M x x N x x Since
L L L L
E M converges we have that E M x x converges and so E N x x
L L
x wehave that E N converges as required x converges Since E N E N
Parts and followby similar arguments whichweomit
d
E E
N then M N where program M Fix x M and N Lemma If M
Fix x N
Pro of The pro of is a generalisation of the argument in the context lemma Weprove the
following hyp othesis by mathematical induction on n that for any context C such that
C M andC N are programs of the same typ e
if C M converges in n steps then C N converges
E C
N follows by Prop osition Just N from which M So we actually provethatM
as in the context lemma Lemma we pro ceed by a case analysis of C There is a
p ossibly empty list of contextual exp eriments E such that one of four cases holds
Case C E V The argument of the context lemma is valid here b ecause it to ok no
account of the structure of programs M and N
Case C E Fix x C For the same reason the argument of the context lemma applies
Case C E Let D be the context E M x where term M is b eing used as a
context we can assume that the b ound variables in context M are distinct from x
M
So C M E M M E M M x E M x M D M and hence D M
N
converges in n steps By IH we have that D N E N M x converges
d
E E
N N N
N we have that M x Since M N x Therefore since E N M x
N N
converges so do es E N N x But C N E N N x so C N converges
L is C L a program Case C E x Trivial b ecause for no term
d
E E
Prop osition If M N then M N
Pro of By considering each rule from Table in turn We consider the rule for Case
terms in detail Wehave M Case M x M x M andN Case N x N x N
E E
N i M N x M
i i i i
d
E
M N
L L
Let x be any closure and for any term L write L for L x We are to prove that
E
N thatis M
E
Case M x M x M Case N x N x N
APPLICATIVE AND COMPATIBLE SIMILARITY
Using Lemmas and wehave
E E
N Case M x M x M Case N x M x M from M
E E
Case N x M x M Case N x N x M N from x M
E E
N Case N x N x M Case N x N x N from x M
Then by transitivity of exp erimental order we have as required Each of the other
cases follows similarly
This prop osition together with Prop osition establishes that exp erimental and con
textual order are the same and hence that contextual order is an op erationally adequate
precongruence
Applicative and compatible similarity
We now de ne the ground preorder applicative similarity which we will prove in x to
b e an indep endentcharacterisation of contextual order
De nition De ne function over ground relations such that
b
M S N i whenever M U there is V with N V and U S V
An applicative simulation is a relation S such that S S De ne ground applica
A
tive similarity to be the union of all applicative simulations De ne applicative
A
to b e the con ned extension of ground applicative similarity similarity
Prop osition
Function is monotone
The identity ground relation is an applicative simulation
If each S is an applicative simulation then so is S S
i
Ground applicative similarity is the greatest xp oint of and is the greatest ap
plicativesimulation
A
M N i there is an applicative simulation S such that M S N
Ground applicative similarity is a preorder
easy to check from the de nition The remaining parts Pro of Parts and are
then follow from Prop osition
De nition An applicative bisimulation is a relation S such that both S and
A
S are applicative simulations Ground applicative bisimilarity is the ground
A A
A A
relation suchthat M N i M N and N M Applicative bisimilarity is
the con ned extension of ground applicative bisimilarity
Clearly applicative bisimilarity is an equivalence relation Furthermore for any programs
A
M and N M N i for some applicativebisimulation S M N S
A
A
Prop osition Ground relations and are b oth op erationally adequate
CHAPTER OPERATIONAL PRECONGRUENCE
Pro of When R stands either for applicative similarityor bisimilarity we need to prove
each of the following prop erties
If M N then N RM
If M V then V RM
M i M R
M i for some canonical V V RM
Toprove parts and it is su cienttoshowthatS is an applicative bisimulation
where S is given by
def
f N M j M N g S
Observe rst that if M N then for an y U M U i N U from Prop osition
c
and that Id S Supp ose N M S and N U Then M U and U Id U Similarly
c
supp ose M N S and M U Then N U and U Id U
For the forwards direction of part weshow that S below is an applicative bisimulation
def
f M j U M U g S
For any pair M N S neither M nor N so S is trivially an applicative bisimu
A
A
implies that pair lation For the backwards direction of either M or M
M is contained in some applicativesimulation Hence as cannot converge neither
can M
The forwards direction of part follows immediately from the de nition of M and part
A
A
For the backwards direction either U M or U M implies that pair U M is
contained in some applicative simulation so M since U
We prove that applicative similarity for M is a precongruence by a typ ed reworking of
Howe s metho d
to be the De nition De ne the con ned relation compatible similarity
least set closed under the following rule
A
c
L M M N
N L
Beware the ambiguous notation the in do es not denote re exive transitive clo
sure
Prop osition
Applicative similarity is natural that is rules Sp ec Weak and Stren are valid
c
Con ned relations and are re exive
A
c
b oth imply Con ned relations and
A
M M then M M and M If M
M N
Sub If x M N and M x x N then M N
APPLICATIVE AND COMPATIBLE SIMILARITY
c
If U M then there is V such that M V and U V
If M N and M U then U N
A
If M N then M N
Applicative similarity is a precongruence
Pro of Applicative similarity is natural b ecause it is the con ned extension of a ground
relation Prop osition
By structural induction on M one can easily verify that M M so is re exive
c
is re exive A corollary is that
Corollary of part and the de nition of compatible similarity
A A
c
N and N M Since is M then for some N M If M
A
M transitive wehave N M Then by de nition wehave M
The pro of is by induction on the depth of inference of x M N By de nition
A
c
there is N such that x N and x N N By Sp ec we have M
A
N N
N x Pro ceed by a structural analysis of M and showineach case N x
M N
N that M x x The result will then follow by part We show three
The other cases are similar cases
Case M x So N N by assumption x Wehave M
Case M y x So N y by re exivity part y Wehave y
Case M y M So N N By IH y y N and x y M
c
M N M N
M x N x So M x N x by de nition Then we have
M N
N M x x by part as required
A
c
M Pro ceed by N and N In each case there must b e a program N such that U
a structural analysis of canonical program U
A
M So N x N with x M N Then N M implies that Case U x
A
M x N with x N N Therefore x M N by part which
c
as required x N implies U
A
N for i Then N M implies Case U hM M i So N hN N i with M
i i
A
that M hL L i with N L So M L by part and U hM M i so
i i i i
c
hL L i as required U
By induction on the depth of inference of M U pro ceeding by analysis of M
Case M canonical Immediate
M
N there x U From M Case M M M We have M x M and M
A
N and N N N Since M x M and are N N such that each M
i i
N wehave x M N by IH By there is N with N x N and M
N N M
N x M N x N x and then byIHwehave U By Sub M x
A
N N
Since N N N x we have N x N N by op erational adequacy
CHAPTER OPERATIONAL PRECONGRUENCE
A A
N N
From N N x N wehave N N by transitivity Finally from U x N
wehave U N by
The other cases follow the same pattern as the ones ab ove
First we prove that the ground restriction of compatible similarity is an applicative
e are to show that simulation Supp ose for any programs M and N that M N W
c
V Supp ose that M U By whenever M U there is V such that M V and U
c
wehave U N By there is V such that N V and U V
A
Supp ose nowthat M N We are to prove that M N whichistosaythatfor
L L L
all closures x programs M x and N x are paired in an applicative simulation
L L
But by Sub and re exivitywehave M x N x and since the ground restriction of
compatible similarity is an applicativesimulation we are done
A corollary of and is that applicative and compatible similarityare the same
con ned relation We know applicative similarity is a preorder from Prop osition
We know that rule Comp holds for compatible similarity from part Therefore by
Prop osition applicative similarity is a precongruence
Contextual order and applicative similarity
Contextual order equals applicative similarity Our pro of is essentially a typ ed reworking
of Howe s Theorem
Lemma Ground applicative similarity equals the ground restriction of contextual
order
Pro of Let S b e the ground restriction of contextual order
def
C
N g S f M N j M
C A A
N If and showthat M S it is enough to consider any M N Toseewhy
A A
C M for some context C thenC M C M since is a precongruence and therefore
as required C M
For the reverse inclusion it su ces by co induction to show that symmetric S is an
C
applicative simulation Supp ose then that M N S and that M V Since M N
there must b e U such that N U or else the trivial context C would distinguish M
b
and N Based on the following case analysis of V weshow that V S U as required for S
to b e an applicative simulation
Case V Inl M Since U and V have the same typ e U must take one of the
forms Inl N or Inr N We can rule out the latter or else the context C
Case x L y where L is some convergent program such as Lift x x would
C
distinguish M and N Let context D be Case x x y M Wehave D M D N
C
N By op erational since is a congruence Moreover D M M and D M
adequacy and determinacy the reduction relation and its inverse are in
A A A A
cluded in Hence wehave M D M and D M N Since S wehave
CONTEXTUAL ORDER AND APPLICATIVE SIMILARITY
b
M N S and therefore V S U
b
Case V x M This time U must take the form x N Toshow x M S x N
L L
it su ces to provethat M x N x S for arbitrary program L Let context
C
L L
x andD N N x D b e L Wehave D M D N Since D M M
L L
wemay obtain M x N x S as in the previous case
Case V Lift M Again we may conclude that U takes the form Lift N To show
C
b
N Supp ose then for an that Lift M S Lift N it su ces to prove that M
arbitrary context C thatC M and C N are programs of the same typ e and that
C M converges We must show that C N converges to o We may supp ose that
C M and C N are of lifted typ e otherwise by the convergence theorem for M it
follows that b oth always converge Set context D to be Seq x C x We have
C
that D M C M and D N C N As b efore we have that C M D M
C C C
and D N C N Since M N we have also D M D N and therefore by
C
erges so do es C N Hence transitivity that C M C N Therefore if C M conv
b
wehave established Lift M S Lift N as required
The other cases followby similar reasoning
A C
Theorem Op erational Extensionality
Pro of Wemust show
A C
N N i M M
for arbitrary M and N The backwards direction follows by the same argument as used
A
S As for the forwards direction wemust show in the previous lemma to prove that
that
A
L L
x N x M
L
x Set context D to be x for any closure x L L and we have
n n
A C
D N and since D N by precongruence By Lemma wehave D M D M
A
L L L L
x as required N x x wehave M x and D N N D M M
In summary this chapter considered two indep endent de nitions of op erational order
contextual order and applicative similarity and proved that they are in fact the same
relation henceforth denoted by We use for op erational equivalence equal to
also known as contextual equivalence or applicative bisimilarity Both characterisations
are of interest By de nition contextual order is p erhaps the simplest congruence induced
by the evaluation relation By de nition applicative similarity is a greatest xp oint
giving rise to a powerful principle of co induction used extensively in Chapter Given
op erational extensionality Howe s metho d and the context lemma metho d amount to the
same thing a pro of that op erational order is an op erationally adequate precongruence
In a calculus where op erational extensionality fails the two metho ds would b e of separate
interest
CHAPTER OPERATIONAL PRECONGRUENCE
Chapter
Theory of the metalanguage
We conclude the three chapters on M by obtaining results ab out evaluation and equiva
lence needed in connection with the denotational semantics of H in the next chapter We
make extensive use in this chapter of the principal results obtained in Chapters and
that no term of a certainly convergent typ e can diverge and that op erational equivalence
that is contextual equivalence or applicative bisimilarity is an op erationally adequate
congruence
In x we prove basic facts ab out op erational equivalence these facts parallel the axioms
of LCF but are simpler in that there are divergent terms only at lifted typ es This
is a consequence of the convergence theorem proved in Chapter via Mendler s calculus
def
X X develop ed in Chapter x investigates empty and one p oint typ es in M
def
and resp ectively There are no programs of typ e Anytwo programs of typ e
are op erationally equivalent x sketches the standard construction of iterated sum and
pro duct typ es from binary sums binary pro ducts and the empty and one p ointtyp es We
conclude the chapter byinvestigating typ es of b o oleans and natural numb ers in x and
x resp ectively
Laws of op erational equivalence in M
Recall that op erational equivalence and order and equal the con ned relations of
A
A
resp ectively Weknow that b oth these applicative bisimilarity and similarity and
are op erationally adequate precongruences Let ground equivalence be the ground
restriction of op erational equivalence
Prop osition All the laws from Table hold for op erational equiv alence
Exhaustion law
Congruence
Canonical Freeness
Beta strictness and eta laws
Co induction
CHAPTER THEORY OF THE METALANGUAGE
Exhaustion Law
If M is a program and is certainly convergentthen V M V
If M is a program then either M or N M Lift N
Congruence
b
If M N then M N
Canonical Freeness
b
If U V then U V
Beta Laws
M M
x x Split hM M i x x N N
M
Case Inl M x N x N N x
M
Case Inr M x N x N N x
N
x M N M x
M
Seq Lift M x N N x
Fix x M
Fix x M M x
Elim Intro M M
Strictness Law
Seq x N
Eta Laws
Split M x x hx x i M
Case M x Inl x x Inr x M
x Mx M if x fv M
Seq M x Lift x M
Intro Elim M M
Co induction
is a ground relation S such that whenever A bisimulation up to
M S N either M N or there are canonical programs U and V
b
such that M U N V and U S V
Any bisimulation up to is a subset of ground equivalence
Table Laws of equivalence in M
LAWS OF OPERATIONAL EQUIVALENCE IN M
Pro of Supp ose that M is a program and that is certainly convergent By the
convergence theorem Theorem M which is to say V M V By op erational
adequacy wehave M V
Supp ose that M is a program Either M or M In the rst case we are done In
the second case there is some canonical program V such that M V Since V by
insp ection of the typing rules there must be a program N such that V Lift N By
op erational adequacy wehave M V Lift N as required
Congruence for applicative bisimilarity is a corollary of Prop osition in whichwe
showed that applicative similarity and compatible similarity coincide The congruence
rule in Table is the same as the Comp rule in Table The Precong rule in
Table is a common way to de ne precongruence We showed in Prop osition
that for any preorder such as applicative similarity or bisimilarity rule Comp is valid i
rule Precong is valid
Note rst a corollary of Prop osition for any canonical programs U and V
b
U V implies U V
L L L
b
x Wehave U V which x V x that U We are to showforany closure
L L L
x x V x U is to say that for all closures Since U and V are canonical terms
L L
x are canonical programs Therefore the corollary of Prop osition x andV U
b
L L
x as required V x applies and wehave U
Each law takes the form M M First we show that each law holds for
L R
ground equivalence when is empty For each beta lawwe can check by insp ection that
M M In the strictness law M andM Therefore by op erational adequacy
L R R L
wehave M M
L R
There are ve eta laws We can prove each lawbya case analysis of program M based
R
on the exhaustion law In the lawforSplit there must b e programs M such that M
i R
hM M i By the beta law for Split we have M Split hM M i x x hx x i
L
M M
x M Recall the general convention that b ound variables such x hx x i
R
as x and x in the table are distinct In the law for abstraction there must be a
term M such that M y M We know that x fv M So by the beta law for
R R
x
y M The other cases are similar abstraction M x M
R L
Second we prove each law for non empty Supp ose we are to prove a law of the form
L L L
M M We are to show for any closure x that M x M x Terms
L R L R
are identi ed up to alpha conversion so we may assume that any bound variable in M
L
is distinct from x Therefore the closure can be distributed into the bodies of or M
R
terms M and M to obtain programs M and M to which the law with empty can
L R L R
b e applied
Supp ose S is a bisimulation up to We will prove S where S is S the
relational comp osition of S and Since S S we will have established S as
required
Supp ose then that M S N and that We show that S is an applicative simulation
d
M U We must nd V such that N V and U S V Since M S N there must be
programs M and N such that M M M S N and N N Since M U and M M
CHAPTER THEORY OF THE METALANGUAGE
b
there must b e a canonical program U such that M U and U U Since M U we
have M U whichwithM S N implies there must b e canonical U and V such that
b
S V and V N Since V N there must be canonical V such that M U U
and V V From N V and N N there must b e a canonical V such that N V
b
N V and V V In all wehave
b
b b
U U U S V V V
b
b b b b
and therefore by canonical freeness that U S V Now for any con ned relations R
c d
b d
and R it is not hard to verify that R R RR Hence wehave U S V whic h completes
the pro of that S is an applicative simulation Since S is symmetric we have shown
that S as required
Empty and one p oint typ es in M
def def
Let typ e X X and in the following prop ositions we prove that is
empty and that has just one element
Prop osition There is no program of typ e
Pro of First consider the relation on programs of typ e inductively de ned by the
following two rules
M N
M N
Intro M Intro M
M N
By de nition includes the reduction relation but also reduces b eneath Intro s Since
is a partial function so is In Prop osition weshowed that each reduction of
an M program can b e matched by one or more reductions of its translation into
According to Table the translation Intro M is program Intro M we can
strengthen Prop osition to the following
If M and M N in M then M N in
e have a corollary analogous to the normalisation result for M Theorem Hence w
Let an normal form b e a program M of typ e such that M
For each M there is an normal form N such that M N
Any normal form is also an normal form so it must lo ok like Intro M where M
is also an normal form So then each normal form contains another normal form
that contains one less Intro If there is an normal form it must contain a nite number
of Intro s Therefore if there is an normal form at all there must b e one containing no
Intro s But this is imp ossible as each normal form is also an normal form and
hence must contain an outermost Intro Hence there can be no normal forms and
therefore no programs of type atall
Prop osition There is a canonical program oftyp e suchthatforany program
M oftyp e M
ITERATED PRODUCTS AND SUMS IN M
def
Pro of Let x x Toshow that is unique up to it su ces to show that relation
S is a bisimulation
def
S f M j M g
Supp ose that M S Since is certainly convergent there must be some canonical
term x N such that M x N Since x x we must show that
b
L L
S x N that is for all programs L that x x N x S But this is vacuously
true as there are no programs suchas L
Iterated pro ducts and sums in M
De ne iterated pro duct and sum notations by induction on n
if n
def
if n
n
if n
n
if n
def
if n
n
if n
n
Prop osition Supp ose and Then
n n
there are canonical terms hM M i and In M for i n and terms
n
i
Split M x x N and Case M x N x N such that
n n n
M x x N
M i n
n n
i i
Split M x x N
hM M i
n
n
M
M x N i n
i
i i i
In M
Case M x N x N
n n
i
M hM M i
n
M M
n
Split M x x N N x x
n
n
M In M
i
M
Case M x N x N N x
i
n n i
Pro of We can de ne the canonical terms as follows
if n
def
n
M if n
hM M i
n
n
hM hM M i i if n
n
M if n
def
n
n
Inl M if n i
M In
i
n
In M if n i Inr
i
CHAPTER THEORY OF THE METALANGUAGE
Given these de nitions it is straightforward to de ne terms Split M x x N and
n
Case M x N x N and to prove the desired prop erties
n n
Bo oleans in M
We de ne a b o olean typ e Bo ol and two Bo ol programs btt c and b c
def
Bo ol
def
btt c Inl
def
b c Inr
Prop osition Any program of typ e Bo ol either equals btt c or b c Moreover btt c
do es not equal b c
Pro of Supp ose M is a program of typ e By the exhaustion law M either equals
Inl N or Inr N for some program N By Prop osition program N equals so M
either equals Inl orInr that is btt c or b c
Toshowthatbtt c do es not equal b c pro ceed bycontradiction and supp ose that Inl
b
Inr By canonical freeness Inl Inr But there is no rule in Table which could
b
derive Inl Inr since the two terms have di erent outermost constructors Inl and
Inr Contradiction
We de ne the notation if M then N else N to mean Case M u N u N where variable
u fv N The if notation has the following prop erties
i
Prop osition
The following typ e assignment rule is valid
M Bo ol N i
i
if M then N else N
The op erational b ehaviour is characterised by the following rules
M b c M btt c
if M then N else N N if M then N else N N
The following equational laws are valid
if btt c then N else N N
if b c then N else N N
if M then N else N N
Pro of Immediate In the rst rule since M btt c wehave M Inl L for some
u N N The second rule program L So Case M u N u N Case Inl L u N
follows by a similar argument These equations are simple consequences of the b eta
and eta laws in Table
NATURAL NUMBERS IN M
Natural numbers in M
We de ne a typ e Num and a Num program bnc for each natural number n N
def
Num X X
def
b c Intro Inl
def
bn c Intro Inr bnc
Prop osition Any Num program equals bnc for some natural number n N More
over bnc do es not equal bn c forany n N
Pro of The second part follows easily from canonical freeness For the rst part we de ne
another extension of the reduction relation on Num programs
M N M N
M N
Intro M Intro M Intro Inr M Intro Inr N
M N
Evidently is a partial function and is contained in It is normalising in the sense that
if M is a Num program there exists an normal form N such that M N The pro of
is by app eal to the emb edding of M in from Chapter similar to Prop osition
If Num program M is normal then it either takes the form Intro Inl N where N
or Intro Inr N where N is also normal Hence for any normal form M there is a
c by induction on the number of Intro s in M Moreover for number n such that M bn
an arbitrary Num program M there is an normal form N suchthat M N and hence
anumber n such that M bnc
Let abbreviate the partial function The next prop osition provides a general
metho d for de ning partial functions by recursion
Prop osition For eachtermF that satis es
F
there is a term Rec F such that
Rec F
and if F is a program
Rec F M F Rec F M
for any program M
Pro of Let terms REC F and Rec F with typ es and resp ectively be as
follows
def
REC F Fix fx Lift x Seq fx g F g x
def
Rec F x Seq REC F g F g x
CHAPTER THEORY OF THE METALANGUAGE
One can easily verify that Rec F is of the desired typ e Now supp ose that M is a program
of typ e Wemay calculate as follows
REC F Fix fx Lift x Seq fx g F g x
Lift x Seq REC F g F g x
Lift Rec F
Rec F M x Seq REC F g F g x M
Seq REC F g F g M
Seq Lift Rec F g F g M
F Rec F M
Hence w ehave the desired reduction b ehaviour
In order to de ne the elementary arithmetic op erations needed in the denotational seman
tics of H we need the following programs
def
Succ x Num Lift Intro Inr x
def
Pred x Num Case Elim x u Lift b c y Lift y
def
Iter Z F Rec g Num y Num
Case Elim y u Lift Z y Seq gy x Fx
Prop osition
The typ e assignments
Succ Num Num
Pred Num Num
and
Z F
Iter Z F Num
are valid
These programs have the following reduction behaviour where Z and
F
Succ bnc Lift bn c
Pred b c Lift b c
Pred bn c Lift bnc
Iter Z F b c Lift Z
Z F bnc x F x Iter Z F bn c Seq Iter
Pro of By insp ection From the b eta rules of Table We calculate the last two
prop erties For any m wehave
Iter Z F bmc Case Elim bmc u Lift Z y Seq Iter Z F y x Fx
When m wehave
Iter Z F bmc Lift Z
and when m n
Iter Z F bmc Seq Iter Z F bnc x F x
NATURAL NUMBERS IN M
as required The other equations are no harder to prove
We can de ne some simple arithmetic op erations
def
Num
M b cN Iter M Succ N
def
Num
M b cN Iter M Pred N
def
Num
M b cN Iter b c x xb cM N
def
Iszero M Case Elim M y btt c y b c
def
M b cN Seq N b cM x if Iszero x then Lift b c else Lift btt c
def
M b cN Seq M b cN b if b then Lift b c else
Seq N b cM b if b then Lift b c else Lift btt c
Prop osition
bmcb cbnc Lift bm nc
bmcb cbnc Lift bm nc
bmcb cbnc Lift bm nc
Iszero b c btt c
bn c b c Iszero
bmcb cbnc Lift btt c if m n
bmcb cbnc Lift b c if n m
bmcb cbmc Lift btt c
bmcb cbnc Lift b c if m n
Pro of From the previous prop osition by inductive arguments
CHAPTER THEORY OF THE METALANGUAGE
Chapter
An op erational theory of
functional programming
The second half of this dissertation which begins with this chapter examines a small
functional language called H and considers how theories for various forms of functional
I O can be based on a theory of H This chapter shows how to develop an op erational
theory of programming for H Two parallel theories are develop ed one based directly
on an op erational semantics of H the other based on a denotational semantics of H in
M The former has b een develop ed from rst principles in this dissertation whereas the
latter dep ends ultimately on Mendler s result assumed in Chapter This chapter is one
of the most imp ortant in the dissertation b ecause of its p otential applications far b eyond
the study of functional I O
H is essentially a fragment of Haskell H contains the basic data and control constructs of
lazy functional programming natural numbers b o oleans functions recursion and lazy
algebraic typ es but is small enough that its theory can b e develop ed in this chapter We
omit from H many constructs needed for practical programming but which are irrelevant
to the study of functional I O such as p olymorphic typ es typ e inference mo dules and
realistic arithmetic
In x we de ne syntax and typing rules for the ob ject language H The abstract syntax
of H is rather restrictive compared to Haskell notation We show informally how certain
Haskell notations maybeinterpreted as H programs This interpretation allows us to use
of theories of functional Haskell notation in Chapters and for H in our development
I O In a study like this which is meant to be relevant to practical programming it is
go o d metho dology to work with as realistic a notation as p ossible Another merit of using
Haskell notation is that programs can b e typ e checked and executed
Wefollow the pattern set by Plotkin and givetwo semantics for H a deterministic
lazy op erational semantics and a denotational semantics in terms of the metalanguage M
The style of denotational semantics is greatly in uenced bythework of Moggi
and Pitts on the computational calculus as a semantic metalanguage The most
striking di erence between the computational calculus and M is that the former s se
Wehave used Mark Jones Gofer system and also an implementation of H on top of Standard ML
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
mantics is ultimately domain theoretic whereas the latter s semantics is entirely op era
tional We nish x by proving a corresp ondence theorem b etween the op erational and
denotational semantics that evaluation of any H term e is exactly matched byevaluation
of its M denotation e
We base our theory of functional programming on equivalence of H terms x develops
notions of ground and con ned relations We develop two relations op erational equiv
alence and denotational equivalence x develops op erational equivalence directly from
the op erational semantics of H Weprove that op erational equivalence for H is a congru
e s metho d just as we proved the same result for M in Chapter x ence using How
de nes denotational equivalence as the equivalence relation induced on H terms from the
denotational semantics in M The main result of the section is a soundness theorem that
denotational equivalence implies op erational equivalence Weleave op en the converse full
abstraction
In x we state a collection of basic programming laws and show that they are valid for
b oth op erational and denotational equivalence x shows how a further collection of laws
and pro of principles follow from the basic programming laws of the previous section The
theory of functional programming develop ed in x and x is entirely op erational in
that it rests on op erational equivalences de ned from the op erational semantics of either
M or H
Finally x considers what happ ens to the theory of functional programming if an ex
ception mec hanism is added to H The resulting language is called HX and is needed in
Chapter in the study of Landin stream I O
H a small functional language
Table shows the abstract syntax of H essentially a fragment of Haskell The only
construct of H not present in Haskell is call by value function application e e Call
by value is included in H b ecause of the control it gives over evaluation order suchcontrol
is useful when I O is expressed using side e ects
We assume countably in nite sets of typ e variables and term variables ranged over by
y z resp ectively We assume there is a countably metavariables X Y Z and x
in nite set of value constructors ranged over by metavariable K There are six cate
gories of abstract syntax typ es algebraic datatyp es data clauses dc
canonical terms c terms e p q andcase clauses cc Binding o ccurrences of typ e
and value variables are indicated in Table The notations and conventions concerning
variables substitution and alpha conversion stated in x apply to H
We let metavariable ranges over the set of literals ftt g N We use metavariable
to range over the arithmetic op erators f g An H literal or op erator is written
as or the corresp onding canonical program in the metalanguage is written b c or b c
resp ectively The H typ e Int includes just the natural numb ers we call this typ e Int for
the sake of compatibility with Haskell
Supp ose that is the algebraic typ e data X dc dc Then de ne functions
n
H A SMALL FUNCTIONAL LANGUAGE
Syntax
X typ e variable
j Bool b o oleans
j Int natural numbers
j functions
j algebraic datatyp e
data X dc dc algebraic datatyp e n
n
Con dc Con dc i i j
i j
X b ound in each dc
i
dc K data clause m
m
c literal ftt g N
j x e abstraction x b ound in e
j K e e constructor application m
m
e x value variable
j c canonical expression
j if e then e else e conditional
e arithmetic f g j e
j e e call by name application
j e e call by value application
j rec x e recursion x bound in e
j case e of cc cc case expression n
n
cc K e case clause
Table Syntax of H
Con dc Rank K andArity K as follows
def
Con K K
m
def
Rank K i if K Con dc
i
def
Arity K m if dc K
i m
Op erations Rank and Arity are well de ned functions b ecause the side condition on alge
braic typ es in Table requires that no two data clauses contain the same constructor
We place a further well formedness condition on algebraic typ es in any algebraic typ e
dc for each clause dc K in each typ e we insist data X dc
n i m j
that the typ e variable X o ccurs p ositively in each and that the only typ e variable free
j
in is X Atyp e variable X o ccurs p ositively in a typ e i each o ccurrence of X in
j
is to the left of an even numb er of function s The only reason we make this restriction
is so that the typ es of H can be mapp ed into well formed typ es of M We implicitly
assume that any H typ e wedealwithiswell formed Just as in M the set of well formed
typ es is closed under substitution
Typ e assignment in H
An environment is a list of variables paired with closed typ es written
x x where the variables are pairwise distinct We adopt the same no
n n
We followHaskell in using the symb ol for typ e assignment
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Typ e Assignment
n N
fT F g
n Int
Bool
x x
e Bool e e
if e then e else e
e Int e Int
f g
e e Int
e Int e Int
f g
e e Bool
x e
e e
x e
e e
x e
e e
rec x e
e e
e X j m
j j
data X dc dc
n
dc K
i m
K e e
m
e
m
K e K
m
e cc dc X i n
i i
data X dc dc
n
case eofcc cc
n
Canonical Terms
If c Bool then b ftt g c b
If c Int then n N c n
If c then e c x e
If c then K e e c K e e
m m
Table Typ e assignment rules for H
H A SMALL FUNCTIONAL LANGUAGE
tational conventions as de ned for M environments at the b eginning of x
De nition The H typ e assignment relation e is inductively de ned by
the rules in Table which make use of an auxiliary relation cc dc de ned in the
rule marked with Each rule b ears the implicit side condition that any environments
appearingintherulearewell formed
Prop osition
If e and e then
If e and then e
e
If x e and e then e x
If e then fv e Dom
The statements ab out canonical terms in Table are true
If e then ftv e
Pro of Easy rule inductions Part dep ends on the fact that no typ e variable can o ccur
free in a typ e in the range of an environment and that the environments used to derive
the typ e assignment relation must b e well formed
We de ne classes of programs and con ned terms
De nition
A program is a term e such that e for some necessarily unique typ e
The typ e is called the typ e of e which is called a program
A con ned term is a pair e such that there is a necessarily unique typ e
such that e The typ e is called the typ e of e and is called the
environment of e
Just as in M the syntax of H carries enough typ e information that any program has a
unique typ e We often omit typ e information from terms
Interpreting Haskell in H
We show in this section how some Haskell notations absent from H can nevertheless be
interpreted as derived forms of H For tutorial and reference material on Haskell we refer
the reader to the Haskell rep ort and to Fasel and Hudak s tutorial We shall use a
Haskell like notation for the programming examples in the remainder of this dissertation
We do not attempt a formal description of how Haskell can be interpreted in H but
instead give a series of illustrative examples Descriptions of similar interpretations are
eyton Jones textb o ok discusses in detail how Miranda well known for instance P
can b e translated into a calculus notation of ab out the same level as H
We will use the letters a b and c to stand for closed typ es of H Typ e and value de nitions
in Haskell may b e p olymorphic that is they may dep end on one or more typ e parameters
Weinterpret such de nitions in H as de ning families of closed typ es or terms indexed by
the typ e parameters Here are three examples the second of which also illustrates how
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
id a a
id x x
fst a b a
fst a b a
snd a b b
snd a b b
const a b a
const a b a
curry a b c a b c
curry f a b f a b
uncurry a b c a b c
uncurry f ab f fst ab snd ab
b c a b a c
f g x f g x
a head a
head x xs x
tail a a
tail x xs xs
a a a
ys ys
x xs ys x xs ys
map a b a b
map f
map f x xs f x map f xs
iterate a a a a
iterate f x x iterate f f x
foldr a b b b a b
foldr f z z
foldr f z x xs f x foldr f z xs
Table Some standard functions
H A SMALL FUNCTIONAL LANGUAGE
def
id x x
def
fst ab case ab of Tuple a b a
def
snd ab case ab of Tuple a b b
def
const a b a
def
curry f a b f a b
def
uncurry f ab f fst ab snd ab
def
f g x f gx
def
head xs case xs of
Nil
Cons x xs x
def
tail xs case xs of
Nil
Cons x xs xs
def
rec app xs ys case xs of
Nil ys
Cons x xs Cons x app xs ys
def
map rec map f xs case xs of
Nil Nil
Cons x xs Cons fx map f xs
def
rec iterate f x Cons x iterate f fx iterate
def
foldr rec foldr f z xs case xs of
Nil z
Cons x xs f x foldr f z xs
Table Interpretations of Haskell functions in H
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Haskell style algebraic datatyp es are interpreted in H
Interpretation in H Polymorphic Haskell de nition
def
a
type Endo a a a Endo a a
def
a
List data X Nil Cons a X data List a Nil Cons a List a
id a a
def
a
id x a x
id x x
We use notation in typ ewriter fontforH literals for instance interpret True False
and as the b o olean literals tt and the numeric literals and and the
addition op erator resp ectively
Write a as an abbreviation for List a De ne
def
Nil empty list
def
e e Cons e e e non empty list
n n
def
Cons e e in x cons e e
De ne a series of p olymorphic tuple typ es indexed by m
data Tuplem a am Tuplem a am
De ne the following abbreviations for tuples where m
def
a a Tuplem a a m tuple typ e
m m
def
e e Tuplem e e tuple
m m
def
x x e y case y of tupled abstraction
m
Tuplemx x e y fv e
m
The tuple typ e is called the unit typ e
To discuss teletyp e I O we need to supp ort the Haskell typ e Char of ASCI I characters
For the b ene t of simplicity at the cost of cavalier software engineering we de ne Char
ell character notation to be n where n is the ASCI I to be Int We interpret any Hask
co de for the character For instance a and n are interpreted as and
resp ectively We interpret the Haskell functions ord Char Int and chr Int Char as
the identity function id We de ne a typ e of strings as in Haskell
type String Char
To illustrate how Haskell function de nitions are to be interpreted in H we show in
Table some standard function de nitions Each of these function de nitions can be
def
interpreted as an H de nition as shown in Table Let H program rec x x We
use to represent unde ned values as in head or tail The case clauses in Table
use a derived syntax
def
Kx x e K x x e
n n
Indentation is used instead of symbols to separate multiple case clauses Using the
theory develop ed later in this chapter one can check that any of the de nitions in Table
when treated as an equation between programs is provable from the corresp onding H
de nition in Table
H A SMALL FUNCTIONAL LANGUAGE
Expressions in Haskell maycontain lo cal de nitions An expression ewherex e isto
be interpreted as the H expression x e e AHaskell list comprehension of the form
e x e is interpreted in H as map x e e
Op erational semantics of H
The op erational semantics of H is deterministic and lazy in the sense that constructors
of algebraic typ es do not evaluate their arguments
De nition Left and right exp eriments LE and RE resp ectively are functions
from terms to terms de ned by the grammar at the top of Table A left exp eriment
de nes the leftmost p osition in a term where a reduction may o ccur A right exp eriment
de nes a p osition in a term where a reduction may o ccur provided terms to its left are
canonical These are analogous to exp eriments in M De nition
The reduction and evaluation relations for H are the binary relations on H programs
and resp ectively de ned inductively by the rules Table
Prop osition
If e and e e then e
If e and e e then e
If e e and e e then e e
If e c and e c then c c
The canonical terms are the normal forms of reduction
If e c then e c
Supp ose e e Then for any c e c implies e c
e c i e c
Pro of Similar to the pro of of Prop osition
We de ne terminology for termination of evaluation
De nition Supp ose e is a program Say that e converges and write e i there is a
necessarily unique canonical program c such that e c Conversely saythate diverges
and write e i e do es not converge
a a
Recall the program de ned earlier to be rec x x at each closed typ e a We have
a a
hence there is a divergent program at every closed typ e of H
Denotational semantics of H
The denotational semantics of the ob ject language H is given using M as a metalanguage
given by Pitts Classical domain The semantics is based on that of Tiny ML
theoretic semantics use explicit environments to mo del the binding of ob ject
variables to their values These environments are typically denoted by the metavariable
and are not to be confused with the environments denoted by here Following
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Reduction Semantics
e j e j e j case ofcc cc LE if thene else e j
n
RE j x e
iftt then e else e e
if then e else e e
e
x e e e x
c
x e c e x
rec x e
rec x e e x
cc K e
Rank K
case K e e of cc cc e e e
m n m
e e e e
LE e LE e RE e RE e
Evaluation Semantics
c c
e tt e c e e c
if e then e else e c if e then e else e c
e e
e e
e
x c e x e e
e e c
c
e x e e c e x c
e e c
rec x e
e x c
rec x e c
e K e e cc K e e e e c
m m
Rank K
case e of cc cc c
n
Table Op erational semantics of H
H A SMALL FUNCTIONAL LANGUAGE
Pitts we can take advantage of variable binding in the metalanguage and do without such
environments by mapping ob ject variables to metavariables
We structure the denotational semantics to re ect the distinction made by the op erational
semantics b etween canonical and non canonical terms Computation is characterised op
erationally either as a series of reductions or as a single evaluation Canonical terms
whichare not reducible are the answers from computation Non canonical terms which
are reducible represent a computation which dep ending on the ob ject language may
terminate and return an answer or may diverge or may engage in other activity such as
I O We parameterise the denotational semantics on a metalanguage typ e constructor T
Let and Val that ob ey the following typing rules with asso ciated op erations
M M T v N T
Val M T Let v M in N T
Free o ccurrences of v in N become b ound in Let v M in N Typ e T represents
computations of typ e To compute Val N simply return the answer N To compute
Let v M in N rst compute M If M returns an answer L pro ceed to compute
L
N v to obtain the answer from the whole computation In principle the bene t of
this parametric approach is that the same translation rules can be used with di erent
interpretations of the structure T Val Let For instance in x wegiveaninterpretation
where computations may raise an exception instead of returning an answer
This parametric approach to denotational semantics was pioneered by Moggi
who de ned a metalanguage called computational calculus into which ob ject lan
guages are translated and whose own semantics is given by a computational mo del
a category with a strong monad and other prop erties Pitts prop oses the devel
evaluation logic for reasoning ab out the denotations of ob ject programs in opment of
computational calculus Crole and Pitts investigate how to obtain the power
of general recursion in computational calculus We have adopted Moggi s parametric
approach but instead of using a metalanguage with a general categorical mo del we use
M a metalanguage based on Plotkin s but with a sp eci c op erational semantics and
e ectively a term mo del Rather than work in a general categorical framework wework in
a pragmatic op erational framework in order to study a particular programming language
and its extensions for I O
Wenowgive a denotational semantics of H in which computations may either diverge or
return an answer to re ect the op erational semantics of Table
De nition Make the following de nitions of parameters T Val and Let
def
T
def
Val N Lift N
def
Let v M in N Seq M v N
Given these parameters de ne denotations for the abstract syntax of H inductively ac
cording to the rules in Table
to eachtyp e anM typ e
to each data clause dc anM typ e dc
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Typ es
def
X X
def
Bool Bo ol
def
Int Num
def
T T
def
data X dc dc X dc dc
n n
Data clauses
def
K T T
m m
Canonical terms
def
j x ej x T e
def
j b c j
def
jK e e j Intro In h e e i
m m
Rank K
Terms
def
x x
def
c Val jcj
def
if e then e else e Let v e in if v then e else e
def
e e Let v e in Let v e in
Seq v b cv x Val x
def
e e e e
def
e e Let v e in Let v e in v Val v
def
rec x e Fix x e
def
case eofcc cc Let v e in
n
Case Elim v v cc v cc
n n
v v
n
Case clauses
def
u u where m Arity K K e Split v u u e
m m
v
Table Denotational semantics of H
H A SMALL FUNCTIONAL LANGUAGE
to each canonical term c an M term jcj
to each term e an M term e
to each case clause cc anM term cc
The b ound variables v v u used to construct M denotations are assumed to b e distinct
i i
from anyvariables o ccurring free in ob ject language terms We use a derived notation for
function application
def
M N Let f M in fN
The notation asso ciates to the left that is M M M means M M M
Finally if H environment x x then is the M environment
n n
x T x T
n n
Op erations Val and Let are simply an alternative syntax for primitives Lift and Seq of M
we use these abstract op erations so that the rules of Table can b e re used with other
interpretations of T Val and Let
The only rules to makeany assumptions ab out the typ e constructor T in the denotational
and arithmetic By translating semantics rules in Table are the ones for recursion
rec to Fix and using Seq in the translation of arithmetic we are assuming that T has the
prop erty that for any T is a lifted typ e
Prop osition
If is an H typ e then is a well formed M typ e
If e then e T
e e
If x e and e then e x e x
e e e e
n n
Pro of Part follows by structural induction on given the restriction on the form of
algebraic typ es in H Part is by induction on the depth of inference of e Part
is by induction on the depth of inference of x e and is straightforward
since the denotational semantics is comp ositional Part follows by an induction on n
Example denotations
We rep eat the de nition in H of the function head from Table
def
cc Nil
def
cc Cons x xs x
def
head xs case xs of cc cc
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
To illustrate the denotational semantics we calculate the following denotations
X T TX
Cons Nil
Val Intro Inr h Nil i
Val Intro Inr hValb c Val Intro Inl i
head Val xs case xs of cc cc
Val xs Let v xs in
Int Int
Case Elim v v cc v cc
v v
Int
cc Split v
v
Int
cc Split v u u x xs x u u
v
x xs xs Val x Val xs x
We will return to these examples when we lo ok at denotational equivalence in x
Op erational corresp ondence between the semantics
First we show op erational prop erties of the denotational constants that is the M typ e
constructor T and op erations Val Let used in the denotational semantics
Lemma
For any T program M eitherM or M Val N for some program N
The op erational b ehaviour of the denotational constants ob eys the following rules
M
M Val L
L
Let v M in N
Let v M in N N v
Pro of From the de nitions of Val and Let
We conclude this section on the de nition of H with the pro of of a close corresp ondence
between the op erational b ehaviour of each H program and its denotation
Prop osition For any H program e canonical H program c and canonical M
program V wehave
If e c then e c
If e V then there is c suchthat V c and e c
Therefore e c i e c
Pro of Part is proved by induction on the depth of inference of e c We show two
example cases
Case e e where e c is canonical Wehave e Valjcj which is canonical so e e
e Wehave c m m where each e m By app eal to the induction Case e e
i i
hyp othesis and Prop osition wehave the following
GROUND AND CONFINED RELATIONS IN H
e Let v e in Let v e in Seq v b cv x Val x
Let v Val bm c in Let v e in Seq v b cv x Val x
Let v e in Seq bm cb cv x Val x
Let v Val bm c in Seq bm cb cv x Val x
Seq bm cb cbm c x Val x
Seq bm m c x Val x
Val bm m c
c
Part is proved by induction on the depth of inference of e V pro ceeding by a
structural analysis of e Again weshowtwo example cases
Case e is canonical Wehave e V where V e and e e
Case e e e Wehave
e Let v e in Let v e in Seq v b cv x Val x
Pro ceed by a case analysis of the evaluation behaviour of e and e From
Lemma either e or N e Val N for each i But neither e nor
i i i i
e can diverge or else by Lemma e would diverge So there are N such
i
and e Val jm j Then by a calculation that e Val N By IH wehave e m
i i i i i i
similar to the one given for part wehavethat e Val jm m j and also that
e m m as required
Ground and con ned relations in H
Before developing op erational and denotational equivalence we need several preliminary
de nitions This section reworks basic notions of ground and con ned relations develop ed
rst in x Recall the notions of program and con ned term from De nition
De nition
A ground relation R is a binary relation b etween programs of the same typ e
A con ned relation R is a binary relation between con ned terms of the same
typ e and environment Write eRe to mean that e e R
We state a sense in which a ground relation resp ects the op erational semantics Recall
yp e equals rec x x from x that program of anyt
De nition A ground relation R isop erationally adequate i for all programs
e e and canonical programs c
If e e then e Re
If e c then cRe
e i eR
e i for some canonical c cRe
b
The rst half of Table shows inference rules for constructing the con ned relation R
from any con ned relation R The second half of the table shows six inference rules that
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
b
Rules of R
b b
x R x R
e R e i
e Re i
i
i
i
i
b
b
e e R e e
if e then e else e R if e then e else e
x eRe e R e i
i
i
b b
x e R x e e e R e e
x eRe e Re i n
i
i
b b
rec x e R rec x e K e e R K e e
n
n
i n e R e K e eRe cc K e cc
i i i i i
i i i
b
case e of cc cc R case e of cc cc
n
n
Prop erties of Con ned Relations
eRe
Weak
eRe
eRe
Stren
eRe
x e Re e
Sp ec
e e
e x Re x
eRe
Precong
C e RC e
b
e R e
Comp
eRe
x e Re e Re
Sub
e e
e x Re x
b
Table De nition of R and rules concerning con ned relations
GROUND AND CONFINED RELATIONS IN H
maybevalid for a con ned relation
De nition
b
If R is a con ned relation then con ned relation R is de ned by the rules in the
rst half of Table
The inference rules Weak Stren Sp ec Precong Comp and Sub are de ned in
the second half of Table
A con ned relation is natural i the rules Weak Stren and Sp ec are valid
A con ned relation is a precongruence i the rule Precong is valid A congru
ence is a con ned relation that is b oth a precongruence and an equivalence relation
As usual the rules in Table b ear the implicit side condition that any sentence in a rule
is well formed In rule Stren for instance from the lower sentence eRe we may
deduce that fv e e Dom and then from the upp er sentence eRe that
Dom Dom and hence that fv e e Dom
Prop osition
If rules Sp ec and Precong are valid for a transitive con ned relation then rule
Sub is valid to o
If R is a preorder rule Comp is valid i rule Precong is valid
Pro of Similar to that of Prop osition
We will often need to induce a con ned relation from a ground relation and vice versa
as follows
De nition
Let be an environment x x Then a closure is an iterated substi
n n
p p
n
tution x x where each p is a program The order of substitution
n
i i
do es not matter b ecause the variables are disjoint and each p is closed
i
The con ned extension of a ground relation R is the con ned relation R such
G
that
p p p
eRe i for all closures x e x R e x
G
If R is a con ned relation then its ground restriction is the ground relation
f p q j pRqg
If R is a con ned relation write pRq to mean that pair p q is in the ground
restriction of R
Prop osition The con ned extension of a ground relation is natural that is rules
and Sp ec are valid Weak Stren
Pro of Similar to that of Prop osition
Wetypically use the same symb ol for a ground relation and its con ned extension If R
G
d b
is a ground relation and R is its con ned extension we sometimes write R for R G
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Op erational equivalence for H
In this section we de ne a notion of applicative bisimilarity directly from the op erational
semantics of H In x we will develop a second notion of equivalence from the de
notational semantics This op erational theory of H parallels the theory of applicative
bisimilarity for M develop ed in Chapter Just as in that chapter one can also develop a
theory of contextual equivalence for H and establish an op erational extensionality result
but for the purp ose of studying functional I O applicative bisimilarity is su cient Webe
gin with applicative similarity the ground preorder from whichwe shall de ne op erational
equivalence
De nition De ne function over ground relations such that
b
c with q c and c S c p S q i whenever p c there is
q q p q p
An applicative simulation is a relation S such that S S De ne ground applica
tive similarity v to be the union of all applicative simulations De ne applicative
similarity v to b e the con ned extension of ground applicative similarity
We can restate the de nition of as follows
Lemma For any ground relation S p S q i
whenever p then q
whenever p x e then q x e
p p
and for all programs p e x e x S
whenever p K p p then q K q q
n n
and p q S for each i
i i
Prop osition
Function is monotone
The identity ground relation is an applicative simulation
If each S is an applicative simulation then so is S S
i
Ground applicative similarity is the greatest xp oint of and is the greatest ap
plicativesimulation
M v N i there is an applicative simulation S suchthat M S N
Ground applicative similarity is a preorder
Pro of Parts and are easy to check from the de nition The remaining parts
then follow from Prop osition
De nition An applicative bisimulation is a relation S such that both S and
S are applicative simulations Ground applicative bisimilarity is the ground
relation v v Applicative bisimilarity is the con ned extension of ground
applicative bisimilarity
OPERATIONAL EQUIVALENCE FOR H
Clearly applicative bisimilarityis an equivalence relation We have for any programs e
and e e e i for some applicative bisimulation S eS e
Prop osition Ground relations v and are b oth op erationally adequate
Pro of Similar to the pro of of Prop osition
We adopt v and as op erational order and equivalence resp ectively on H
Applicative similarity is a precongruence
We prove that applicative similarity for H is a precongruence by a typ ed reworking of
Howe s metho d just as weproved that applicative similarityfor M was a precongruence
in Chapter
De nition De ne the con ned relation compatible similarity v to b e the least
set closed under the rule
c
e v e e v e
e v e
Again despite the notational ambiguity v is not de ned to be the re exive transitive
closure of v
Prop osition
Applicative similarity is natural that is rules Sp ec Weak and Stren are valid
c
Con ned relations v and v are re exive
c
Con ned relations v and v b oth imply v
If e v e and e v e then e v e
e e
x then e x v e and e v e Sub If x e v e
c
If c v e then there is c such that e c and c v c
If e v e and e c then c v e
e v e If e v e then
Applicative similarity is a precongruence
Pro of The pro of takes exactly the same form as Prop osition We omit the details of
parts to which are almost identical to the corresp onding parts of Prop osition
c
In each case there must b e a program e such that c v e and e v e Pro ceed by
a structural analysis of canonical program c
So e Then v e implies that e as required Case c
Case c x e So e x e with x e v e Then e v e implies
that e x e with x e v e Therefore x e v e by part
c
whichimpliesc v x e as required
p p So e K q q with p v q for each i Then e v e Case c K
n n i i
implies that e K q q with q v q for each i By part we have p v q
i i
n
i i
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
c
for each i and therefore c v K q q as required
n
By induction on the depth of inference of e c pro ceeding by analysis of e
Case e canonical Immediate
Case e e e We have e with c From e e v e there are
i i
e i such that e v e and e e v e By IH we have v e hence
i i
i i i
v e From v e by de nition there are e such that v e and
i
i i
e e v e By wehave e So e e and v e e
i
i
by op erational adequacy Therefore c v e by transitivity and c v e by
e
x c From e v e there are e Case e e e Wehave e x e ande
i
i such that e v e and e e v e Since e x e and e v e
i
i
we have x e v e by IH By there is e with e x e and
e e e
e c v e x x e v e By Sub e x v e x then by IH we hav
e e
Since e e e x we have e x v e e by op erational adequacy From
e e
x wehave e e v e wehave e by transitivity Finally from c v e x v e
c v e by
c
Case e e e We have e x e e c and e x c From e v e
there are e i such that e v e and e e v e Since e x e
i
i i
and e v e wehave x e v e by IH Since e c and e v e wehave
c v e by IH By there is e with e x e and x e v e
c
By there is c with e c and c v c Therefore c v c by By Sub
c c c c
e x v e x then by IH we have c v e x Since e e e x
c
we have e x v e e by op erational adequacy From e e v e we have
c c
e x v e by transitivity Finally from c v e x wehave c v e by
K e We have e K p p for some j and Case e case e of
i i j n
e p p c From e v e there are e suchthat e v e and case e of
j n i
i i
ehave K p p v e K e v e Since e K p p and e v e w
j n i j n
i
by IH By there are p such that e K p p andp v p By Sub wehave
j i
n
i i
p p so since e p p c wehave c v e p p e p p v e by
j n j n
n n
j j
v p p we have e p p e of K e IH Since case e
i
n n
j j i
by op erational adequacy By transitivity and case e of K e
i
i
wehave c v e
We omit the case for conditionals which has the same structure as the cases ab ove
restriction of compatible similarity is an applicative First we prove that the ground
simulation Supp ose for any programs e and e that e v e We are to show that whenever
c
e c there is c suchthate c and c v c Supp ose that e c By wehave c v e
c
By there is c such that e c and c v c
e v e whichisto say that Supp ose now that e v e We are to prove that
p p p
for all closures x programs e x ande x are paired in an applicative simulation
p p
But by Sp ec and re exivitywe have e x v e x and since the ground restriction of
compatible similarity is an applicativesimulation we are done
A corollary of and is that applicative and compatible similarityare the same con ned relation just as in Prop osition
DENOTATIONAL EQUIVALENCE FOR H
To summarise this section we havetaken op erational equivalence on H to b e applicative
bisimilarityshown it to b e an op erationally adequate congruence
Denotational equivalence for H
Two ob ject terms are denotationally equal just when their denotations in the metalanguage
are equal We take equivalence in the metalanguage to be a op erational equivalence on
M the equivalence characterised as contextual equivalence and applicative bisimilarity
in Chapter
De nition De ne denotational equivalence to b e the con ned relation
def
e e e e
Since op erational equivalence on M is a natural equivalence relation so is denotational
equivalence on H Next we prove that denotational equivalence resp ects the op erational
semantics of H
Prop osition Denotational equivalence is op erationally adequate
Pro of There are four parts to the de nition of op erational adequacy De nition to
establish We app eal to op erational adequacy for ground applicative bisimilarityin M
Prop osition and the corresp ondence theorem between the op erational and denota
tional semantics of H Prop osition Weleave part to the end of the pro of
Part Supp ose e c Then e c from the corresp ondence theorem So e c
from op erational adequacy of M Then e c by de nition
Part We have b een using the symb ol to stand for Fix x x in M and rec x x in
H Note that where is interpreted in H and M in the left and right hand
sides resp ectively
e i c e c
i c e c Corresp ondence
i e
e Op erational adequacy for M i
i e since
i e
Part
e i c e c
i c e c Corresp ondence
i c e c Op erational adequacy for M
i c e c
c In the former case e Finally part Supp ose e e Either e or for some c e
to o so e e by part In the latter case e c to o and e e c by part
Moggi de nes a notion of computational mo del based on a categorical strong monad
as a general framework for reasoning with denotational semantics We make an analogous
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
de nition of a computational monad in the context of M
De nition A computational monad is a structure T Val Let where T maps
any M typ e toanM typ e T and Val and Let are M op erations that ob ey the typing
rules
M T v N T M
Val M T Let v M in N T
together with the equations
M T v N T
M
Let v Val M in N N v
M T
Let v M in Val v M
M T v M T v M T
Let v Let v M in M in M
Let v M in Let v M in M
and the injectivity requirement that Val M Val N i M N
Lemma stated prop erties of the evaluation behaviour of the op erations used in the
denotational semantics as a preliminary to proving the corresp ondence theorem between
the op erational and denotational semantics for H In the following lemma weprove equa
tional prop erties of the op erations used in the denotational semantics as a preliminary to
proving the main result of this section that denotational equivalence implies op erational
equivalence
Lemma The structure T Val Let that parameterised the denotational semantics
of H in De nition is a computational monad
Pro of The typing rules in De nition followby insp ection The rst two equations
follow from the b eta and eta laws for lifting in Table Asso ciativityof Let follows from
proving the following equation
Seq Seq M v M v M Seq M v Seq M v M
It su ces to prove the equation for empty from which case the non empty case follows
Lift L for some program L In By the exhaustion lawinTable either M orM
the rst case b oth sides equal by the strictness lawinTable In the second case by
L L
the b eta law from we have lhs Seq M v v M andrhs Seq M v M v
lhs since v fv M Finally the injectivity requirement follows from canonical freeness
in Table
An example calculation
Before pro ceeding we show an example calculation Recall the denotations calculated on
page We show that head by calculating as follows where we use
DENOTATIONAL EQUIVALENCE FOR H
only the laws of M from Chapter the equational laws of a computational monad and
substitution prop erties from Prop osition
head Let v head in v
Let v Val xs case xs of cc cc in v
xs case xs of cc cc
xs Let v xs in
Case Elim v v cc v cc
v v
Let v in
Case Elim v v cc v cc
v v
Let v Val Intro Inr h Nil i in
Case Elim v v cc v cc
v v
Case Inr h Nil i v cc v cc
v v
h Nil i
cc v
v
Nil i u u x xs xs u u Split h
x xs x Nil
Let v x xs x in v Nil
Let v Let v x xs x in v in v Nil
x xs x in v Nil Let v
Let v xs in v Nil
xs Nil
The calculation is extremely detailed but each step is trivial A more tractable way to
reason ab out H programs is to app eal to a set of laws formulated at the ob ject level
such as those we develop in x Suchlaws can b e proved correct by translation into the
metalanguage and then used for ob ject level reasoning
The other point to make ab out this calculation is that it holds for any computational
monad not just the one in De nition An imp ortant reason for parameterising the
semantics is so that program calculations that do not dep end on the detail of a particular
computational monad can b e proved once and for all
Denotational implies op erational equivalence
Weprove a soundness theorem that denotational implies op erational equivalence Com
pleteness or full abstraction whether op erational implies denotational equivalence is left
as an op en problem
Prop osition
b
If c c then c c
For all programs e e ife e then e e
If e e then e e
Pro of Part Wehave Valjcj Valjc j and therefore by the monadic injectivity
requirement wehave jcj jc j By analysis of the typ e of c and c there are three cases to consider
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Case c c Wehave b c b c therefore as required
Case c x e c x e We have x e x e and there
fore x T e e by canonical freeness in M as required
e e c K e e Supp ose c and c are of typ e in environment Case c K
m
n
Wehave
Intro in h e e i Intro in h e e i
m
Rank K Rank K
n
Therefore wehave K K and m n and for each i e e by canonical
i
i
freeness in M as required
For part we prove that the ground restriction S of denotational equivalence is an
applicative bisimulation
def
S f e e j e e g
that it is an applicative simulation Supp ose Since S is symmetric we need only prove
then that e c By the corresp ondence theorem wehave e c Since e e there
is a V suchthat e V and c V by op erational adequacy for M By corresp ondence
there is a canonical c such that e c and V c Wehavethat c c and hence
b
b
Hence by wehave c c whichistosay c S c as required that c c
p p p
Part We are to show for any closure x that e x e x But from
p p
e e and Sp ec we have e x e x By substitution lemma we have
p p
e x e x Then by part we are done
A theory of H programming
We state a collection of programming laws in Table In his seminal study of program
ming languages Strachey b egins his discussion of expressions and evaluation with the
de nition that the characteristic feature of an expression is that it has a value What
wehave called canonical expressions corresp ond to what Strachey calls values Expressions
in H are more expressivethanStrachey s in that the former include recursive expressions
and hence non termination Nonetheless what wehave named Strachey s prop ertyinTa
ble conveys the essence of Strachey s view that every H program either equals or
some canonical program
We show in this section that all these laws hold for both op erational and denotational
equivalence except the principle of bisimulation up to which has only b een proved for
op erational equivalence We conjecture without pro of that it holds also for denotational
equivalence We b egin with op erational equivalence
Prop osition All the laws from Table hold for op erational equivalence
Strachey s prop erty
congruence
canonical freeness
b eta strictness and eta laws
co induction
ATHEORYOFH PROGRAMMING
Pro of The pro of is similar to that of Prop osition
Weprove the same results apart from the co induction principle for denotational equiv
alence
Prop osition All the laws from Table hold for denotational equivalence with
in place of
Strachey s prop erty
congruence
canonical freeness
b eta strictness and eta laws
Pro of By de nition of evaluation any program e either diverges or evaluates to some
canonical program c Then by op erational adequacy in the rst case e and in the
second e c
b
Given e e we are to show e e Pro ceed byananalysisofwhich rule
b
in Table derived e e In each case from the rule s anteceden ts we can use the
denotational semantics rules in Table to prove e e as required
Canonical freeness for denotational equivalence was proved as Prop osition
Eachlawtakes the form e e By de nition we are to show for all closures
L R
L L L
x Terms are identi ed up to alpha conversion so we may x e x that e
R L
is distinct from variables in the listx For assume that any b ound variable in e or e
R L
L L
eachbetalawwecancheckby insp ection that e x e x For each eta lawwe
L R
can check the same prop erty but with app eal also to the laws of canonical programs in
L
x Therefore for any Table For each strictness law e we can check that e
R L
one of the strictness laws the required equation follows from op erational adequacy of M
As a simple example of co induction here is a pro of of an equivalence of two streams
used byPaulson to illustrate Scott induction Since the two streams are unb ounded
structural induction would fail on this example
Prop osition For any f and x
iterate f fx map f iterate f x
De nitions of iterate and map are shown in Table
Pro of Consider relation S de ned as follows
S f iterate f fx map f iterate f x j f x g
We shall prove that the union S is a bisim ulation up to and hence that S
Supp ose then that e and e are arbitrary programs such that eS e Set
e iterate f f fx
e map f iterate f fx
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Strachey s Prop erty
Any program e either equals or some canonical term c
Congruence
If e b e then e e
Canonical Freeness
If c c then c b c
Beta Laws
ifT then e else e e
ifF then e else e e
e
x e e e x
c
x e c e x
case K e e of K e K e e e e
i m n m
n
i
rec x e
rec x e e x
Strictness Laws
LE
RE
Eta Laws
if c then e else e e
x c x c if x fv c
case cof K x x e e if no x fv e
i i
Arity K
i
Co induction
A bisimulation up to is a ground relation S such that whenever eS e
e c either e e or there are canonical programs c c such that
b
c and c S c e
Any bisimulation up to is a subset of ground equivalence
Table Laws of equivalence in H
DERIVED LAWS OF PROGRAMMING
and we can calculate as follows
e iterate f fx
fx e
e map f iterate f x
map f x iterate f fx
fx e
Since f x f x and e S e we have f x e S f x e Since e and e were
arbitrary wehave established that S is a bisimulation up to Hence S and the
prop osition follows
Derived laws of programming
In this section we derive further laws of program equivalence Their derivations dep end
only on the laws in Table not including co induction so they hold for both op era
tional and denotational equivalence It is reassuring that these familiar results functional
extensionality and structural induction can b e derived simply from the theory of H
Functional extensionality
Two canonical functions are equal if whenever they are applied to the same argument they
yield the same answer
Prop osition If x c x c x then c c
Pro of We may assume that the canonical terms c and c take the forms x e and
x e We have x x e x x e x Therefore by the b eta law for
functions wehave x e e Then by precongruence c c
This principle do es not extend to non canonical functions Counterexample let programs
f and g of typ e be and x resp ectively Then x f x g x yet f g
since f whereas g
Structural induction for sum of pro duct typ es
We can derive a familiar structural induction principle for a class of sum of pro duct typ es
De nition A sum of pro ducts typ e is an algebraic datatyp e such that
data X dc dc and each clause dc takes the form K where each
n i m i
X or is closed is either the variable
For each suchtyp e de ne a program size
size Int
def
size rec size x case x of cc cc
n
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
where for each i ifdc K we de ne cc to b e
i m i
def
cc Kx x t t
i m m
size x if X
def
j j
where each t
j
otherwise
De ne Finite to b e the following set of programs
def
g Finite fe j e is a program and n N size e n
Finally saythata program e is nite i e Finite
Prop osition Supp ose is a sum of pro duct typ e data X dc dc If
n
is a predicate on nite programs that is Finite then the following induction
principle is valid
for each clause dc K e e K e e
i m m m m
if X
def
j
where eachset
j
fe j e g otherwise
j
Finite
Pro of Supp ose that the rule hyp othesis holds By mathematical induction on n we prove
then e This amounts to a pro of that for all programs e that if size e n
Finite as required
Supp ose then that size e n for some e By Strachey s prop erty we have that either
e or e equals a canonical program But size by the strictness law for case
expressions so e must be canonical So there must be a clause K such that
m
e K e e From size K e e n wehave
m m
size e if X
j j
where for each j t t t n
j m
otherwise
Wecanshow that each e Either X or not If X from the equation ab ove
j j j j
we have that size e m for some m n So byIHwe have e Otherwise
j j j
weknowthate since e K e e and hence e
j j m j j
Since each e we can conclude that e from the rule s hyp othesis Hence wehave
j j
that every program e of any size is contained in So Finite and wehaveveri ed
the induction principle
The list typ e is a sum of pro ducts typ e in the sense of De nition Given a closed typ e
a of list elements the general de nition of size given there sp ecialises to the following
a
size a Int
def
a
size rec size x case x of
Nil
Consxxs size xs
a
The set Finite consists of those a programs xs such that size xs n for some
n We call such a list nite Lists or are nite lists
HX AN EXTENSION OF H WITH EXCEPTIONS
rec xs xs which are all distinct since Cons is lazy are not nite in this sense
Prop osition yields the following structural induction principle for nite lists
Nil x a xs Consxxs
a
Finite
a
where Finite is a predicate on lists
HX an extension of H with exceptions
We develop our theory of functional programming by considering what happ ens when
we add an exception mechanism to H Languages such as Standard ML and Mo dula
have an exception mechanism which is typically used by programmers to express error
conditions We use exceptions in Chapter to mo del the demand for function arguments
yp e during evaluation In HX a computation of typ e can either return a value of t
diverge or raise an exception For the sake of simplicity we consider a language with just
one exception the canonical term bang Raising an exception is represented by a program
evaluating to bang which is present at every typ e For the sake of brevity we say the
program has banged Program bang bangs In general if a program needs to evaluate
several subterms b efore terminating and evaluation of any one of the subterms bangs
then the whole program bangs The only exemptions from this rule are programs of the
form e e If evaluation of e returns an answer or div erges then evaluation of the
whole program do es so to o But if evaluation of e bangs then the whole program b ehaves
the same as e The op erator called biased choice is a new primitivein HX itisa
version without parameters of exception handling mechanisms found in Standard ML and
Mo dula
De nition
The abstract syntax of HX is based on the same syntactic categories as H de ned by
the H rules from Table augmented by the two new syntactic rules from Table
De ne the predicate Mute c on canonical terms to hold i for no typ e do es
c bang
The typ e assignment relation e is inductively de ned bythe H rules from
Table together with the new typ e assignment rules from Table
X are the binary relations on HX pro The reduction and evaluation relations for H
grams and resp ectively de ned inductively by the reduction and evaluation
rules resp ectively from Tables and The rule for call by value reduction in
Table is to apply only when Mute c and the rule for call by value evaluation
only when Mute c
Make the following de nitions of parameters T Val and Let
def
T
def
Val N Lift Inl N
def
Let v M in N Seq x u Case u v N w Lift Inr
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Syntax
c bang exception
e e e biased choice
Typ e Assignment
e e
bang
e e
Reduction Semantics
e e
bang e e
e e e e
c e c if Mute c
LE bang bang if LE bang
RE bang bang if RE bang
Evaluation Semantics
e bang e e e
e bang LE e
e e bang
LE e bang
e x e e bang e e
e e bang
e bang e c
e c Mute c
e e c
e e c
Denotational Semantics
bang T
alt T T T
def
bang Lift Inr
def
alt x y Seq x u Case u v Val v w y
def
bang bang
def
e e alt e e
Table HX an extension of H with exceptions
HX AN EXTENSION OF H WITH EXCEPTIONS
Given these parameters together with the p olymorphic M constants bang and alt
de ned in Table inductively de ne the denotational semantics of HX according
to the rules in Tables and denotational rule c Valjcj from Table is to
apply only when Mute c
Prop osition
The typ e assignment laws of Prop osition remain valid for HX except for the
statements ab out canonical terms in Table
In HX wehave the following prop erties of canonical terms
Bool
g c b a If c Bool then either c bang or b ftt
Int
b If c Int then either c bang or n N c n
c If c then either c bang or e c x e
d If c then either c bang or K e e c K e e
m m
The op erational semantics laws of Prop osition remain valid for HX
The denotational semantics laws of Prop osition remain valid for HX
Pro of Part follows by easy rule inductions similar to Prop osition Part
follows by insp ection of the typ e assignmen t rules Pro of of part is similar to that for
Prop osition follows by simple inductions as b efore
As in H we can prove a close corresp ondence b etween evaluation of an HX program and
its denotation
Prop osition
For any T program M either M orM bang or M Val N for some program
N
The op erational b ehaviour of the denotational constants ob eys the following rules
M M
Let v M in N alt MN
M Val L M Val L
L
Let v M in N N v alt MN Val L
M bang M bang
Let v M in N bang Let v M in N N
If e c then e c
If e V then there is c suchthat V c and e c
e c i e c
Pro of Parts and followby calculating from the de nitions of Val Let bang and
alt
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Part follows by induction on the depth of inference of e c as b efore Part follows
by induction on the depth of inference of e V We show one case to illustrate the pro of
Supp ose that e is e e and that e V where f g Wehave
Let v e in Let v e in Seq v b cv x Val x V
Since each e isaprogramoftyp e T Num by parts and either e m e
i i i i
Val bm c or e bang Since e converges to V by app eal to part we have only
i i
three cases to consider
a e bang and e bang V
b e Val bm c e bang and e bang V
c e Val bm c e Val bm c
and e Seq bm cb cbm c x Val x V
In each case we exhibit c such that e c and c V
a Take c to b e bang By IH wehave e bang and then e e bang
b Take c to b e bang By IH wehave e m and e bang so then e e bang
By IH we have e m and e m soe m m But c Take c to be m m
Seq bm cb cbm c x Val x m m soV m m as required
Finally part is a corollary of parts and
We now rework the theory of op erational equivalence We conjecture that denotational
equivalence could b e reworked similarly butleave this as future work
De nition We adopt all the de nitions to do with con ned relations and applica
tive similarity and bisimilarity from x and x for HX with the following amendment
b
In HX given a con ned relation R the con ned relation R is de ned bythe rules in the
rst half of Table together with the additional rules
e Re
i
i
b
b
R e e e e
bang R bang
Prop osition
All the prop erties of con ned and ground relations proved in x remain true
Applicative bisimilarity is an op erationally adequate precongruence
Pro of By reworking x and x We omit the details
As b efore we take op erational equivalence to be applicative bisimilarity We nish this
section on HX byreworking the laws of programming intro duced in x
Prop osition The laws of H programming in Table hold for HX given the
following mo di cations
The b eta law for call by value application has a new side condition that Mute c
There are new b eta laws for biased choice
bang e e c e c if Mute c
HX AN EXTENSION OF H WITH EXCEPTIONS
Each eta law has a new side condition that Mute c
There is a new eta law for biased choice
e bang e
There are additional strictness laws
e
LE bang bang
RE bang bang
Pro of By reworking the pro of of Prop osition
Weleave a study of denotational equivalence for HX as future work The following result
stated without pro of would b e useful
Prop osition The structure T Val Let intro duced in De nition is a compu
tational monad
This implies for instance that the example calculation shown earlier for H is valid for HX to o
OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
Chapter
Four mechanisms for teletyp e I O
In this chapter we consider four widely implemented mechanisms for I O in functional
languages side e ecting I O continuation passing I O synchronised stream I O and
Landin stream I O In the context of interaction with a teletyp e wesketch the semantics
of side e ecting I O and give detailed semantics for the other three mechanisms These
semantics are based on the theory of H and HX develop ed in the previous chapter Tele
typ e I O is a very limited mo del of I O in which the computer interacts with a keyb oard
and a character based printer By concentrating on a simple I O mo del the contrasts
between the four I O mechanisms can b e clearly seen
W e do not describ e related work on using functional languages with more complex I O
mo dels in any detail For the record the interested reader is referred to pap ers on the
following topics asynchronous interrupts p olling the keyb oard real time
behaviour interaction with a le system concurrency
controlling a window system writing an op erating system shell in a functional
language and even the whole op erating system
Surveys by Hudak and Sundaresh Jones and Sinclair and Perry cover much
of this previous work
In x wesketchhow the denotational semantics of H can b e extended to accommo date
side e ecting I O and discuss why side e ecting I O is unsuitable for lazy languages
We base the semantics of the remaining three I O mechanisms on the language HX de ned
in Chapter Continuation passing and synchronised stream I O can b e de ned in terms
e need the exception mechanism in HX We use of H but for Landin stream I O w
exceptions to represent demand for a value in the input stream
Weuseastyle of op erational semantics for the remaining three mechanisms that was rst
used by Holmstr om in his semantics of PFL Holmstr om used a continuation passing
style to embed CCS like op erations for communication and concurrency in a functional
language Starting with an evaluation relation for the host language he de ned the
meaning of the emb edded op erations in the style of a lab elled transition system as used
in CCS A lab elled transition system is a way to formalise the idea that an agent
such as a functional program engaged in I O can p erform an action such as input or
output of a character and then b ecome a successor agent This style of semantics is
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
attractive for at least three reasons First it can mo del a wide variety of nondeterministic
and concurrent computation witness the CCS scho ol of concurrency theory Second the
evaluation relation for the host language is unmo di ed any prop erty of the host language
without I O will still hold after the I O mechanism has b een added Third the metho d
complements an op erational language de nition such as that of Standard ML In x
we de ne notions of lab elled transition system and bisimilarity on programs of HX
We go further than Holmstr om bydeveloping an equational theory of functional I O based
on lab elled transitions We adopt bisimilarity from CCS as equivalence on programs
engaged in I O Bisimilarity is an equivalence on agents induced by their op erational
behaviour two agents are bisimilar i whenever one can p erform an action the other can
to o such that their two successors are bisimilar
Bisimilarity should not be confused with applicative bisimilarity Each is de ned as the
greatest xp oint of a certain functional but they relate di erent kinds of b ehaviour Ap
plicative bisimilarity as develop ed for M in Chapter and for H and HX in Chapter
relates the evaluation b ehaviour of two programs Bisimilarity as de ned in this chapter
relates the observable communication b ehaviour of two programs Bisimilaritywas intro
duced in Milner s theory of CCS whereas applicative bisimilarity comes from Abramsky s
theory of lazy calculus which itself followed CCS in part
Both Landin stream and synchronised stream I O are based on stream transformers func
tionsoftyp e inp out whereinp and out are typ es of input and output values re
sp ectively x de nes some general op erations on stream transformers x gives lab elled
transition system semantics for Landin stream I O and explains why HX rather than H
is needed x and x give lab elled transition system semantics for synchronised stream
and continuation passing I O resp ectively The main result of the chapter is prov ed in
x that there are bisimilarity preserving translations b etween the three mechanisms for
teletyp e I O other than side e ecting I O In this sense wehaveproved that the three are
of equal expressiveness The mappings between Landin stream and continuation passing
I O are original whilst those b etween synchronised stream and continuation passing I O
were discovered during the design of Haskell buthave not hitherto b een veri ed
formally x concludes the chapter with a discussion of the practical use of the three
mechanisms suitable for lazy languages
Side e ecting I O
The original functional language LISP had a side e ecting I O mechanism
Side e ecting I O which is by far the most widely used mechanism for functional I O
p ersists in LISP and is found in other eager languages suchasScheme and Standard ML
o accommo date side e ecting I O in H we add two new non canonical op erations to the T
syntax of expressions
e read read a character
j write e write a character
SIDE EFFECTING I O
with typing rules and informal intended meanings as follows
e Int
read Int
write e
read means
input a character n from the keyb oard and return n
means write n
output character n to the printer and return unit
For example here is a program to read two characters and output their sum
main
main write read read
Pardon the cavalier identi cation of Int and Char from x
The denotational semantics of H can be extended to accommo date side e ecting I O
Apart from diverging there are three things a computation can do return an answer ask
for a numb er as input or output a number Accordingly a computation of H typ e can b e
mo delled as a term of M typ e T with the following de nition of T and corresp onding
de nitions of Val M read and write e
def
Num X X ftv T X Num X
def
Val M Lift Intro In M
def
read Lift Intro In v Num Val v
def
write e Let v e in Lift Intro In hv i
This construction is based on Moggi s notes Exercise but the basic idea
go es back at least as far as Plotkin s Pisa notes Example in Chapter This is
just a sketch of the semantics we omit the de nition of Let An analogous ob ject level
construction is detailed in Chapter Were M to be used in a detailed study of side
e ecting I O it would need to be extended to include mutually recursive M typ es such
as
We do not pursue this construction here because the primary fo cus of this dissertation
as is well known side e ecting I O do es not is the study of I O for lazy languages and
combine well with lazy languages There are three reasons why the use of side e ecting
I O in a lazy language is problematic First to use side e ecting I O in a lazy language a
programmer must b e concerned with evaluation order which b ecause of laziness is harder
to predict than in an eager language Side e ecting I O comp els programmers to think
ab out something that otherwise they can usually leave to the implementation
The other two reasons dep end on the observation that Strachey s prop erty no longer holds
Recall Strachey s prop erty from Chapter that every program either equals or some
canonical program Strachey s prop erty holds for H and HX but not for H extended
with side e ecting op erators Programs read and write n neither diverge nor equal any
canonical program
The second reason then that side e ecting I O do es not combine well with lazy languages
is that adding side e ects makes a functional language harder to reason ab out Pro ofs of
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
programs are harder to construct b ecause there are more cases to consider than covered
by Strachey s prop erty This applies to eager languages as well With side e ecting I O
one would exp ect that every program either diverges or converges to a canonical program
or do es some I O to b ecome another program Howtoformulate precisely and verify this
exp ectation is an op en question
The third reason concerns e ciency Strachey s prop erty is imp ortant if languages with
call by name semantics for function application are to be implemented e ciently Such
languages are only practical if call by need semantics coincides with call by name If
so call by name applications can be implemented by call by need as in graph reduction
wo semantics for an application for instance Recall the di erence b etween the t
x e e In b oth regimes the function b o dy e is evaluated Under call by name seman
tics each time need arises for the value of x argument e is evaluated Under call by need
semantics the rst time need arises for the value of x argument e is evaluated but its
value is retained and re used if subsequently the value of x is needed
We give an informal pro of that if evaluation is deterministic and op erational equivalence
satis es op erational adequacy and Strachey s prop erty then call by name semantics co
incides with call by need Consider an application x e e If the value of e do es not
t since the argument e is never evalu dep end on x then clearly b oth regimes are equivalen
ated Otherwise supp ose that the value of x is demanded one or more times By Strachey s
prop erty the argument e either equals or some canonical form so by op erational ade
quacy e either diverges or converges to a unique canonical form c If e diverges then so
do es the evaluation of e under either regime Otherwise e deterministically converges to
c so it makes no di erence whether the second and subsequent uses of the argument use
a retained copyofc or evaluate e again
If evaluation of argument e causes side e ects then call by need may not b e equivalentto
call by name and hence an imp ortant implementation technique is invalid Witness the
examples x x x read or x x x write A where we de ne e e to
mean y e e with y fv e Under call by name execution of the two examples
involves two inputs and two outputs resp ectively under call by need execution of the
programs involves just one input and one output resp ectively
In summary side e ecting I O and lazy languages do not combine well b ecause of the dif
culty of predicting the order of side e ects of constructing pro ofs of program prop erties
and b ecause call by need semantics ceases to b e a correct implementation technique This
explains why in practice side e ecting I O is only used with call by value languages such
as LISP or ML
hey s prop ertyby no means rules out pro ofs of program prop erties The absence of Strac
but weleave further investigation of H with side e ecting I O as future work We conjec
ture that the b eta eta strictness precongruence and canonical freeness laws of Table
remain valid The only work to present a theory of I O in a call by value language is the
algebra for FL programs develop ed by Williams and Wimmers FL uses what is es
sentially side e ecting I O One might disp ense with call by name semantics and develop
an equational theory for call by need semantics and hence accommo date side e ecting
I O Such a theory would have to make a distinction b etween the rst and subsequent us
ages of b ound variables which on the face of it is much more complicated than the theory
LABELLED TRANSITION SYSTEMS AND BISIMILARITY
of call by name and call by value applications develop ed in Chapter Previous work on
the semantics of call by need in deterministic functional languages has fo cused on prov
ing that an op erational semantics using call by need correctly implements a denotational
semantics sp ecifying call by name
In his unpublished dissertation Redelmeier sketched an I O scheme that used side
e ecting op erators together with data dep endencies to ensure a predictable order of side
e ects His idea was that each side e ecting op erator to ok an additional state parameter
and returned a new state He argued that if each state value has a unique successor then
there is a predictable thread of side e ects His scheme was unsatisfactory b ecause he
did not suggest how to guarantee single threading Hudak and Sundaresh describ e
a variant known as systems I O Peyton Jones and Wadler have implemented
in the Glasgow Haskell compiler using side e ecting op erators with the I O mechanism
an e cient translation to C Their work is a clear advance on Redelmeier s They have
ingeniously used an abstract data typ e based on a monad to guarantee single
threading Their scheme app ears to b e a promising technique for e cient implementation
of I O in lazy languages
Lab elled transition systems and bisimilarity
Weadoptlab elled transition systems and bisimilarity from the theory of CCS to
give semantics for Landin stream continuation passing and synchronised stream I O
In eachmechanism for functional I O there is a single ob ject language typ e whose programs
can be executed to interact with the teletyp e We call this typ e the execution typ e
A program is executable i it is of this typ e For instance executable programs using
Landin stream I O are of the stream transformer typ e Char Char We formalise
the execution of programs as a lab elled transition system
De nition The set of actions ranged over by is pro duced by the following
grammar
n input character n N
n output character n N j
A lab elled transition system is a family of binary relations indexed by actions f g
suchthatifp q then p and q are HX programs of an execution typ e
WeuseHX rather than H for reasons explained in x The intuitive meaning of transition
n
p q is that program p can input the character n from the keyb oard to b ecome program
n
q Similarly theintuitive meaning of transition p q is that program p can output the
character n to the printer to b ecome program q If p q for some and q wesay that
p has a transition
We de ne bisimilarity and prove the standard results
De nition De ne function h i to be the function over binary relations on HX pro
grams such that phS iq i
whenever p p there is q with q q and p S q
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
whenever q q there is p with p p and p S q
A bisimulation is a binary relation on agents S suchthat S hSi Bisimilarity
is the union of all bisimulations
Prop osition
Function h i is monotone
The identity relation is a bisimulation
If each S is a bisimulation then so is S S
i
If S is a bisimulation then so is S
Bisimilarity is the greatest xp ointof h i and is the greatest bisimulation
h that pS q p q i there is a bisimulation S suc
Bisimilarityisanequivalence relation
Pro of Parts follow easily from the de nition The remaining parts then follow
from Prop osition
Stream transformers
Manymechanisms for functional I O or for concurrency have b een based on stream
transformers A stream is a list typ e whose cons op eration is lazy such as in H
Stream transformers in H have the general typ e
type ST inp out inp out
The idea is simple a stream transformer maps a stream of values of typ e inp into a
stream of values of typ e out This mapping represents an interactive computing device
that consumes values of typ e inp and pro duces values of typ e out Intuition if the device
has b een o ered the sequence of values in in for consumption applying the stream
n
transformer to the stream in in yields a stream containing the sequence
n
of values the device can pro duce The list cons op eration has to have lazy semantics
so that the partial list in in do es not simply equal which explains why
n
stream based I O is not typically used with languages like ML or Scheme where cons is
eager Implementations of stream based I O typically represent the unde ned value
at the end of a partial list as a memory cell that can b e instantiated to hold the next input
tuitively correct character and to p oint to a fresh unde ned value Such a technique is in
but we leave op en the question of how to verify formally that it correctly implements
the semantics to be given here None of the work on veri cation of functional language
implementation has considered I O
Stream transformers for stream based I O havetypically b een written using explicit con
struction of the output list and explicit examination of the input list Such a
programming style can be hard to read We can avoid explicit mention of input and
output lists by using the following combinators to construct stream transformers
getST inp ST inp out ST inp out
STREAM TRANSFORMERS
putST out ST inp out ST inp out
nilST ST inp out
getST k xs case xs of x xs k x xs
putST x f xs x f xs
nilST xs
Thompson and Cupitt suggest other combinators for I O which are related to
the monadic style discussed in Chapter
A programmer would use the combinators ab ove to construct stream transformers to
give semantics to stream based I O we use combinators giveST nextST and skipST The
intention is that giveST feeds an input value to a stream transformer nextST tests whether
a stream transformer can pro duce an output value without any further input and skipST
consumes an output value from a stream transformer
data Maybe a Nothing Just a
giveST inp ST inp out ST inp out
nextST ST inp out Maybe out
skipST ST inp out ST inp out
giveST c f xs f c xs
nextST f case f of
Nothing
x xs Just x
skipST f xs tail f xs
The Haskell committee discovered the technique of using a mo ck argument to test
whether a stream transformer is ready to pro duce output Of course if the next output
alue in its input stream nextST f will from a stream transformer f dep ends on the next v
lo op
The following prop osition relates the six combinators intro duced in this section
Prop osition For all programs u inp v out k inp ST inp out
and f ST inp out
giveST u getST k ku
nextST putST v f Just v
nextST nilST Nothing
skipST putST v f f
Pro of Straightforward calculations
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
Landin stream I O
Landin suggested that streams would b e used to mo del input output if ALGOL
included such The simplest kind of stream transformer used for I O is one that maps a
stream of input characters to a stream of output characters We call such a mechanism
Landin stream I O Executable programs are stream transformers of typ e LS
type LS ST Char Char
A second kind of stream based I O synchronised stream I O is discussed in x We use
the combinators of x to sp ecify an intended meaning for Landin streams
getST k means
input a character n from the keyb oard and then execute k n
putST n f means
output character n to the printer and then execute f
nilST means
terminate immediately
We wish to give a semantics for each I O mechanism in terms of the op erational semantics
of H Given a function f LS we are to compute whether f can output a character with
no further input or whether f needs an input character b efore pro ducing more output or
whether f can terminate More precisely we need a function ready of the following typ e
data RWD out R W out D
ready ST inp out RWD out
and satisfying the equations
f W n ready putST n
ready getST k R
ready nilST D
WeshowthatinH there is no such program Consider programs e and e of typ e LS
e getST x putST nilST
e putST nilST
It is not hard to see that for any xs the following equations hold
if x xs xs x xs
e xs
otherwise
e xs
and hence that e v e and e v e by Strachey s prop erty To see why there can be
ve we assume there is and derive no function ready that ob eys the equations shown ab o
a contradiction We have ready e R and ready e W and R v W But
e v e so by precongruence for v Prop osition we have ready e v ready e
Contradiction
John Hughes showed me this argument in
LANDIN STREAM I O
Intuitively the problem is that in H there is no way to tell whether an expression dep ends
on the value of one of its sub expressions suchasanelement of the input stream Wecan
remedy this by adding an exception mechanism to H as we didinthe previous chapter
We nd that the ab ove argument do es not hold in HX In HX we have that e and e
are incomparable b ecause e bang bang e bang and bang and are
incomparable
Roughly sp eaking to tell whether an expression dep ends on the value of one of its sub ex
pressions replace the sub expression with bang and use the handler op erator to see if
ws the whole expression bangs We can de ne ready in HX as follo
ready f
case f bang of
D
x W x
R
and from the theory of HX from the previous chapter it follows that the conditions ab oveon
ready are satis ed Intuitively sp eaking the exception bang in HX provides a computable
test of whether a function needs the value of its argument
The semantics of Landin streams can be given for LS programs in HX as the lab elled
transition system de ned by the following tworules
ready f Wv v n
ready f R
n
n
f giveST n f
f skipST f
The following lemma shows that this formal semantics correctly re ects the informal in
tended meanings given for Landin stream programs apart from termination which we
have not formalised
Lemma
ready getST k R
ready putST v k Wv
n
getST k k n
n
putST v p p if v n
Pro of Parts and follow from the de nitions of ready getST and putST For parts
and we can calculate the following transitions
n
getST k giveST n getST k
n
putST v p skipST putST v p
These together with Prop osition establish the required results
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
Synchronised stream I O
In the synchronised stream mechanism for functional I O the stream transformer pro
duces a stream of requests and consumes a stream of acknowledgements The requests and
acknowledgements are in one to one corresp ondence the computing device sp eci ed by
a stream transformer alternates between pro ducing an output request and consuming an
input acknowledgement It is the programmer s burden to ensure that the value of eachre
quest do es not dep end on the corresp onding acknowledgement Synchronised streams were
rst rep orted as the underlying implementation technique for Karlsson s Nebula op erating
system They were indep endently discovered by Stoye whose terminology we
adopt and O Donnell They are the underlying mechanism of the Kent Applicative
Op erating System KAOS and of I O in Haskell
with Here is the typ e SS of executable programs in the setting of teletyp e I O together
intended meanings of some example programs
type SS ST Ack Req
data Req Get Put Char
data Ack Got Char Did
putST Get getST k means
input a character n from the keyb oard and then execute k Got n
getST k means putST Put n
output character n to the printer and then execute kDid
nilST means terminate immediately
A wide range of imp erative activity can b e expressed using this mechanism as illustrated
by the Haskell I O mechanism We de ne an auxiliary function for use in examining the
acknowledgement obtained from a Get request
outGot Ack Char
outGot Got x x
The semantics of synchronised streams can b e given for SS programs in H or in HX as the
lab elled transition system de ned by the following tworules
nextST f Just r r Put v v n
nextST f Just r r Get
n
n
f giveST Got n skipST f
f giveST Did skipST f
Unlike Landin streams there is no need for the problematic ready op eration b ecause of the
synchronisation b etween input and output Just as for Landin streams we state a lemma
to show that this formal semantics correctly re ects the informal intended meanings given
for synchronised stream programs apart from termination
Lemma Supp ose k Char SS and h SS are programs de ne programs f and g to
be
def
f putST Get getST ack k outGot ack
def
g putST Put v getST ack h
CONTINUATION PASSING I O
Then wehave
nextST f Just Get
nextST g Just Put v
n
f k n
n
g h if v n
Pro of Parts and follow from the de nitions of nextST putST and getST For parts
and we can calculate the following transitions and equations using Prop osition
n
f giveST Got n skipST f
getST ack k outGot ack giveST Got n
k outGot Got n
k n
n
g giveST Did skipST g
giveST Did getST ack h
h
Continuation passing I O
In continuation passing I O the executable typ e is an algebraic typ e with a constructor
corresp onding to each kind of expressible imp erative activity In the case of teletyp e I O
wehave
data CPS INPUT Char CPS
OUTPUT Char CPS
DONE
Holmstr om s PFL was the rst functional language to take the continuation passing
mechanism as primitive In earlier work Karlsson programmed continuation passing op
erations on top of a synchronised stream mechanism A similar datatyp e was used by
Plotkin in the Pisa notes as semantics for side e ecting I O as discussed in x The
mechanism is called continuation passing b ecause of the similaritybetween the argument
to INPUT and continuations as used in denotational semantics
The intended meaning of CPS programs is easily given
INPUT k means
input a character n from the keyb oard and then execute k n
p means OUTPUT n
output character n to the printer and then execute p
DONE means
terminate immediately
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
Landin stream I O
ready f R ready f Wv v n
n n
f giveST n f f skipST f
Synchronised stream I O
nextST f Just r r Put v v n nextST f Just r r Get
n n
f giveST Got n skipST f f giveST Did skipST f
Continuation passing I O
p OUTPUT v q v n p INPUT k
n n
p k n p q
Table Three mechanisms for functional I O
These intended meanings are re ected in the following two rules which de ne a lab elled
transition system for CPS programs in either H or HX
p OUTPUT v q v n
p INPUT k
n
n
p k n
p q
Maps between three of the mechanisms
Wegave detailed semantics for each of the mechanisms apart from side e ecting I O There
are two main results in this section First we show that if two executable programs are
op erationally equivalent then they are bisimilar The force of this result is that the theory
of op erational equivalence from Chapter can b e used to prove prop erties of the execution
behaviour of executable programs Second we show that each of the three mechanisms
has equivalent expressivepower in the following sense If p is an executable program with
resp ect to one mechanism then for each other mechanism there is a function f suchthat
p are f p is an executable program with resp ect to the other mechanism and p and f
bisimilar
We work in HX so that all three mechanisms are supp orted in the same language as
discussed in x the semantics of Landin streams based on the ready function cannot b e
programming in H
De nition The teletyp e transition system is a family of binary relations on HX
programs indexed by actions f g and is de ned by the rules in Table
Lemma If p and p q then q Typ e is one of LS SS or CPS dep ending
on whether transition p q was derived from one of the Landin stream synchronised
stream or continuation passing rules resp ectively inTable
Pro of By insp ection
Given its simple sequential nature one would exp ect the semantics of teletyp e I O to b e
MAPS BETWEEN THREE OF THE MECHANISMS
determinate The following result makes this precise
Prop osition For any program p p p and p p implies p p
Pro of By insp ection of each of the inference rules
Given this determinacy bisimilarity can alternatively b e characterised in terms of traces
If s is a nite sequence of actions say that s is a trace of program p i there
n
n
p Two programs are trace equivalent i are programs p with p p
n i
they have the same set of traces
In a nondeterministic calculus like CCS trace equivalence do es not in general imply bisim
ilarity Given the determinacy result ab ove however it is not hard to show that the two
equivalences coincide We omit the pro of but see Milner s b o ok for a more general result
Chapter
Op erational equivalence applicative bisimilarity implies bisimilarity
Prop osition If p and q are programs of the same typ e then p q implies p q
Pro of Recall that wetake on terms of HX to b e applicative bisimilarity as de ned in
Chapter It su ces to show that ground applicative bisimilarityon HX is a bisimulation
Supp ose that p q for programs p and q We pro ceed by a case analysis of If
is not one of the three typ es CPS SS or LS then p and q have no transitions so p q
trivially Otherwise there are three cases to consider In each case we establish condition
of the de nition of bisimulation that whenever p p there is q suchthat q q
and p q Condition follows by a symmetric argument
Case CPS
n
Then q INPUT k with k k from If p p then p INPUT k and p k n
n
applicative bisimilarity So q k n p
n
If p p then p OUTPUT v p and v n Then q OUTPUT v q withv v and
n
p q So v n and wehave q q p
Case SS
n
If p p then nextST p Just r r Get and p giveST Got n skipST p
n
Since p q we have nextST q Just r with r r so r Get We have q
skipST q and the latter equals p since op erational equivalence is a giveST Got n
congruence
n
If p p then nextST p Just r r Put v v n and p giveST Did skipST p
Again since p q wehave nextST q Just r with r r sor Put v with v v
n
Therefore q giveST Did skipST q which equals p so v n
The case for LS follows by a similar argument and is omitted
On the other hand bisimilarity do es not imply op erational equivalence of H programs
Prop osition There are program pairs p and q in each of the typ es CPS LS and
SS suchthat p q but not p q
Pro of Witness program pair OUTPUT Done and in typ e CPS and pair putST nilST
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
and in eachofthetyp es SS and LS
Intuitively the pro of dep ends on op erational equivalence distinguishing more junk
programs than bisimilarity Given a richer I O mo del there would be more signi
cant distinctions Supp ose we extended the CPS algebraic typ e with a new constructor
Par CPS CPS CPS withintended meaning that Par p q is to b e the parallel execution
of programs p and q asinPFL Thenifp q programs Par p q and Par q p would b e op
erationally inequivalent b ecause Par is the constructor of an algebraic typ e but bisimilar
b ecause as in CCS b oth lead to the parallel execution of p and q
Before proving the second main result of this section we imp ort the pro of technique of
bisimulation up to from CCS
De nition A binary relation on programs S is a bisimulation up to i pS q
implies
whenever p p there is q with q q and p S q
whenever q q there is p with p p and p S q
Prop osition If S is a bisimulation up to then S
Pro of Wefollow the pro of on page of Milner s b o ok
Weprove that S is a bisimulation from which it follows by re exivityof that S
S is the relational comp osition of S and Supp ose that p S q that is for
some p and q wehave p p S q q Supp ose that p p we are to exhibit q such
that q q and p S q We make a sequence of deductions there is p such that
p p with p p there is q such that q q and p S q and there is q such
that q q and q q Altogether we have p p S q q hence p S q by
transitivityof This shows that S is a bisimulation
Supp ose there are programs p and q suchthatwe wish to prove p q Prop osition
says that one pro of would be to nd a bisimulation containing the pair p q On the
other hand Prop osition says that another pro of is to nd a bisimulation up to that
contains the pair It is often simpler to do the latter The same idea is used extensively
in the theory of CCS and calculus
We show in Table functions ss cps and cps ss to map between the typ es SS and
CPS and functions ls cps and cps ls to map b etween the typ es LS and CPS Functions
ss cps and cps ss are similar to translations discovered by the Haskell committee
The following is the main theorem of the chapter that the three I O mechanisms of
Table are isomorphic in a certain sense
Theorem
For any SS program f f ss cps f
For any CPS program p p cps ss p
For any LS program f f ls cps f
For any CPS program p p cps ls p
MAPS BETWEEN THREE OF THE MECHANISMS
Translations between SS and CPS
ss cps SS CPS
cps ss CPS SS
ss cps f
case nextST f of
Nothing DONE
Just r case r of
Get INPUT v ss cps giveST Got v skipST f
Put v OUTPUT v ss cps giveST Did skipST f
cps ss p
case p of
INPUT k putST Get getST ack cps ss k outGot ack
OUTPUT v q putST Put v getST ack cps ss q
DONE nilST
outGot Ack Char
outGot Got v v
Translations between LS and CPS
ls cps LS CPS
cps ls CPS LS
ls cps f
case ready f of
R INPUT v ls cps giveST v f
W v OUTPUT v ls cps skipST f
D DONE
cps ls p
case p of
INPUT k getST v cps ls k v
OUTPUT v q putST v cps ls q
DONE nilST
Table Translations in HX between three styles of I O
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
Pro of Weprove that S given by
def
S f f ss cps f j f is an SS programg
is a bisimulation which is to say that S hS i Let f be any SS program and we have
that ss cps f CPS Hence the synchronised stream rules apply to f and the continuation
passing rules to ss cps f We are to show that f ss cps f hS i We pro ceed by
analysis of the evaluation b ehaviour of nextST f There are ve cases to consider
i nextST f or nextST f bang
ii nextST f Nothing
iii nextST f Just r and either r or r bang
iv nextST f Just r and r Get
v nextST f Just r and r Put v
Here are the p ossible transitions from programs f and ss cps f
i iii There are no transitions from either ss cps f or f
iv The only transitions of f are of the form
n
f giveST Got n skipST f forany n
Wehave ss cps f INPUT c ss cps giveST Got c skipST f
So the only transitions of ss cps f are of the form
n
ss cps f ss cps giveST Got n skipST f for any n
n
when f giveST Did skipST f v There is no transition from f unless v n
Wehave ss cps f OUTPUT v ss cps giveST Did skipST f
So there is no transition from ss cps f unless v n
n
when ss cps f ss cps giveST Did skipST f
In each case the conditions for f ss cps f hSi are satis ed
It su ces to show that S given by
def
S f p cps ss p j p is a CPS program g
is a bisimulation up to Supp ose that pair p cps ss p is in S We have p CPS and
cps ss p SS so the continuation passing rules apply to p and the synchronised stream
p then cps ss p q and rules to cps ss p We are to show that whenever p
p S q and vice versa We pro ceed by analysis of the evaluation b ehaviour of p There
are ve cases to consider
i p
ii p bang
iii p DONE
iv p INPUT k
v p OUTPUT v q
Here are the p ossible transitions from programs p and cps ss p
i iii Neither program has any transitions
MAPS BETWEEN THREE OF THE MECHANISMS
n
iv The only transitions of p are of the form p k n for any n
We have cps ss p putST Get getST ack cps ss k outGot ack So
using Lemma the only transitions of cps ss p are of the form
n
cps ss p cps ss k n for any n
n
q when v n v The only transitions of p are p
Wehave cps ss p putST Put v getST ack cps ss q
n
So using Lemma the only transition from cps ss p is cps ss p cps ss q
In each case the conditions for bisimulation up to are satis ed since
Weprove that S given by
def
S f f ls cps f j f is an LS programg
is a bisimulation that is S hS i Supp ose that pair f ls cps f ininS Wehave f LS
and ls cps f CPS so the Landin stream rules apply to f and the continuation passing
rules to ls cps f We are to show that f ls cps f hSi We pro ceed by analysis of
There are four cases to consider the evaluation b ehaviour of ready f
i ready f
ii ready f D
iii ready f R
iv ready f Wv
A fth p ossibility that ready f bang is imp ossible given the form of ready Here are
the p ossible transitions from programs f and ls cps f
i ii No transitions are p ossible from either f or ls cps f
n
f for any n iii The only transitions of f are of the form f giveST n
Wehave ls cps f INPUT c ls cps giveST c f
So the only transitions of ls cps f are of the form
n
ls cps f ls cps giveST n f forany n
n
when f skipST f iv There is no transition from f unless v n
Wehave ls cps f OUTPUT v ls cps skipST f
So there is no transition from ls cps f unless v n
n
ls cps skipST f when ls cps f
In each case the conditions for f ls cps f hSi are satis ed
It su ces to show that S given by
def
S f p cps ls p j p is a CPS program g
is a bisimulation up to Supp ose that pair p cps ls p is in S We have p CPS and
cps ls p LS so the continuation passing rules apply to p and the Landin stream rules to
cps ls p We are to show that whenever p p then cps ls p q and p S q
and vice versa We pro ceed by analysis of the evaluation b ehaviour of p There are ve
cases to consider
i p
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
ii p bang
iii p DONE
iv p INPUT k
v p WRITE v q
Here are the p ossible transitions from programs p and cps ls p
i iii Neither of the programs has any transitions
n
iv The only transitions of p are of the form p k n for any n
Wehave cps ls p getST c cps ls kc
So using Lemma the only transitions of cps ls p are of the form
n
cps ls p cps ls k n for any n
n
v The only transition of p is If p q if v n
Wehave cps ls p putST v cps ls q
So using Lemma the only transition of cps ls p has the form
n
cps ls p cps ls q if v n
In each case the conditions for bisimulation up to are satis ed
Discussion
The main contribution of this chapter is to the semantics of functional I O Wesketched
a semantics of side e ecting I O but gave three ob jections to its use with lazy languages
We considered three mechanisms suitable for lazy languages and gave an op erational
semantics for each We showed how the notion of bisimilarity from CCS is a suitable
equivalence on programs engaged in I O We gave translations b etween the three styles
some of which are well known but gave the rst formal pro ofs that the translations are
correct
Our de nition of bisimilarityisvery simple but for two reasons one might wish to develop
although each of the three I O mechanisms has a notion of program it further First
termination wehave not mo delled termination in the lab elled transition system Hence a
program that immediately terminates is bisimilar to one that diverges Second we have
assumed that teletyp e input is observable Consider two Landin stream programs f and
g
f xs
g xs case xs of
xs g xs
H
Given an input stream g unravels it forever whereas f lo ops immediately Wehave f g
HX
but f g and f g b ecause g forever inputs characters whereas f diverges One
might argue that they have indistinguishable behaviour b ecause neither ever pro duces
output On the other hand it seems reasonable to distinguish them on the ground that
teletyp e input is observable to the op erating system if not always to the end user These
DISCUSSION
points bring out the simplicity of our mo del but then it is meant to be simple so as to
emphasise the di erences b etween the I O mechanisms
We conclude the chapter with some remarks ab out the practical use of the three mech
anisms for I O in lazy languages As they stand the three mechanisms are rather to o
low level for large scale programming As we discussed in Chapter various sets of com
binators have b een put forward as high level programming mo dels Wewill
investigate a set of combinators in the monadic style in Chapter
Landin stream I O is go o d for teletyp e I O but do es not scale to a practical I O scheme
construct elegant parsers that such as Haskell I O Several pap ers have shown how to
act on a lazy input stream of characters It is not clear that such parsing
techniques can be based so simply on synchronised stream or continuation passing I O
Landin stream programs can be written in either Haskell or Hop e C This is
b ecause the I O mechanisms of both provide op erations to obtain a lazy input stream
and hence Landin stream I O can b e simulated
A gulf separates the source level semantics of the stream based I O mechanisms given
here from their imp erative implementation The gulf is particularly wide b etween
of lazy input streams The only other semantics of the semantics and implementation
stream based I O is Thompson s trace based work which is domain theoretic and also
distant from practical implementations In contrast the op erational semantics we gave
for continuation passing I O corresp onds fairly closely to an interpretive implementation
It is an op en question how to relate abstract sp eci cations of I O to e cient
implementations using side e ects
The principal merit of synchronised stream I O over continuation passing I O is that the
former can e ciently simulate the latter via a function such as cps ss Simulation of
the former by the latter using a function such as ss cps is inevitably ine cient
Synchronised stream programs su er from problems relating to synchronisation b etween
input and output streams that do not arise with continuation passing High level
combinators such as for monadic I O can be implemented on top of either mechanism
If all user programs are to use suchcombinators there seems to b e no reason to
cho ose synchronised stream I O as the underlying mechanism rather than continuation passing I O
CHAPTER FOUR MECHANISMS FOR TELETYPE I O
Chapter
Monadic I O
The three I O mechanisms of Chapter suitable for lazy languages express I O at a low
level This chapter explains how to program I O using higher level combinators in a style
which we call here monadic I O The combinators are higher level in two senses the
intended meaning of an executable program is a computation that p erforms I O and then
returns a value rather then just p erforming I O as in Chapter the combinators of
monadic I O can b e implemented using the lower level op erations given in Chapter
This chapter discusses monadic I O in the context of a sp eci c application of functional
programming Over the last decade the Medical Research Council MRC at the Western
General Hospital in Edinburgh has built a prototyp e machine to assist the analysis of
cervical smears The prototyp e w as implemented using dedicated hardware C and
assembly co de A team has started work on re implementing the machine using a commer
cial workstation coupled with software partly written in a lazy functional language
The aim is to compare the gains of using a functional language such as reliability easeof
programming and the p otential for pro ofs of program prop erties with the losses suchas
computational sp eed The system captures digitised images from a microscop e that sights
amovable stage holding a microscop e slide The intention is that the system will b e able
to lo cate automatically the ob jects on the slide that are most likely to be pre cancerous
cells and then display these to a cytologist for diagnosis If successful the computerised
system will allow a cytologist to pro cess slides faster than by simply using a microscop e
The MRC team have been prototyping the software for their system using a mixture of
C and a lazy functional language the prototyp e uses Gofer and the nal implementation
will use Haskell Basic image pro cessing will b e done byaserver running mature C co de
while the top level main program will b e new software written in the functional language
ts of the MRC computer This chapter addresses two questions How can the I O requiremen
b e programmed in a lazy functional language How can prop erties of programs engaged
in I O b e proved In the context of a simpli ed mo del of the MRC apparatus we suggest
howtheMRC computer could b e programmed in a monadic style We consider a simple
I O transformation suggested by Ian Poole of the MRC and prove it correct We show
how the monadic style applies to teletyp e I O because the simpli ed I O mo del for the
MRC computer includes teletyp e I O
x is a tutorial on the monadic style of functional I O It de nes a simpli ed mo del of
CHAPTER MONADIC I O
MRC I O and p oses a veri cation problem x and x de ne monadic typ es in H to
mo del teletyp e I O and the MRC I O mo del resp ectively the latter typ e is de ned in
terms of the former Finally inx weverify the motivating example
All the programs given in this chapter are written in Haskell notation They are to be
understo o d as H not HX programs as discussed in x All the programs have b een
typ e checked and executed in Mark Jones Gofer system Program equivalence written
is op erational equivalence as de ned in Chapter
Programming I O in a monadic style
In the monadic style of functional I O there is a unary typ e constructor IO suchthat a
program closed term of typ e IO a is intended to mean
a computation which may p erform I O and may return an answer of typ e
a
Let a computation be a program of typ e IO a for some a To program teletyp e I O
as discussed in Chapter in a monadic style one might use the following op erations for
constructing computations
input IO Char
output Char IO
return a IO a
IO a a IO b IO b
Here is the intended meaning of computations constructed from these op erations
input means input a character v from the keyb oard and return v
output v means output character v to the printer and return
return v means immediately return v
p f means execute computation p call the answer x and then execute compu
tation fx
An example reading a line from the keyb oard
Let us consider the problem of writing a program to read from the keyb oard a string that
is a list of characters terminated by the newline character n We wish to derive a
function gettingLine with the following prop erties
gettingLine String IO String
gettingLine vs means having already read the string vs continue reading char
acters un til a newline app ears
We can de ne gettingLine in terms of the basic computation constructors
gettingLine vs
input v
PROGRAMMING I O IN A MONADIC STYLE
if v n then return vs else gettingLine v vs
We can read o an interpretation of this de nition from the intended meaning of the basic
constructors computation gettingLine vs means input a character call it v then if v
is a newline return the accumulated string vs or else add v to the accumulated string
and rep eat Here then is a solution to the problem
getLine IO String
getLine gettingLine return reverse
Function reverse a a is the list reversal function from the Haskell prelude The
string returned by gettingLine needs to b e reversed b ecause the characters are accu
mulated in reverse order to avoid rep eated list concatenations
Derived op erations
Sometimes two computations are to be comp osed in sequence but the answer from the
rst is to b e discarded This pattern can b e captured by the combinator
IO a IO b IO b
p q p const
where const is the Haskell version of the K combinator shown in Table Commonly
the rst argumentto is a task a program of typ e IO A task is a computation that
returns no informativeanswer up on termination The simplest task is skip given by
skip return
Sequential comp osition of a list of tasks is achieved bythesequence combinator
sequence IO IO
sequence foldr skip
The Haskell list combinator foldr is also shown in Table It is not hard to verify the
following equations
sequence skip
sequence p ps p sequence ps
Here is a combinator that lifts a curried function f of two arguments to one that given
two computations runs them in sequence to obtain twoanswers x and y and then returns
the answer fxy
lift a b c IO a IO b IO c
lift f p q p x q y return f x y
write a generalisation of sequence that runs a list of computations We can use lift to
in sequence and then returns the list of their answers as its own answer
accumulate IO a IO a
accumulate foldr lift return
CHAPTER MONADIC I O
More examples printing a line and prompting
To illustrate how sequence can b e used here is a program to output a string of characters
followed by newline to the printer
putLine String IO
putLine vs sequence output v v vs output n
This is an example of a Haskell list comprehension which can be translated into H as
discussed in x In general we call functions such as putLine that have some typ e
a a IO b for m parametric computations
m
Finally here is a parametric computation to print a prompt and then read a line of input
String IO String askFor
askFor xs putLine xs getLine
Adding exceptions to the programming mo del
Exp erience of programming I O in imp erative languages suggests that an exception han
dling mechanism is a go o d way to represent and pro cess unexp ected events suchaserrors
For instance the implementors of DEC SRC s Taos op erating system argue that repre
senting op erating system errors by exceptions is b etter than returning an error co de as
in AT T s Unix for instance b ecause it is imp ossible for an exception to b e ignored
Exceptions can b e incorp orated into the monadic programming mo del bycho osing a typ e
Exn of exceptions and extending the intended meaning of a program of typ e IO a to b e
a computation which may p erform I O actions and may return an answer
of typ e a ormay fail with an exception value of typ e Exn
For the purp ose of this chapter we take Exn to be String but in general it can be any
typ e There are two new primitiveways of constructing computations
raise Exn IO a
try IO a Exn IO a IO a
raise exn means immediately raise the exception exn
try p f means execute computation p if an answer v is returned then return v
and ignore f otherwise if p raises an exception exn executef exn
The intended meanings of the other primitive computation constructors are unchanged
except for sequen tial comp osition
p f means execute computation p if an exception exn is raised then immedi
ately raise exn and ignore f otherwise if an answer x is returned pro ceed to execute
computation fx
The exception mechanism in HX was obtained by extending the syntax and semantics of
H Here instead we obtain an exception mechanism byobject level programming with
no extension of H
PROGRAMMING I O IN A MONADIC STYLE
Biased choice is a derived form of exception handling where the exception value is
ignored
p q means execute computation p ifananswer x is returned then return x and
ignore q otherwise if p raises an exception execute q
IO a IO a IO a
p q try p const q
An example using exceptions
Here is a parametric computation that if given a numeral a list of digits returns the
corresp onding numb er Otherwise it raises the exception parseInt
parseInt String IO Int
parseInt cs
toDigit cs accumulate map
return foldr reverse
where
Int Int Int
toDigit Char IO Int
u t t u
toDigit v
if isDigit v then return ord v ord
else raise parseInt
This function can b e used to construct a computation that returns the number found by
reading a line from the keyb oard and treating it as a numeral and rep eating if necessary
until a legitimate numeral has b een typ ed
getInt IO Int
getInt
getLine vs parseInt vs
putLine Oops That wasn t a number getInt
A Simpli ed MRC Computer
We suggest a simpli ed version of a programming mo del suitable for the Edin burgh MRC
team We wish to program a computer equipp ed with a teletyp e interface but also an
external microscop e assembly consisting of a microscop e an image capture device and
a movable stage containing a microscop e slide Supp ose there is a typ e Image whose
values represent digitised images obtained from the external assembly De ne a typ e of
two dimensional co ordinates to represent p ositions on the slide
type Coord Int Int
We assume that the stage can be moved to a certain range of co ordinates and also to a
val of the slide distinguished parked p osition for insertion and remo
CHAPTER MONADIC I O
The functional program is to run on the main computer but we assume there is an in
dep endent I O controller resp onsible for stage motion We prop ose to use the teletyp e
programming mo del with exceptions extended with the following computation construc
tors
moveStage Coord IO
capture IO Image
parkStage IO
moveStage co means instruct the I O controller to move the stage to co ordinate
p osition co and immediately return
capture means wait for the I O controller to signal that the stage is stationary
capture an image im and return answer im but if the stage is parked raise the
exception capture
parkStage means instruct the I O controller to park the stage and immediately
return
This mo del motivated the development of this chapter The MRC team use an I O mo del
based on this simpli ed mo del but considerably extended
Here is a veri cation problem suggested by Ian Poole of the MRC We b egin with a
parametric computation called jobA Its parameters are a list of co ordinates cos and
an image analysis function f of typ e Image Char where for any image im character
fim represents some human readable result computed ab out im The purp ose of jobA is
to capture images from each of the co ordinates in cos apply the analysis function to each
of the images and output the results
jobA Image Char Coord IO
analyse Image Char IO
jobA f cos sequence moveStage co analyse f co cos
analyse f capture im output f im
Although the main computer together with the I O controller p ermit a limited amountof
concurrency one sees that the program ab ove do es not admit any concurrent activity In
pro cessing each co ordinate the main computer tells the I O controller to move the stage
but then immediately tries to capture an image and so undergo es a p erio d of idle waiting
Similarly the I O controller remains idle while the main computer is pro cessing the image
captured at each co ordinate
Clearly the program would run faster if immediately after capturing an image at one
ve the stage to the next co ordinate during the co ordinate the I O controller could mo
same time as the main computer pro cesses the current image Program jobB is intended
to do this
jobB f skip
jobB f co cos moveStage co jobC f cos
jobC f cos sequence each f co co cos analyse f
each f co capture im moveStage co output f im
PROGRAMMING I O IN A MONADIC STYLE
The problem is to verify that parametric computations jobA and jobB are equivalent The
remaining sections of the chapter develop a theory of the simpli ed MRC programming
mo del by mo delling its semantics within H We conclude in x by proving that jobA
and jobB give rise to equal computations
History
The rst serious use of the monadic style was in the Kent Applicative Op erating System
KAOS lines of Miranda written by John Cupitt KAOS has a typ e
constructor interact a typ e sysErr and the following op erations in Haskell notation
return a interact a
comp a interact b interact a interact b
raise sysErr interact a
catch sysErr interact a interact a interact a
Cupitt s interact and sysErr corresp ond to IO and Exn resp ectively and his four op er
ations corresp ond to return raise and try resp ectively Cupitt based his combina
tors on a more complex set intro duced by Thompson for programming teletyp e I O
Cupitt s combinators were programmed on top of a primitivesynchronised stream mecha
nism He found that these combinators hid manylow level details in his implementation
Indep endently of Cupitt the author prop osed an extension of Holmstr om s PFL called
PFL whic hwas based on a continuation passing mechanism and prop osed the use
of a typ e constructor and op erations corresp onding to IO return and called Beh
Ret and resp ectively These high level op erations were de ned in terms of the low
level continuation passing mechanism Unlike Cupitt s scheme PFL was never fully
implemented This dissertation grew out of the e ort to make sense of the intuitive ideas
develop ed in PFL
This style of I O has come to b e called monadic b ecause the structure IO return
can b e mo delled categorically as a strong monad As discussed in Chapter Moggi
advo cated use of such structures to parameterise denotational descriptions of program
ming languages Inspired by Moggi Wadler has advo cated use of monads in the
functional language itself rather than the denotational metalanguage to express imp er
ative activity such as teletyp e I O or interaction with a mutable store The op erations
return and used here corresp ond to Wadler s op erations unit and bind resp ectively
A monadic approach to I O is b eing develop ed in the Glasgow Haskell compiler
Although many lines of co de in this style have been written and it has b een realised
that the combinators should ob ey the monadic laws there has been no previous work
on reasoning ab out programs engaged in monadic I O The remainder of this chapter
develops a theory of monadic I O
CHAPTER MONADIC I O
data TT a Read Char TT a
Write Char TT a
ReturnTT a
thenTT TT a a TT b TT b
p thenTT f
case p of
Read g Read v g v thenTT f
Write v q Write v q thenTT f
ReturnTT x f x
Table Aweak monad for teletyp e I O
A weak monad for teletyp e I O
Our theory of the MRC programming mo del is based on mo delling the semantics of compu
tations as functional programs within H This mo del admits pro ofs of program prop erties
but it is not b eing prop osed as the basis of an e cient implementation Peyton Jones and
Wadler discuss implementation techniques for monadic I O
We b egin by de ning what we mean by a monad in the context of H
De nition For any structure M returnM thenM where M is a typ e constructor and
programs returnM and thenM havetyp es
returnM M
thenM M M M
de ne three prop erties as follows
M b eta returnM x thenM f fx for all suitably typ ed f and x
M asso c p thenM f thenM g p thenM x fx thenM g
for all suitably typ ed p f and g
M eta p thenM returnM p for all suitably typ ed p
A weak monad is such a structure M satisfying M b eta and M asso c a monad is a
weak monad M satisfying M eta
Perhaps surprisingly the weak monad laws su ce for the purp oses of this chapter In
fact the full eta law fails for the twoweak monads we consider We b egin with a weak
monad TT given in Table whichmodelsteletyp e I O It is a version of the constructor
mo del side e ecting I O Roughly sp eaking any program of typ e T prop osed in x to
TT a consists of a string of Reads and Writes terminated bya ReturnTT
Lemma Structure TT ReturnTT thenTT isaweak monad
Pro of TT b eta follows immediately by de nition of thenTT For TT asso c it su ces
A WEAK MONAD FOR TELETYPE I O
to show that relation S is a bisimulation up to where S is de ned as follows
def
S f p thenTT f thenTT g p thenTT x fx thenTT g
j p TT a f a TTb g b TTcg
def
c
S S f c c j c S c g
To show that S is a bisimulation up to it su ces to consider any pair e e in S
e or that there are canonical programs c and c such and show that either e
b
that e c for each i andthatc S c Accordingly let e p thenTT f thenTT g
i i
and e p thenTT x f x thenTT g By Strachey s prop erty one of the following
cases holds
p
p returnTT x
p Write v q or
p Read h
In each case we can establish the required conditions
e e
e ReturnTT x thenTT f thenTT g f x thenTT g
e ReturnTT x thenTT x f x thenTT g f x thenTT g
So e e By Strachey s prop erty either e e and we are done or there
b
b
is a canonical program c with e e c In the latter case wehave c c so c S c
as required
In this case wehave
e Write v q thenTT f thenTT g
Write v q thenTT f thenTT g
Write v q thenTT f thenTT g
e Write v q thenTT x f x thenTT g
Write v q thenTT x f x thenTT g
De ne programs e and c to b e
i
i
q thenTT f thenTT g e
e q thenTT v fv thenTT g
c Write v e
c Write v e
b
S c as required Wehave e c and since v v and e S e wehave c
i i
In this nal case case when p Read h either h equals or some canonical function
CHAPTER MONADIC I O
v e If h wehave
e Read thenTT f thenTT g
Read v v thenTT f thenTT g
Read v thenTT g
Read v thenTT g
Read v
e Read thenTT v fv thenTT g
Read v v thenTT v fv thenTT g
Read v
Since e e Read v and we are done Otherwise h v e and
ehave w
e Read v e thenTT f thenTT g
Read v e thenTT f thenTT g
Read v e thenTT f thenTT g
e Read v e thenTT v fv thenTT g
Read v e thenTT v fv thenTT g
De ne programs c and c to b e
i
i
c v e thenTT f thenTT g
c v e thenTT v fv thenTT g
c Read e
c Read e
c b
Wehave c S c so c S c But then e c for each i and c S c so the pro of for
i i
Read is complete
The TT eta lawisinvalid Here is a counterexample
Read thenTT ReturnTT Read v v thenTT ReturnTT
Read v
Read
If we had a theory of H which satis ed v theintention of the Haskell designers
we could in fact establish TT eta For the purp oses of this chapter the lackof TT eta is
unimp ortant all we need are the weak monad laws TT b eta and TT asso c
A weak monad for the MRC computer
The MRC programming mo del has three comp onents teletyp e I O interaction with the
external microscop e assembly and exception handling We mo del the state of the external
microscop e assembly as a pair consisting of the current p osition either parked or at a
particular co ordinate and a function representing the image that could b e captured from
any of the co ordinate p ositions
type State Maybe Coord Coord Image
data Maybe a Nothing Just a
A WEAK MONAD FOR THE MRC COMPUTER
either a c b c Either a b c
either f g ab case ab of
Left a f a
Right b g b
get IO State
put State IO
get s ReturnTT Left s s
put s s ReturnTT Left s
return x s ReturnTT Left x s
raise x s ReturnTT Right x s
raise p f s p s thenTT uncurry either f
try p f s p s thenTT uncurry either return f
output v s Write v ReturnTT Left s
input s Read v ReturnTT Left v s
parkStage get s put Nothing snd s
moveStage co get s put Just co snd s
capture get s case fst s of
Just co return snd s co
Nothing raise capture
Table Deriving the MRC programming mo del
Mo delling image capture from the loaded slide by a xed function is extremely crude as it
ignores the e ects of random camera noise sloppy stage mechanics and the settings of the
fo cus lamp ob jective lens and lter It is a kind of idealised microscop e mo del whose
principle virtue is simplicity We use the typ e of strings to mo del exceptions
type Exn String
Intuitively we exp ect a computation of typ e IO a to act on some initial state by p erforming
some teletyp e I O b efore returning an answer either a value of typ e a or an exception of
typ e Exn We formalise this exp ectation in the following de nition of IO
type IO a PrimIO Either a Exn
type PrimIO a State TT a State
data Either a b Left a Right b
Recall that programs of typ e TT a can b e thought of as a string of I O op erations ending
with a result of typ e a Therefore a program p IO a is a function that given an initial
state s returns ps TT Either a Exn State which consists of a string of I O op erations
ending with a result of typ e Either a Exn State Such a result consists of a nal state
paired with either a normal result of typ e a or an exception of typ e Exn This de nition of
IO allows us to mo del the semantics of computations within H and hence use our theory
of H to reason ab out programs In practice wewould almost certainly want to implement
the IO typ e more e ciently
CHAPTER MONADIC I O
WeshowinTable how to program the various monadic op erations given our de nition
of the IO typ e These programs should be understo o d as formal sp eci cations of the
programming mo del s semantics rather than as realistic implementations There are three
auxiliary op erations either get and put Functions uncurry fst and snd are given in
Table
Wecanverify the laws IO b eta and IO asso c as follows
Lemma Structure IO return is a weak monad
Pro of The pro ofs are by routine equational reasoning IO b eta follows from the follow
ing calculations
return v f s return v s thenTT uncurry either f raise
returnTT Left v s thenTT uncurry either f raise
either f raise Left v s
fvs
To establish IO asso c rst note the following facts
either f g v h either x f x h x g x h v
raise v f raise v
e thenTT x e x e thenTT e
The third of these a sp ecialised functional eta law can be established by a simple co
induction We can b egin the pro of of IO asso c as follows for any suitably typ ed programs
p f g and s
p f g s p s thenTT uncurry either f raise
thenTT uncurry either g raise
p s thenTT y uncurry either f raise y
thenTT uncurry either g raise
wsete to b e a sub expression of the right hand side No
e uncurry either f raise y thenTT uncurry either g raise
From the facts noted earlier we can calculate as follows
e either f raise fst y snd y thenTT uncurry either g raise
either f raise fst y g snd y
either x f x g x raise x g fst y snd y
either x f x g raise fst y snd y
uncurry either x f x g raise y
Hence via the sp ecialised functional eta law noted ab ovewehave
p f g s p s thenTT y e
p s thenTT uncurry either x f x g raise
p x f x g s
as required for IO asso c
PROOF OF THE MOTIVATING EXAMPLE
analyse Image Char IO
jobA f cos sequence moveStage co analyse f co cos
analyse f capture im output f im
jobB f skip
jobB f co cos moveStage co jobC f cos
jobC f cos sequence each f co co cos analyse f
each f co capture im moveStage co output f im
Table The motivating example
Here is a counterexample to IO eta Let p be const ReturnTT Wehaveforanystate
s
p return s ReturnTT thenTT uncurry either return raise
uncurry either return raise
either return raise
but ps ReturnTT which do es not equal in H nor in Haskell b ecause all datatyp e
constructors are lazy
From IO b eta and IO asso c we can verify the following facts ab out and skip
Lemma
p q r p q r
skip p p
It would help subsequent calculations if we had the right cancellation law
p skip p
whichwould make skip a monoid but this fails b ecause of the same counterexample
urthermore q return is another counterexample wehave q skip as IO eta F
skip return return The problem is that in H and in Haskell
Undesirable unde ned elements cause the failure of all these eta and right cancellation
laws TT eta fails in H b ecause x and IO eta fails in both H and Haskell
b ecause ReturnTT However for the purp ose of proving our motivating example
all we need is a sp eci c right cancellation law Lemma
Pro of of the motivating example
We conclude this c hapter byverifying the example set in x on page We b egin with
a lemma each part of which can easily proved by equational calculations from what we
have already proved ab out the IO typ e
Lemma
output v skip output v
CHAPTER MONADIC I O
capture f Just co g f gco Just co g
moveStage co p mb g p Just co g
output v p mb g Write v p mb g
analyse f p Just co g Write f gco p Just co g
each f co p Just co g Write f gco p Just co g
We need a lemma to relate jobA and the auxiliary function jobC
Lemma F or any nitecos Coord co Coord f Image Char g Coord Image
and mb Maybe Coord
jobA f co cos mb g jobC f cos Just co g
Pro of By app eal to the structural induction principle Prop osition as sp ecialised to
lists in x wemayprovethelemmaby induction on the structure of nite list cos We
have
lhs moveStage co analyse f jobA f cos mb g
analyse f jobA f cos Just co g
Write f gco jobA f cos Just co g
cos co cos In Nowwe consider the two p ossible forms of cos i cos and ii
case i it is not hard to check that b oth sides equal
Write f gco skip Just co g
In case ii wemay calculate as follows
rhs each f co jobC f cos Just co g
Write f gco jobC f cos Just co g
Write f gco jobA f co cos Just co g IH
lhs
In b oth cases then wehave the desired equivalence
The lemma would fail if we were to remove the constraint that cos be nite Consider
cos which is not nite in the sense of x Program jobA f co mb g
Write f gco but jobC f Just co g
Finally we can verify our motivating example
Prop osition For any nite list cos function f and initial state mb g
jobA f cos mb g jobB f cos mb g
cos or cos co cos In the rst case Pro of If cos is nite then either
lhs skip mb g rhs In the second case we have rhs moveStage co
jobC f cos mb g jobC f cos Just co g jobA f co cos mb g by the
previous lemma and hence lhs rhs
Chapter
Conclusion
Summary
Wehaveshown how a theory of functional programming can b e develop ed from structural
op erational semantics and applicative bisimulation Wemight reckon this a CCS view of
calculus We develop ed parallel theories of a metalanguage for denotational semantics
M and a small functional language H essentially a fragmentofHaskell Co induction in
the form of bisimulation up to equivalence was found to b e useful in circumstances where
domain theoretic Scott induction mighthave b een needed
We identi ed four basic mechanisms for teletyp e I O We gave a lab elled transition se
mantics for three of these mechanisms and de ned a notion of bisimilarity We proved
that the three mechanisms are of equivalent expressivepower in the theoretical sense that
there are bisimulation preserving translations b etween the three Actual implementation
of these translations would not necessarily b e e cient however
Weadvo cated a monadic style of functional I O in the context of an application of func
tional programming to medical electronics at the Edinburgh Medical Research Council
MRC Motivated by a veri cation example suggested by the MRC we develop ed a
theory of monadic I O as an extension of our theory of functional programming By mo d
elling the semantics of the MRC programming mo del within H wewere able to verify the
example via functional programming techniques
Ever since McCarthy referred to the I O op erations in LISP as pseudo functions
functional I O has b een viewed with suspicion The work of this dissertation is imp ortant
b ecause it is the rst to show how a theory of functional programming can be smo othly
for functional I O and veri cation of extended to admit b oth an op erational semantics
programs engaged in I O
To nish o the dissertation for each chapter we sketch p ossible future work and o er
some further appraisal
CHAPTER CONCLUSION
A calculus of recursive typ es
It is a pity that the convergence theorem for M Theorem must dep end ultimately
on Mendler s theorem which this dissertation takes on trust It would b e interesting
to investigate whether the M convergence theorem could b e based on Wraith s enco ding
of recursivetyp es in the Girard Reynolds calculus itself Another direction to pursue
would b e to see whether Mendler s original strong normalisation pro of could b e extended
to cop e with the new constants The author has recently obtained a result analogous to
Theorem for a variantof M via a form of Tait s metho d up on which Mendler s pro of
for was based
A metalanguage for semantics
M is based on Plotkin s domain theoretic metalanguage for semantics in which
ts a particular domain construction Missing from b oth M and Plotkin s eachtyp e represen
metalanguage is a typ e of nondeterministic computations corresp onding to a p owerdomain
construction Howe s original pap er showed that applicative bisimulation can be
applied to a nondeterministic calculus It maybeworthwhile to investigate how to extend
M with a typ e representing nondeterministic computations
Op erational precongruence
Chapter studied how the context lemma and Howe s metho d can b e used to provethat
an op erationally de ned equivalence relation is a precongruence We exp ect that Howe s
metho d will be extremely useful for constructing op erational theories of programming
languages The context lemma was simple and ingenious in the setting of combinatory
logic but much care app ears to b e needed to generalise it to a more complex calculus
suchasM
Theory of the metalanguage
elop ed an equational theory for M and proved results ab out certain typ es needed Wedev
for the denotational semantics of H Although not itself domain theoretic the theory
of M can b e compared to axiomatisations of domain theory or functional programming
such as Edinburgh LCF Cambridge LCF and Thompson s logic for Miranda
There are two ma jor di erences First only M typ es of the form contain a
divergent term Hence the presentation of the theory in Table is simpler than in
LCF or Thompson s logic Second there is no principle of Scott induction for M Smith
h a principle in an op erational setting On the other has shown how to derive suc
hand wehave found in this dissertation that co induction is su cienttoprove theorems
such as Lemma that in LCF would probably have required Scott induction
AN OPERATIONAL THEORY OF FUNCTIONAL PROGRAMMING
infix
abstype a Job JOB of unit a
with
fun exec JOB f f
fun unit x JOB fn x
fun JOB f q JOB fn exec q f
fun getStr n JOB fn input std in n
fun putStr s JOB fn output std out s
end
fun p q p fn u q
fun gettingLine s
getStr fn c
if c n then unit s else gettingLine s c
val getLine gettingLine
val main
putStr First name getLine fn first
putStr Second name getLine fn second
putStr Hello first second n
Table A monadic style of I O for Standard ML
An op erational theory of functional programming
The most original asp ect of the theory of functional programming in Chapter is that
it is entirely grounded in op erational semantics Note that the theory of op erational
equivalence in H do es not dep end in any way on Mendler s normalisation theorem
although the theory of denotational equivalence do es
It is interesting to compare the op erational and denotational theories of H The denota
tional semantics is go o d for comparing di erent ob ject languages e g H HX andH with
side e ecting I O while the op erational semantics allows a simple derivation of program
equivalence
A useful future pro ject would b e to extend H to include eager algebraic typ e constructors
and a more realistic exception mechanism Such an extension would contain a non trivial
fragment of core Standard ML
I O Four mechanisms for teletyp e
The main question left op en in this chapter is howtointegrate an op erational semantics of
side e ecting I O into a theory of functional programming Several authors have suggested
op erational semantics for ML extended with side e ecting op erators for concurrency
however there has been little work on equational theories for functional languages
extended with concurrency
CHAPTER CONCLUSION
Monadic I O
This chapter showed how to construct a crude mo del of the MRC computer within H and
use functional programming techniques to verify a simple prop erty Before this sort of
metho dology is to b e of any practical use there needs to b e a go o d deal more mechanised
supp ort for pro ofs of programs and exp erience of sp ecifying more realistic systems
One can view monadic I O as a controlled form of side e ecting I O the monad of side
e ects has moved from the denotational semantics into the typ e system of the ob ject
language
This dissertation has not discussed practical implementation though of course the op er
ational semantics rules give some clues There is ongoing work at Glasgow on e cient
implementation of monadic I O Veri cation of an I O mechanism is a problem not
previously examined in work on veri ed functional implementations
The monadic style is not con ned to lazy languages Table shows how monadic I O can
side e ecting I O In naive exp eriments b e implemented in core Standard ML on top of
this style of monadic I O was ab out six times slower than side e ecting I O It should
b e p ossible to improve this p erformance if the Job typ e were taken as primitive whichis
more or less what is prop osed in the Glasgow Haskell compiler If side e ects are
allowed only in the implementation of the Job typ e reasoning ab out monadic programs
in ML should not b e much harder than reasoning ab out monadic programs in Haskell
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Notation Index
Chapter One
X Y Z typ e variable
f g u v w x y z term variable
t phrase of abstract syntax
ftv t typ e variables free in t
fv t term variables free in t
t
t X substitution for a typ e variable
t
t x substitution for a term variable
t t syntactic identity up to alpha conversion
bv t term variables b ound in t
C D context
C t instantiation of a context
R R comp osition of relations R and R
R transitive closure of relation R
R re exiv e transitive closure of relation R
Id identity relation
R inverse of relation R
Chapter Two
typ e in
recursivetyp e in
L M N term in
M typ e assignmentin
M N reduction in
M N simulation in
Chapter Three
typ e in M
recursivetyp e in M
U V canonical term in M
NOTATION INDEX
L M N term in M
environmentinM
M typ e assignmentinM
E exp erimentinM
M N reduction in M
M V evaluation in M
M convergence in M
M divergence in M
translation of M into
Chapter Four
V canonical context
D
C x substitution for contexts
E contextual exp eriment
adivergent program in M
b
R a con ned relation generated from R
C
contextual order in M
E
exp erimental order in M
functional inducing applicative similarityin M
A
applicative similarityin M
A
applicative bisimilarityin M
compatible similarityin M
Chapter Five
op erational equivalence in M
op erational order in M
bbc representation of b o olean b ftt g in M
bnc representation of number n N in M
b c representation of op erator in M
Chapter Six
typ e in H
algebraic t yp e in H
dc data clause in H
c canonical term in H
e term in H
cc case clause in H
representation of literal in H
environmentinH
e typ e assignmentinH
NOTATION INDEX
adivergent program in H
LE left exp erimentinH
RE right exp erimentinH
e e reduction in H
e c evaluation in H
e convergence in H
e divergence in H
M N translation of H function application into M
translation of H into M
b
R a con ned relation generated from R
functional inducing applicative similarityin H
op erational equivalence in H
v op erational order in H
v compatible similarityin H
denotational equivalence in H induced from M
Chapter Seven
teletyp e action
h i functional inducing bisimilarity
lab elled transition
bisimilarity
Chapter Eight
IO typ e of computations
input input a character
output output a character
return return a value
sequential comp osition
restricted sequential comp osition
sequence iterated sequential comp osition
lift lift a curried function
accumulate accumulating sequential comp osition
raise raise an exception
try handle an exception
biased choice
Coord typ e of co ordinates
moveStage move microscop e stage
capture capture an image
parkStage park microscop e stage
TT typ e of teletyp e I O
State typ e of state
Maybe optional typ e
NOTATION INDEX
Exn typ e of exceptions
PrimIO primitivetyp e of computations
Either sum typ e
either sum discrimination
get return the current state
put replace the current state
Index
concurrency actions
constructor for adequacy xi
in MRCmodel op erational
Kahn networks advice to the reader
Concurrent Clean xi ALGOL
con ned extension alpha conversion
con ned relation in
con ned term
bisimilarity
congruence
applicative
context
compared with applicative bisimilar
canonical
ity
instantiation
ground applicative
context lemma
simplicity of
contextual equivalence
bisimulation
con textual order
applicative xi
convergence
and nondeterminism
denotational equivalence
bisimulation up to
divergence
Bo oleans in M
Budgets xi
eager language
emptytyp e in M
call by name
environment
call by need xi
evaluation
call by value
evaluation contexts
canonical freeness
exceptions
CCS
in HX
in monadic I O
certainly convergenttyp es
execution typ e
co induction
exp erimental order
and bisimulation
exp eriments
and Scott induction
left and right
principle of
combinators for I O
FL
history
FP
monadic programming
Fudgets xi
stream transformers
functional extensionality
compatible relation
computation
closure
computational monad
Gofer x
INDEX
lamb da calculus GPM
computational grammar of
lazy H terms
p olymorphic H typ es
simply typ ed M contextual exp eriments
untyp ed M exp eriments
lazy language M terms
LCF vii M typ es
LISP terms
typ es Lucid
ground relation
Miranda
ground restriction
ML x
Haskell vii ix x
Lazy
Standard
Mo dula
committee
monad T of
interpreted in H
exceptions
Hop e
lifting
Howe s metho d
side e ects
hyp othesis of this dissertation
MRC computer
I O mechanism
name convention in
continuation passing
Nebula
semantics
numb ers in M
FL histories
one p ointtyp e in M
semantics
op erational equivalence
hiatons
op erational extensionality
I O trees
op erational order
Landin stream
semantics
PCF
monadic
PFL
for Standard ML
p olymorphic de nitions
side e ecting
POP
semantics
p ositivetyp es
streams for eager languages
p ossibly divergenttyp es
synchronised stream
precongruence
Dialogue typ e
preorder
semantics
program
systems
initial typ es
reduction
input p olling
relational op erations
io not gettingLine
iterated pro ducts and sums in M
Scheme
Silage
KAOS
similarity
applicative lab elled transition system
INDEX
compatible
ground applicative
simulation
applicative
SKIM I I
Strachey s prop erty
stream
implementation of input
recursivetyp e for
strong normalisation
for Mendler s calculus
for
structural induction
substitution
syntax conventions
task
terminal typ es
trace
equivalence and bisimilarity
semantics
typ e assignment
variable convention