This is a technical rep ort A mo died version of this rep ort will b e published in Handbook of

Algebra Vol edited by M Hazewinkel

Some Asp ects of Categories in Computer Science



P J Scott

Dept of Mathematics

University of Ottawa

Ottawa Ontario CANADA

Sept

Contents

Intro duction

Categories Lamb da Calculi and FormulasasTyp es

Cartesian Closed Categories

Simply Typ ed Lamb da Calculi

FormulasasTyp es the fundamental equivalence

Some Datatyp es

Polymorphism

Polymorphic lamb da calculi

What is a Mo del of System F

The Untyp ed World

Mo dels and Denotational Semantics

CMonoids and Categorical Combinators

Church vs Curry Typing

Logical Relations and Logical Permutations

Logical Relations and Syntax

Example ReductionFree Normalization

Categorical Normal Forms

P category theory and normalization algorithms

Example PCF

PCF

Adequacy

Parametricity

Dinaturality

Reynolds Parametricity

Linear Logic

Monoidal Categories

Gentzens pro of theory

Gentzen sequents



Research partially supp orted by NSERC Canada

Girards Analysis of the Structural Rules

Fragments and Exotic Extensions

Top ology of Pro ofs

What is a categorical mo del of LL

General Mo dels

Concrete Mo dels

Full Completeness

Representation Theorems

Full Completeness Theorems

Linear Lauchli Semantics

Feedback and Trace

Traced Monoidal Categories

Partially Additive Categories

GoI Categories

Literature Notes

Intro duction

Over the past years category theory has b ecome an increasingly signicant conceptual and

practical to ol in many areas of computer science There are ma jor conferences and journals de

voted wholly or partially to applying categorical metho ds to computing At the same time the

close connections of computer science to logic have seen categorical logic develop ed in the s

fruitfully applied in signicant ways in b oth theory and practice

Given the rapid and enormous development of the sub ject and the availability of suitable gradu

ate texts and sp ecialized survey articles we shall only examine a few of the areas that app ear to

the author to have conceptual and mathematical interest to the readers of this Handb o ok Along

with the many references in the text the reader is urged to examine the nal section Literature

Notes where we reference omitted imp ortant areas as well as the Bibliography

We shall b egin by discussing the close connections of certain closed categories with typ ed lamb da

calculi on the one hand and with the pro of theory of various logics on the other It cannot b e

overemphasized that mo dern computer science heavily uses formal syntax but we shall try to tread

lightly The socalled CurryHoward isomorphism which identies formal pro ofs with lamb da

terms hence with arrows in certain free categories is the cornerstone of mo dern programming

language semantics and simply cannot b e overlo oked

o

Notation We often elide comp osition symb ols writing g f A C for g f A C whenever

f A B and g B C To save some space we have omitted large numb ers of routine diagrams

which the reader can nd in the sources referenced

Categories Lamb da Calculi and FormulasasTyp es

Cartesian Closed Categories

Cartesian closed categories cccs were develop ed in the s by F W Lawvere Law Law

Both Lawvere and Lamb ek L stressed their connections to Churchs lamb da calculus as well as

to intuitionistic pro of theory In the s work of Dana Scott and Gordon Plotkin established

their fundamental role in the semantics of programming languages A precise equivalence b etween

these three notions cccs typ ed lamb da calculi and intuitionistic pro of theory was published in

Lamb ek and Scott LS We recall the appropriate denitions from LS

Ob jects Distinguished Arrows Equations

!

A

Terminal A f

A

f A

AB

o

A B A hf g i f

1

1

AB

o

Pro ducts A B A B B hf g i g

2

2

f g

C A C B

o o

h h hi h

1 2

hf g i

C A B

h C A B

A A

o o

i f Exp onentials B ev B A B ev hf

1 2 AB

f

C A B

o o

ev hg i g

1 2

f

A

A

C B

g C B

Figure CCCs Equationally

Denition

i A cartesian category C is a category with distinguished nite pro ducts equivalently binary

pro ducts and a terminal ob ject This says there are isomorphisms natural in A B C

H om A fg

C

H om C A B H om C A H om C B

C C C

ii A cartesian closed category C is a cartesian category C such that for each ob ject A C

A

the functor A C C has a sp ecied right adjoint denoted That is there is an

isomorphism natural in B and C

A

H om C A B H om C B

C C

For many purp oses in computer science it is often useful to have categories with explicitly given

strict structure along with strict functors that preserve everything on the nose We may present

such cccs equationally in the spirit of multisorted universal algebra The arrows and equations

are summarized in Figure These equations determine the isomorphisms and In this

presentation we say the structure is strict meaning there is only one ob ject representing each of

A A

the ab ove constructs A B B The exp onential ob ject B is often called the function space of

A and B In the computer science literature the function space is often denoted A B while the

f

A

arrow C B is often called currying of f

Remark Following most categorical logic and computer science literature we do not assume

cccs have nite limits Law LS AC Mit in order to keep the corresp ondence with

simply typ ed lamb da calculi cf Theorem b elow Earlier b o oks cf Mac do not always

follow this convention

Let us list some useful examples of cartesian closed categories for details see LS Mit

Mac

Example The category Set of sets and functions Here A B is a chosen cartesian pro duct

ev

A A

and B is the set of functions from A to B The map B A B is the usual evaluation map

f

A

B is the map c a f c a while currying C

An imp ortant subfamily of examples are Henkin models which are cccs in which the terminal

ob ject is a generator Mit Theorem More concretely for a lamb da calculus signature

with freely generated typ es cf Section b elow a Henkin model A is a typ eindexed family of

A

sets A fA j a typ e g where A fg A A A A A which forms a ccc with

1

resp ect to restriction of the usual ccc structure of Set In the case of atomic base sorts b A is

b

some xed but arbitrary set A ful l type hierarchy is a Henkin mo del with full function spaces ie

A

A A

op

C

Example More generally the functor category Set of presheaves on C is cartesian closed

Its ob jects are contravariant functors from C to Set and its arrows are natural transformations

op

C

b etween them We sketch the ccc structure given F G Set dene F G p ointwise on

F A

ob jects and arrows Motivated by Yonedas Lemma dene G A N ath F G where

A F

h H omA This easily extends to a functor Finally if H F G dene H G by

a h c H ha c

C C

A

Functor categories have b een used in studying problematic semantical issues in Algollike lan

guages Rey Ol OHT Ten as well as recently in concurrency theory and mo dels of

calculus CSW CaWi Sp ecial cases of presheaves have b een studied extensively Mit LS

op

C

Let C b e a p oset qua trivial category Then Set the category of Kripke mo dels over C

may b e identied with sets indexed or graded by the p oset C Such mo dels are fundamental

in intuitionistic logic LS TrvD and also arise in Kripke Logical Relations an imp ortant

to ol in semantics of programming languages Mit OHT OHRi

Let C O X the p oset of op ens of the top ological space X The sub category S hX of

sheaves on X is cartesian closed

op

C

is the category of M Let C b e a monoid M qua category with one ob ject Then Set

sets ie sets X equipp ed with a left action equivalently a monoid homomorphism M

E ndX where E ndX is the monoid of endomaps of X Morphisms of M sets X and Y

are equivariant maps ie functions commuting with the action A sp ecial case of this is

when M is actually a group G qua category with one ob ject where all maps are isos In

op

C

that case Set is the category of Gsets the category of p ermutational representations of

G Its ob jects are sets X equipp ed with left actions G S y mX and whose morphisms

are equivariant maps We shall return to these examples when we sp eak of Lauc hli semantics

and Full Completeness Section

Example CPO Ob jects are p osets P such that countable ascending chains a a a

0 1 2

have suprema Morphisms are maps which preserve suprema of countable ascending chains in

particular are order preserving This category is a ccc with pro ducts P Q ordered p ointwise

P

and Q H omP Q ordered p ointwise In this case the categories are CPOenrichedie

the homsets themselves form an CPO compatible with comp osition An imp ortant sub ccc is

CPO in which all ob jects have a distinguished minimal element but morphisms need not

preserve it

The category CPO is the most basic example in a vast research area domain theory which

has arisen since This area concerns the denotational semantics of programming languages

and mo dels of untyp ed lamb da calculi cf Section b elow See also the survey article AbJu

Example Coherent Spaces and Stable Maps A Coherent Space A is a family of sets satisfying

i a A and b a implies b A and ii if B A and if c c B c c A then B A

In particular A Morphisms are stable maps ie monotone maps preserving pullbacks and

ltered colimits That is f A B is a stable map if i b a A implies f b f a

ii f a f a for I directed and iii a b A implies f a b f a f b

iI i iI i

This gives a category Stab Every coherent space A yields a reexivesymmetric undirected

graph jAj where jAj fa j fag Ag and a b i fa bg A Moreover there is a bijective

corresp ondence b etween such graphs and coherent spaces Given two coherent spaces A B their

pro duct A B is dened via the asso ciated graphs as follows jA B j with jA B j

AB

jAj jB j fg jAj fg jB j where a a i a a b b i

AB A AB

A

b b and a b for all a jAj b jB j The function space B S tabA B of stable

B AB

maps can b e given the structure of a coherent space ordered by Berrys order f g i for all

a a A a a implies f a f a g a For details see GLT Tr This class of domains

led to the discovery of linear logic Section

Example Per Models A partial equivalence relation p er is a symmetric transitive relation

2

A Thus is an equivalence relation on the subset D om fx A j x xg A P set

A A A A

is a pair A where A is a set and is a p er on A Given two P sets A and B

A A A B

a morphism of P sets is a function f A B such that a a implies f a f a for all

A B

a a A That is f induces a map of quotients D om D om which preserves the

A A B B

asso ciated partitions

P Set the category of P sets and morphisms is a ccc with structure induced from S et we

A

A

where dene A B where a b a b i a a and b b and B

AB AB A B B

A

f g i for all a a A a a implies f a g a We shall discuss variants of the ccc

B A B

structure of P Set in Section b elow with resp ect to reductionfree normalization

Other classes of Per mo dels are obtained by considering p ers on a xed functionally complete

partial combinatory algebra for example built over a mo del of untyp ed lamb da calculus cf Section

b elow The prototypical example is the following category P er N of p ers on the natural

f

numb ers The ob jects are p ers on N Morphisms R S are equivalence classes of partial

recursive functions Turingmachine computable partial functions N N which induce a total

map on the induced partitions ie for all m n N mRn implies f m f n are dened and

f mS f n Here we dene equivalence of maps f g R S by f g i m n mRn implies

f m g n are dened and f mS g n The fact that P er N is a ccc uses some elementary

recursion theory BFSS Mit AL See also Section

Example Free CCCs Given a set of basic ob jects X we can form F the free ccc generated

X

()

by X Its ob jects are freely generated from X and using and its arrows are freely

generated using identities and comp osition plus the structure in Figure and we imp ose the

minimal equations required to have a ccc More generally we may build F the free ccc generated

G

by a directed multigraph or even a small category G by freely generating from the vertices resp

ob jects and edges resp arrows of G and thenin the case of categories G imp osing the appropriate

equations The sense that this is free is related to Denition and discussed in Example

Cartesian closed categories can themselves b e made into a category in many ways This dep ends

to some extent on how much bi enriched etc structure one wishes to imp ose The following

elementary notions have proved useful We shall mention a comparison b etween strict and nonstrict

cccs with copro ducts in Remark More general notions of monoidal functors etc will b e

mentioned in Section

Denition CART is the category of strictly structured cartesian closed categories with

st

functors that preserve the structure on the nose CART is the category whose cells are

st

cartesian closed categories whose cells are strict cartesian closed functors and whose cells are

natural isomorphisms

As p ointed out by Lamb ek L LS given a ccc A we may adjoin an indeterminate arrow

x

A to A to form a polynomial cartesian closed category Ax over A with the exp ected

universal prop erty in CART The ob jects of Ax are the same as those of A while the arrows are

st

p olynomials ie formal expressions built from the symb ol x using the arrowforming op erations

of A The key fact ab out such p olynomial expressions is a normal form theorem stated here for

cccs although it applies more generally see LS p

Prop osition Functional Completeness For every polynomial x in an indeterminate

x h

A

o

A over a ccc A there is a unique arrow C A such that ev hh xi x where

x x

is equality in Ax

Lo oking ahead to lamb da calculus notation in the next section we write h x so the

x:A

o

equation ab ove b ecomes ev h x xi x The universal prop erty of p olynomial algebras

x:A

x

a

guarantees a notion of substitution of constants A A for indeterminates x in x We

obtain the following

a

Corollary The rule In the situation above for any arrow A A

o

ev h x ai a

x:A

holds in A

The rule is the foundation of the lambda calculus fundamental in programming language theory

It says the following we think of x as the function x x Equation says evaluating

x:A

the function x at argument a is just substitution of the constant a for each o ccurrence of

x:A

x in x However this pro cess is far more sophisticated than simple p olynomial substitution in

algebra In our situation the argument a may itself b e a lamb da term which in turn may contain

other lamb da terms applied to various arguments etc After substitution the right hand side a

of Equation may b e far more complex than the left hand side with many new p ossibilities for

evaluations created by the substitution Thus if we think of computation as oriented rewriting

from the LHS to the RHS it is not at all obvious the pro cess ever halts The fact that it do es is a

basic theorem in the socalled Op erational Semantics of typ ed lamb da calculus Indeed the Strong

Normalization Theorem cf LS p says every sequence of ordered rewrites from left to

right eventually halts at an irreducible term cf Remark and Section b elow

Remark We may also form p olynomial cccs Ax x by adjoining a nite set of inde

1 n

x

i

terminates A Using pro duct typ es one may show Ax x Az for an indeterminate

i 1 n

z

A A

1 n

Polynomial cartesian or cartesian closed categories Ax may b e constructed directly showing

they are the Kleisli category of an appropriate comonad on A see LS p Extensions of

this technique to allow adjoining indeterminates to brations using categorical machinery are

considered in HJ

Simply Typ ed Lamb da Calculi

Lamb da Calculus is an abstract theory of functions develop ed by Alonzo Church in the s

Originally arising in the foundations of logic and computability theory more recently it has b ecome

an essential to ol in the mathematical foundations of programming languages Mit The calculus

itself to b e describ ed b elow encompasses the pro cess of building functions from variables and

constants using application and functional abstraction

Actually there are many lamb da calculityp ed and untyp edwith various elab orate structures

of typ es terms and equations Let us give the basic typ ed one We shall follow an algebraic syntax

as in LS

Denition Typ ed calculus Let Sorts b e a set of sorts or atomic typ es The typed

calculus generated by Sorts is a formal system consisting of three classes Typ es Terms and

Equations b etween terms We write a A for a is a term of typ e A

Types This is the class obtained from the collection of Sorts using the following rules Sorts are

A

typ es is a typ e and if A and B are typ es then so are A B and B We allow the p ossibility

of other typ es or typ eforming op erations and p ossible identications b etween typ es

A

Terms To every typ e A we assign a denumerable set of typ ed variables x A i

i

A

We write x A or x for a typical variable x of typ e A Terms are freely generated from vari

ables constants and termforming op erations We require at least the following distinguished

generators

AB AB

If a A b B c A B then ha bi A B c A c B

1 2

A A

If a A f B B then ev f a B B

AB x:A

There may b e additional constants and termforming op erations b esides those sp ecied

We shall abbreviate ev f a by f a read f of a omitting typ es when clear Intuitively

AB

ev denotes evaluation h i denotes pairing and denotes the function x

AB x:A

where is some term expression p ossibly containing x The op erator acts like a quan

x:A

tier so the variable x in is a b ound or dummy variable just like the x in or

x:A x:A

R

in f xdx We inductively dene the sets of free and b ound variables in a term t denoted

FV t B V t resp cf Bar p We shall always identify terms up to renaming of

b ound variables The expression ax denotes the result of substituting the term a A for

each o ccurrence of x A in if necessary renaming b ound variables in so that no clashes

o ccur cf Bar Terms without free variables are called closed otherwise open

Equations between terms A context is a nite set of typ ed variables An equation in context

is an expression a a where a a are terms of the same typ e A whose free variables are

contained in

The equality relation b etween terms in context of the same typ e is generated using at least

the following axioms and closure under the following rules

i is an equivalence relation

a b

ii whenever

a b

iii must b e a congruence relation with resp ect to all termforming op erations It suces

to consider closure under the following two rules cf LS

A a b

fx g

f a f b

x:A x:A

iv The following sp ecic axioms we omit subscripts on terms when the typ es are obvious

Products

a a for all a

b ha bi a for all a A b B

1

c ha bi b for all a A b B

2

d h c ci c for all c C

1 2

Lambda Calculus

Rule a ax

x:A

A

Rule f x f where f B and x is not a free variable of f

x:A

Remark There may b e additional typ es terms or equations Following standard conven

tions we equate terms which only dier by change of b ound variablesthis is called conversion

in the literature Bar Equations are in contextie o ccur within a declared set of free variables

This allows the p ossibility of empty types ie typ es without closed terms of that typ e This view

is fundamental in recent approaches to functional languages Mit and necessary for interpreting

such theories in presheaf categories for example However if there happ en to b e closed terms a A

of each typ e we may omit the subscript on equations b ecause of the following derivable rule cf

LS Prop p for x and if all free variables of a are contained in

x x

fxg

ax ax

Example Freely generated simply typed lambda calculi These are freely generated from

sp ecied sorts terms andor equations In the minimal case no additional assumptions we

obtain the simply typ ed lamb da calculus with nite pro ducts freely generated by Sorts Typically

however we assume that among the Sorts are distinguished datatypes and asso ciated terms p ossibly

with sp ecied equations For example basic universal algebra would b e mo delled by sorts A with

n

distinguished nary op erations given by terms t A A and constants c A Any sp ecied

term equations are added to the theory as nonlogical axioms

Example The internal language of a ccc A Here the typ es are the ob jects of A where

()

have the obvious meanings Terms with free variables x A x A are p olyno

1 1 n n

x

i

A is an indeterminate lamb da abstraction is given by functional mials in Ax x where

i 1 n

completeness as in Prop osition and we dene a b to hold i a b as p olynomials in AX

X X

where X fx x g

1 n

Remark

A

i Historically typ ed lamb da calculi were often presented with only exp onential typ es B no

pro ducts and the asso ciated machinery Bar Bar This p ermits certain simplications

in inductive arguments athough it is categorically less natural cf also Remark

ii It is a fundamental prop erty that lamb da calculus is a higherorder functional language terms

A

of typ e B can use an arbitrary term of typ e A as an argument and A and B themselves may

b e very complex Thus typ ed lamb da calculus is often referred to as a theory of functionals

of higher type

FormulasasTyp es the fundamental equivalence

Let us describ e the third comp onent of the trio cartesian closed categories typ ed lamb da cal

culi and formulasastyp es The FormulasasTypes view sometimes called the CurryHoward

Formulas A j Atoms j A A j A A

1 2 1 2

Provability is a reexive transitive relation such that

for arbitrary formulas A B C

A A B A A B B

C A B i C A and C B

C A B i C A B

Figure Intuitionistic Logic

isomorphism is playing an increasingly inuential role in the logical foundations of computing

esp ecially in the foundations of functional programming languages Its historical ro ots lie in

the socalled BrouwerHeytingKolmogorov BHK interpretation of intuitionistic logic from the

sGLT TrvD The idea is based on mo delling pro ofs which are programs by functions

ie lamb da terms Since pro ofs can b e mo delled by lamb da terms and the latter are themselves

arrows in certain free categories it follows that functional programs can b e mo delled categorically

In mo dern guise the CurryHoward analysis says the following Pro ofs in a constructive logic

L may b e identied as terms of an appropriate typ ed lamb da calculus where

L

typ es formulas of L

lamb da terms pro ofs ie annotations of Natural Deduction pro of trees

provable equality of lamb da terms corresp onds to the equivalence relation on pro ofs generated

by Gentzens normalization algorithm

Often researchers imp ose additional equations b etween lamb da terms motivated from categorical

considerations eg to force traditional datatyp es to have a strong universal mapping prop erty

Remark Formulas Sp ecications More generally the CurryHoward view identies

typ es of a programming language with formulas of some logic and programs of typ e A as pro ofs

within the logic of formula A Constructing pro ofs of formula A may then b e interpreted as building

programs that meet the sp ecication A

For example consider the intuitionistic f gfragment of prop ositional calculus as in

Figure This logic closely follows the presentation of cccs in Denition and Figure We

now identity FormulasasTyp es the prop ositional symb ols with the typ e constructors

resp ectively We assign lamb da terms inductively To a pro of of A B we assign terms

x A tx B where tx is a term of typ e B with at most the free variable x A ie in context

fx Ag as follows

x A sx B y B ty C

x A tsxy C x A x A

x A x A B x A x A B x B

1 2

z C A tz B

x C a A x C b B

x C ha bi A B y C thy xiz A B

x:A

y C ty A B

z C A t z y z B

1 2

We can now refer to entire pro of trees by the asso ciated lamb da terms We wish to put an equi

valence relation on pro ofs according to the equations of typ ed lamb da calculus Given two pro ofs

of an entailment A B say x A sx B and x A tx B we say they are equivalent if we

can derive s t in the appropriate typ ed lamb da calculus

fxg

Denition Let Calc denote the category whose ob jects are typ ed lamb da calculi and

whose morphisms are translations ie maps which send typ es to typ es terms to terms including

mapping the ith variable of typ e A to the ith variable of typ e A preserve all the sp ecied

op erations on typ es and terms on the nose and preserve equations

Theorem There are a pair of functors C Calc C ar t and L C ar t Calc which

st st

set up an equivalence of categories C ar t Calc

st

The functor L asso ciates to ccc A its internal language while the functor C asso ciates to any

lamb da calculus L a syntactically generated ccc C L whose ob jects are typ es of L and whose

arrows A B are denoted by equivalence classes of lamb da terms tx representing pro ofs

x A tx B as ab ove see LS

This leads to a kind of Soundness Theorem for diagrammatic reasoning which is imp ortant in

categorical logic

Corollary Verifying that a diagram commutes in a ccc C is equivalent to proving an equation

in the internal language of C

The ab ove result includes allowing algebraic theories mo delled in the cartesian fragment Mac

Cr as well as extensions with categorical data typ es like weak natural numb ers ob jects see

Section Theorem also leads to concrete syntactic presentations of free cccs LS

Tay Let Graph b e the category of directed multigraphs ST

Corollary The forgetful functor U C ar t Gr aph has a left adjoint F Gr aph C ar t

st st

Let F denote the image of graph G under F We cal l F the free ccc generated by G

G G

Example Given a discrete graph G considered as a a set F the free ccc generated by the

0 G

0

set of sorts G It has the following universal prop erty for any ccc C and for any graph morphism

0

F G C there is a unique extension to a strict cccfunctor F C

0 F G

0

F

C F

G

0

F

G

0

This says given any interpretation F of basic atomic typ es no des of G as ob jects of C there

0

is a unique extension to an interpretation in C of the entire simply typ ed lamb da calculus

F

generated by G identifying the free ccc F with this lamb da calculus

0 G

0

Remark A Pitts Pi has shown how to construct free cccs syntactically using lamb da

calculi without pro duct typ es The idea is to take ob jects to b e sequences of typ es and arrows

Ob jects Distinguished Arrows Equations

O

A

Initial A O f

A

f A

AB

o

in A A B f g in f

1

1

AB

o

Copro ducts A B in B A B f g in g

2

2

f g

A C B C

o o

h in h in h

1 2

[f g ]

A B C

h A B C

Figure Copro ducts

to b e sequences of terms The terminal ob ject is the empty sequence while pro ducts are given

by concatenation of sequences For a full discussion see CDS This is useful in reductionfree

normalization see Section b elow

Remark There are more advanced and bicategorical versions of the ab ove results We

shall mention more structure in the case of cartesian closed categories with copro ducts in the next

section

Some Datatypes

Computing requires datatyp es for example natural numb ers lists arrays etc The categorical

development of such datatyp es is an old and established area The reader is referred to any of the

standard texts for discussion of the basics eg MA BW Mit Ten General categorical

treatments of abstract datatyp es ab ound in the literature The standard treatment is to use

initial T algebras cf Section b elow or nal T coalgebras for denable or p olynomial

endofunctors T There are interesting common generalizations to lamb da calculi with functorial

typ e constructors Ha Wr categories with datatyp es determined by strong monadsMo

CSp and using enriched categorical structures K There is recent discussion of datatyp es in

distributive categories Co W and the use of the categorical theory of sketches BW Bor

We shall merely illustrate a few elementary algebraic structures commonly added to a cartesian

or cartesian closed category or the asso ciated term calculi

Denition A category C has nite coproducts equivalently binary copro ducts and an initial

ob ject if for every A B C there is a distinguished ob ject A B together with isomorphisms

natural in A B C C

H om A fg

C

H om A B C H om A C H om B C

C C C

1

We say C is bicartesian closed biccc if it is a ccc with nite copro ducts

Just as in the case of pro ducts cf Figure we may present copro ducts equationally as

in Figure and sp eak of strict structure etc In programming language semantics copro ducts

corresp ond to variant types settheoretically they corresp ond to disjoint union while from the

logical viewp oint copro ducts corresp ond to disjunction Thus a biccc corresp onds to intuitionistic

f glogic We add to the logic of Figure formulas and A A together with the

1 2

Not to b e confused with bicategories cf Bor

rules

A

A B C i A C and B C

corresp onding to equations The asso ciated typ ed lamb da calculus with copro ducts is rather

[f g ]

subtle to formulate Mit GLT The problem is with the copairing op erator A B C which

in S ets corresp onds to a denitionbycases op erator

f x if x A

f g x

g x if x B

The correct lamb da calculus formalism for copro duct typ es corresp onds to the logicians natural

deduction rules for strong sums The issue is not trivial since the word problem for free bicccs

and the asso ciated typ e isomorphism problem DiCo is among the most dicult of this typ e

of question andat least for the current state of the artdep ends heavily on technical subtleties

of syntax for its solution see Gh

Just as for cccs we may intro duce various categories of bicccs cf Cu For example

Denition The category B iC ART has cells strict bicartesian closed categories

st

cells functors preserving the structure on the nose and cells natural isomorphisms

One may similarly dene a nonstrict version B iC ART

Remark Every bicartesian closed category is equivalent to a strict one Indeed this is part

of a general categorical adjointness b etween the ab ove categories from a theorem of Blackwell

Kelly and Power See Cubric Cu for applications to lamb da calculi

Denition In a biccc dene Bo ole the typ e of b o oleans

B ool es most salient feature is that it has two distinguished global elements b o olean values

T F B ool e corresp onding to the two injections in in together with the universal prop erty

1 2

of copro ducts In S et we interpret B ool e as a set of cardinality similarly in typ ed lamb da

calculus it corresp onds to a typ e with two distinguished constants T F B ool e and an appropriate

notion of denition by cases In any biccc we can dene all of the classical nary prop ositional

n

logic connectives as arrows B ool e B ool e see LS I A weaker notion of b o oleans in the

category CPO is illustrated in Figure

0 S

Denition A natural numbers object in a ccc C is an ob ject N with arrows N N

a h

which is initial among diagrams of that shap e That is for any ob ject A and arrows A A

there is a unique iterator I N A making the following diagram commute

ah

S

N N

I I

ah ah

a

R

h

A A

A weak natural numbers object is dened as ab ove but just assuming existence and not necessarily

uniqueness of I

ah

f n t

H

H

H

B N

H

Figure Flat Datatyp es in CPO

In the category S et the natural numb ers N S is a natural numb ers ob ject where S n n

C

In functor categories S et a natural numb ers ob ject is given by the constant functor K where

N

0 S

K A N and K f id with obvious natural transformations K K In

N N N N N

CPO there are numerous weak natural numb ers ob jects for example the at pointed natural

numb ers N N fg ordered as follows a b i a b or a where S n n and

S pictured in Figure

Natural numb ers ob jectswhen they existare unique up to isomorphism however weak ones

are far from unique Typical programming languages and typ ed lamb da calculi in logic assume

only weak natural numb ers ob jects

If a ccc C has a natural numb ers ob ject N we can construct parametrized versions of iteration

using pro ducts and exp onentiation in C LS FrSc For example in S et given functions g A

B and f N A B B there exists a unique primitive recursor R N A B satisfying i

g f

R a g a and R S n a f n a R n a These equations are easily represented in any

g f g f g f

ccc with N or in the asso ciated typ ed lamb da calculus eg the numb er n N b eing identied

n

with S In the case C has only a weak natural numb ers ob ject we may prove the existence but

not necessarily the uniqueness of R

g f

An imp ortant datatyp e in Computer Science is the typ e of nite lists of elements of some typ e

A This is dened analogously to weak natural numb ers ob jects

Denition Given an ob ject A in a ccc C we dene the ob ject istA of nite lists on A

with the following distinguished structure arrows nil istA cons A istA istA

satisfying the following weak universal prop erty for any ob ject B and arrows b B and

h A B B there exists an iterator I istA B satisfying in the internal language

bh

I nil b I cons ha w i hha I w i

bh bh bh

Here nil corresp onds to the empty list and cons takes an element of A and a list and concatenates

the element onto the head of the list

Analogously to weak natural numb ers ob jects N we can use pro duct typ es and exp onentiation

to extend iteration on istA to primitive recursion with parameters cf GLT p

n

What nary numerical S et functions are represented by arrows N N in a ccc The answer

of course dep ends on the ccc In general the b est we could exp ect is the following cf LS

Part I I I Section

Prop osition Let F be the free ccc with weak natural numbers object The class of numerical

N

total functions representable therein is properly contained between the primitive recursive and the

Turingmachine computable functions

In general such fastgrowing functions as the Ackermann function are representable in any ccc

with weak natural numb ers ob ject see LS Analogous results hold for symmetric monoidal

and monoidal closed categories PR

The question of strong versus weak datatyp es is of some interest For example although

we can dene addition N N N by primitive recursion on a weak natural numb ers typ e

commutativity of addition follows from having a strong natural numb ers ob ject a weak parametrized

primitive recursor would only allow us to derive x n n x for each closed numeral n but we

cannot then extend this to variables this is similar to consequences of Godels incompleteness

theorem cf LS p Notice that on the face of it the denition of a natural numb ers

ob ject app ears not to b e equational informally uniqueness of the arrow I requires an implication

ah

for al l f N A if f a and f S hf then f I

ah

Here we remark on a curious observation of Lamb ek L Let us recall from universal algebra

3

that a Malcev op erator on an algebra A is a function m A A satisfying m xxz z and

A A

1

m xz z x For example if A were a group m xy z is such an op erator Similarly the

A A

m

A

3

A denition of a Malcev op erator on an ob ject A makes sense in any ccc eg as an arrow A

satisfying some diagrams or equivalently in any typ ed lamb da calculus eg as a closed term

3

m A A satisfying some equations

A

Theorem Lamb ek Let C be a ccc with weak natural numbers N S in which each object

A has a Malcev operator m Then the fact that N S is a natural numbers object is equational ly

A

denable using the family fm j A C g In particular if C F the free ccc with weak natural

A N

numbers object there are a nite number of additional equations as schema that when added to

the original data guarantee that every type has a Malcev operator and N is a natural numbers

object

Polymorphism

The p erplexing sub ject of p olymorphism

C Darwin Life Lett

Although Darwin was sp eaking of biology he might very well have b een discussing computer

science years later Christopher Strachey in the s intro duced various notions of p oly

morphism into programming language design see Rey Mit Perhaps the most inuential

was his notion of parametric polymorphism Intuitively a parametric p olymorphic function is one

which has a uniformly given algorithm at al l types Imagine a generic algorithm capable of b eing

instantiated at any arbitrary typ e but which is the same algorithm at each typ e instance It is

this idea of the plurality of form which inspired the biological metaphor

Example Reverse Consider a simple algorithm that takes a nite list and reverses it

Here lists could mean lists of natural numb ers lists of reals lists of arrays indeed lists of lists of

The p oint is the typ es do not matter we have a uniform algorithm for all typ es Let l ist

denote the typ e of nite lists of entities of typ e We thus might typ e this algorithm

r ev l ist l ist

where r ev a a a a

1 n n 1

A second example discussed by Strachey is

Example Maplist This algorithm b egins with a function of typ e and a nite

list applies the function to each element of the list and then makes a list of the subsequent

values We might represent it as

map l ist l ist

where map f a a f a f a

1 n 1 n

Formulas A v bl j A A j A

1 2

Provability is a relation b etween nite sets of formulas

and formulas

A if A

fAg B

A A B

A B B

A A

A AB

where FV for any formula B

Figure Second Order Intuitionistic Prop ositional Calculus

Many recent programming languages eg ML Ada supp ort sophisticated uses of generic typ es

and p olymorphism The mathematical foundations of such languages were a ma jor challenge in the

past decade and category theory played a fundamental role We shall briey recall the issues

Polymorphic lambda calculi

The logician JY GirardGi Gi in a series of imp ortant works examined higherorder logic

from the CurryHoward viewp oint He develop ed formal calculi of variable typ es the socalled

p olymorphic lamb da calculi which corresp ond to pro ofs in higherorder logics At the same time

he develop ed the pro of theory of such systems J ReynoldsRey indep endentl y discovered the

secondorder fragment of Girards system and prop osed it as a syntax representing Stracheys

parametric p olymorphism

Let us briey examine Girards System F second order p olymorphic lamb da calculus The

underlying logical system is intuitionistic second order prop ositional calculus The latter theory

is similar to ordinary prop ositional calculus except we can universally quantify over prop ositional

variables

The syntax of second order prop ositional calculus is presented in Figure

The usual notions of free and b ound variables in formulas are assumed For example in

is a b ound variable while is free AB denotes A with formula B substi

tuted for free changing b ound variables if necessary to avoid clashes Notice in the quantier

rules that when we instantiate a universally quantied formula to obtain say AB the

formula B may b e of arbitrary logical complexity Thus inductive pro of techniques based on the

complexity of subformulas are not available in higherorder logic This is the essence of the problem

of impredicativity in p olymorphism

We now intro duce Girards second order lamb da calculus We use the notation FV t and

BV t for the set of free and b ound variables of term t resp ectively We write FTV A and

BTV A for the set of free typ e variables and b ound typ e variables of formula A resp ectively

Denition Girards System F

Types Freely generated from typ e variables by the rules if A B are typ es so are A B

and A

A

Terms Freely generated from variables x of every typ e A by

i

Firstorder lamb da calculus rules if f A B a A B then f a B and

x:A

A B

Sp ecically secondorder rules

a If t A then t A where FTV FV t

b If t A then tB AB for any typ e B

Equations Equality is the smallest congruence relation closed under and for b oth lamb das

that is

1 1

a ax and f x f where x FV f

x:A x:A

2 2

t where FTV t B and t B

Equations are the rst order equations while equations are second order

From the CurryHoward viewp oint the typ es of F are precisely the formulas of second order

prop ositional calculus Fig while terms denote pro ofs For example to annotate second order

rules we have

x t A x t A

x t A x tB AB

The equations of course express equality of pro of trees

What ab out p olymorphism Supp ose we think of a term t A as an algorithm of typ e

A varying uniformly over all typ es Then tB AB is the instantiation of t at the sp ecic

typ e B Moreover B may b e arbitrarily complex Thus the typ e variable acts as a parameter

In System F we can internally represent common inductive data typ es within the syntax as weak

T algebras for covariant denable functors T Weakness refers to the categorical fact that these

structures satisfy existence but not uniqueness of the mediating arrow in the universal mapping

prop erty Thus for any typ es A B we are able to dene the typ es N at ListA A B A

B A etc see GLT for a full treatment

Let us give two examples and at the same time illustrate p olymorphic instantiation

Example The typ e of b o oleans is given by

B ool e

It has two distinguished elements T F B ool e given by T x and F y

x: y : x: y :

together with a Denition by Cases op erator for each typ e A D A A B ool e A

A

dened by D uv t tAuv where u v A t B ool e One may easily verify that D uv T u

A A

1 2

and D uv F v where stands for

A

Example The typ e of Church numerals

N at

n n

o o o

The numeral n N at corresp onds at each to nfold comp osition f f where f f f f

0 n

n times and f I d x x Formally it is the closed term n f N at Thus for

f :

n

any typ e B we have a uniform algorithm nB f B B B B Successor is

f :B B

n+1 n

o o

given by S n where n f f f f nf

n:N at f : f : f :

Finally iteration is given by if a A h A A I xAha N at A The reader

ah x:N at

may easily calculate that I a and I n hI n for numerals n

ah ah ah

Let us illustrate impredicativity in this situation Recall the discussion of Church vs Curry

typing Section Notice that for any typ e B nB B nB makes p erfectly go o d sense In

particular let B N at the typ e of n itself This is a welldened term and if we erase all its typ es

n

we obtain the untyped expression nn f f This latter untyp ed term is not typable in simply

typ ed lamb da calculus

Formal systems describing far more p owerful versions of p olymorphism have b een develop ed

For example Girards thesis describ ed the typ ed lamb da calculus corresp onding to order intu

itionist typ e theory socalled F Programming in the various levels of Girards theories fF g

n

n is describ ed in PDM Other systems include Co quandHuets Calculus of

Constructions and its extensions Luo These theories include not only Girards F but also

MartinLof s dep endent typ e theories Ha Indeed these theories are among the most p ower

ful logics known yet form the basis of various pro ofdevelopment systems eg LEGO and Co q

+

LP D

What is a Model of System F

The problem of ndingand indeed dening preciselya mo del of System F was dicult

Cartesian closedness is not the issue The problem of course is the universal quantier clearly

in A the is to range over all the ob jects of the mo del and at the same time should b e

interpreted as some kind of pro duct over al l ob jects Such large pro ducts create havo c as

foreshadowed in the following theorem of Freyd cf Mac Prop osition p

Theorem Freyd A smal l category which is smal l complete is a preorder

Cartesian closed preorders eg complete Heyting algebras are of no interest for mo delling pro ofs

we seek nontrivial categories

Supp ose instead we try to dene a naive settheoretic mo del of System F in which

have their usual meaning and is interpreted as a large pro duct Such mo dels are dened in

detail in RP Pi John Reynolds proved the following

Theorem There is no S et model for System F

There is an elegant categorical pro of in Reynolds and Plotkin RP Let us sketch the pro of which

applies to somewhat more general categories than S et

Let C b e a category with an endofunctor T C C A T algebra is an ob ject A together with

a

an arrow TA A A morphism of Talgebras is a commutative square

T f

TA TB

a b

f

A B

An initial T algebra resp weakly initial T algebra is one for which there exists a unique morphism

resp there exists a morphism to any other T algebra

We shall b e interested in ob jects and arrows of the mo del category C which are denable

ie denoted by typ es and terms of System F There are simple covariant endofunctors T on C

whose action on ob jects is denable by typ es and whose actions on arrows is denable by terms of

System F For example the identity functor T and the functor T B B

for any xed B have this prop erty

Now it may b e shown see RP that for any denable functor T the System F expression

P T is a weakly initial T algebra Suppose the ambient model category C

has equalizers of al l subsets of arrows eg S et has this prop erty Essentially by taking a large

equalizer cf the Solution Set Condition in Freyds Adjoint Functor Theorem Mac p we

could then construct a subalgebra of P which is an initial T algebra Call this initial T algebra I

We then use the following imp ortant observation of Lamb ek

f

Prop osition Lamb ek If T I I is an initial T algebra then f is an isomorphism

Applying this to the denable functor T B B we observe that T I I In

particular let C S et and B B ool e and take the usual S et interpretation of as cartesian

pro duct and as the full function space Notice car dB since there are always the two

I

B

distinct closed terms T and F Hence we obtain a bijection B I for some set I which is

imp ossible for cardinality reasons 2

The search for mo dels of System F led to some extraordinary phenomena that had considerable

inuence in semantics of programming languages Let us just briey mention the history Notice

that the ReynoldsPlotkin pro of dep ends on a simple cardinality argument which itself dep ends

on classical set theory Similarly the pro of of Freyds result Theorem dep ends on using

classical ie nonintuitionistic logical reasoning in the metalanguage This suggests that it is

really the nonconstructive nature of the category Sets that is at fault if we were to work within a

nonclassical universesay within a mo del of intuitionistic set theorythere is still a chance that

we could escap e the ab ove problems but still have a settheoretical mo del of System F And

from one p oint of view that is exactly what happ ened

These ambient categories called top oses LS MM are in general mo dels of intuitionistic

higherorder logic or settheory and include such categories as functor categories and sheaves

on a top ological space as well as Sets Moggi suggested constructing mo dels of System F based

on an internally complete internal full sub category of a suitable ambient top os This ambient

top os would serve as our constructive settheory and function typ es would still b e interpreted as

the full settheoretical space of total functions M Hyland Hy proved that the Realizabili ty

or Eective Top os had nontrivial such internal category ob jects The dicult development

and clarication of these internal mo dels was undertaken by many researchers eg D Scott

M Hyland E Robinson P Rosolini A Carb oni P Freyd A ScedrovA Pitts et al eg

HRR Rob Ros CPS Pi

In a separate development R Seely See gave the rst general categorical denition of a

socalled external model of System F and more generally F The denition was based on the

theory of indexed or bred categories This view of logic was pioneered by LawvereLaw who

emphasized that quantiers were interpretable as adjoint functors PittsPi claried the relation

ship b etween Seelys mo dels and internalcategory mo dels within ambient top oses of presheaves

Moreover he showed that there are enough such internal mo dels for a Completeness Theorem It

is worth remarking that Pitts work uses prop erties of Yoneda emb eddings For general exp ositions

see AL Extensions of settheoretical mo dels to cases where function spaces include partial

functions ie nontermination is in RR

One can externalize these internal category mo dels Hy AL to obtain ordinary categories

And one such internal category in the Realizabili ty Top os the modest sets when externalized is

precisely the ccc category P er N discussed in Section

Prop osition P er N is a model of System F

The idea is that in addition to the ccc structure of P er N we interpret as a large intersection

the intersection of an arbitrary family of p ers is again a p er We shall return to this example in

Section

Ironically in essence this mo del was already in Girards original Phd thesis Gi Later

domaintheoretic mo dels of System F were considered by Girard in Gi and were instrumental

in his development of linear logic

The Untyp ed World

The advantages of typ es in programming languages are familiar and welldo cumented eg Mit

Nonetheless there is an underlying untyped asp ect of computation already going back to the ori

ginal work on lamb da calculus and combinatory logic in the s which often underlies concrete

machine implementations In this early view develop ed by Church Curry and Schonnkel func

tions were understo o d in the oldfashioned preCantor sense of rules as a computational pro cess

of going from an argument to a value Such a functional pro cess could take anything even itself

as an argument Let us just briey mention some key directions see Bar AGM

Models and Denotational Semantics

From the viewp oint of cccs untyp edness amounts to nding a ccc C with an ob ject D satisfying

the isomorphism

D

D D

Thus function spaces and elements are on the same level It then makes sense to dene formal

=

D D

application f g for constants f g D by f g ev f g where D D is the isomorphism

ab ove In particular selfapplication f f makes p erfectly go o d sense

Dana Scott found the rst semantical top ological mo dels of the untyp ed lamb da calculus in

Sc ie nontrivial solutions D to equations of the form in various cccs p erhaps the

simplest b eing in CPO This was part of his general investigations with Christopher Strachey

into the foundations of programming languages culminating in the socalled ScottStrachey ap

proach to the semantics of programming languages Arguably this has b een one of the ma jor

arenas in the use of category theory in Computer Science with an enormous literature For an

intro duction see AbJu Gun Ten

More generally one seeks to nd nontrivial domains D satisfying certain socalled recursive

domain equations of the form

D D

where D is some expression built from typ e constructors The diculty is that the variable

D may app ear b oth co and contravariantly Such recursive dening equations are used to sp ecify

the semantics of numerous notions in computer science from datatyp es in functional programming

languages to mo delling nondeterminism concurrency etc cf also DiCo

The seminal early pap er on categorical solutions of domain equations is the pap er of Smyth and

Plotkin SP More recent work has fo cussed on Axiomatic and Synthetic Domain Theory eg

AbJu FiPl ReSt and use of bisimulations and relationtheoretic metho ds for reasoning

ab out recursive domains Pia These metho ds rely on fundamental work of Peter Freyd on

recursive typ es eg Fre

CMonoids and Categorical Combinators

On a more algebraic level a mo del of untyp ed lamb da calculus is a ccc with up to isomorphism

one nontrivial ob ject That is a ccc C with an ob ject D satisfying the domain equations

D

D D D D

An example of such a D in CPO is given in LS using the constructions of D Scott and

SmythPlotkin mentioned ab ove An interesting axiomatization of such D s comes from simply

D

considering H om D D H om D as an abstract monoid It turns out that the axioms are

C C

easy to obtain take the axioms of a ccc remove the terminal ob ject and erase al l the types That

is following the treatment in LS p

o

Denition A Cmonoid C for Curry Church Combinatory or CCC is a monoid M id

together with extra structure structure h i where are elements of M

1 2 i

is a unary op eration on M and h i is a binary op eration on M satisfying untyp ed versions of

the equations of a ccc cf Figure

ha bi a hh i h

1 1 2

ha bi b

2

hk i k

1 2

h c ci c

1 2

o

for any a b c h k M where we elide the monoid op eration

C monoids were rst discovered indep endently by D Scott and J Lamb ek around The

elementary algebraic theory and connections with untyp ed lamb da calculus were develop ed in

LS and indep endently in Cur where they were called categorical combinators Obviously

C monoids form an equational class thus just like for general cccs we may form free algebras

p olynomial algebras prove Functional Completeness etc The asso ciated internal language is

untyped lambda calculus with pairing operators As ab ove this language is obtained from simply

typ ed lamb da calculus by omitting the typ e and erasing all the typ es from terms

The rewriting theory of categorical combinators has b een discussed by Curien Hardin et al

eg see Har Categorical combinators form a particularly ecient mechanism for implement

ing functional languages for example the language CAML is a version of the functional language

ML based on categorical combinators see Hu Part

The deep est mathematical results to date on the cartesian fragment of C monoids were ob

tained by R Statman St Statman characterizes the free cartesian monoid F in terms of a

representation into certain continuous shift op erators on Cantor space as well as characterizing

the nitely generated submonoids of F and the recursively enumerable subsets of F The latter

two results are based on pro jections of suitably enco ded unication problems

Church vs Curry Typing

The fundamental feature of the untyp ed lamb da calculus is selfapplication The rule xxa

ax is now totally unrestricted with resp ect to typing constraints This p ermits nonhalting

computations for example the term xxxxxx has no normal form and only

def

reduces to itself while the xed p oint combinator Y f xf xxxf xx satises

def

f Yf Y f hence Y f is a xed p oint of f for any f Hence we immediately obtain al l terms

of the untyped lambda calculus have a xed point

Untyp ed lamb da calculus suggests a dierent approach to the typ ed world typing untyp ed

terms The Church view which we have adopted here insists all terms b e explicitly typ ed starting

with the variables On the other hand Curry the founder of the related but older sub ject of

Combinatory Logic had a dierent view start with untyp ed terms but add typ e inference rules

to algorithmically infer appropriate typ es when p ossible Many mo dern typ ed programming

languages eg ML essentially follow this Curry view and use typing rules to assign appropriate

typ e schema to untyp ed terms This leads to the socalled Typ e Inference Problem given an

E r ase

explicitly typ ed language L and typ e erasure function L U where U is untyp ed lamb da

calculus decide if an untyp ed term t satises t E r aseM for some M L It turns out that a

problem of typ e inference is essentially equivalent to a socalled unication problem familiar from

Logic Programming cf Mit Fortunately in the case of ML and other typ ed programming

languages there are known typ e inference algorithms however in general eg for System F F

the problem is undecidable To the b est of our knowledge the ChurchvsCurry view of typ ed

languages has not yet b een systematically analyzed categorically

Logical Relations and Logical Permutations

Logical relations play an imp ortant role in the recent pro of theory and semantics of typ ed lamb da

calculi Mit Plo St Recall the notion of Henkin model Section as a sub ccc of Set

Denition Given two Henkin mo dels A and B a logical relation from A to B is a family of

binary relations R fR A B j a typ e g satisfying we write aR b for a b R

R

1

a a R b b if and only if aR b and a R b for any a a A b b B ie

ordered pairs are related exactly when their components are

For any f A g B f R g if and only if for all a A b B

aR b implies f a R g b ie functions are related when they map related inputs to related

outputs

For each atomic base typ e b x a binary relation R A B Then there is a smallest family

b b b

of binary relations R fR A B j a typ e g dened inductively from the R s by

b

ab ove That is any prop erty relation at basetyp es can b e inductively lifted to a family R at all

higher typ es satisfying ab ove We write aRb to denote aR b for some If A B and R

is a logical relation from A to itself we say an element a A is invariant under R if aRa

The fundamental prop erty of logical relations is the Soundness Theorem Mit St Let

x M denote that M is a term of typ e with free variables x in context Consider

Henkin mo dels A with an assignment function assigning variables to elements in A Let M

A

A

denote the meaning of term M in mo del A wrt the given variable assignment following Mit

we only consider assignments such that x A if x The following is proved by

A i i

induction on the form of M

Theorem Soundness Let R A B be a logical relation between Henkin models

A B Let x M Suppose assignments of the variables are related ie for al l

A B

M M x R x x Then R

i A i B i

A B

In particular if A B and M is a closed term ie contains no free variables its meaning

M in a model A is invariant under al l logical relations This holds also for languages which have

c is invariant for al l such constants c constants at base types by assuming

This result has b een used by Plotkin Statman Sieb er et al Plo Sie St to show

certain elements of mo dels are not lamb da denable it suces to nd some logical relation on

A for which the element in question is not invariant

There is no reason to restrict ourselves to binary logical relations one may sp eak of nary

logical relations which relate n Henkin mo dels St Indeed since Henkin mo dels are closed

under pro ducts it suces to consider unary logical relations known as logical predicates

Example Hereditary Permutations Consider a Henkin mo del A with a sp ecied p er

mutation A A at each base typ e b We extend to all typ es as follows i on pro duct

b b b

typ es we extend comp onentwise A A ii on function spaces extend

1

o o

by conjugation f f where f A We build a logical relation R on A by

letting R the graph of p ermutation A A ie R a b a b Memb ers of R

will b e called hereditary permutations Rinvariant elements a A are simply xed p oints of the

p ermutation a a

Hereditary p ermutations and invariant elements also arise categorically by interpretation into

Z

S et

Z

Prop osition The category S et of left Z sets is equivalent to the category whose objects are

sets equipped with a permutation and whose maps equivariant maps are functions commuting

Z

with the distinguished permutations Invariant elements of A are arrows A S et

Z

Alas S et is not a Henkin mo del is not a generator In the next section we shall slightly

generalize the notion of logical relation to work on a larger class of structures

Logical Relations and Syntax

Logical predicates also called computability predicates originally arose in pro of theory as a

technique for proving normalization and other syntactical prop erties of typ ed lamb da calculi

LS GLT Later Plotkin Statman and Mitchell Plo St Mit constructed logical re

lations on various kinds of structures more general than Henkin mo dels Following Statman and

Mitchell we extend the notion of logical relation to certain applicative typed structures A for which

are welldened M i appropriate meaning functions on the syntax of typ ed lamb da calculus

A

and ii all logical relations R are in a suitable sense congruence relations with resp ect to the syn

tax This guarantees that the meanings of lamb da abstraction and application b ehave appropriately

under these logical relations Following Mit St we call them admissible logical relations

The Soundness Theorem still holds in this more general setting now using admissible logical

relations on applicative typ ed structures see Mit Lemma

Example Let A b e the hereditary p ermutations in Example Consider a free simply

typ ed lamb da calculus without constants Then as a corollary of Soundness we have the meaning

of any closed term M is invariant under al l hereditary permutations This conclusion is itself a

Z

consequence of the universal prop erty of free cartesian closed categories when interpreted in S et

cf Corollary and Lauchlis Theorem

Remark The rewriting theory of lamb da calculi is a prototyp e for Op erational Semantics

of many programming languages recall the discussion after the rule Corollary See also

Section b elow on PCF Logical Predicates socalled computability predicates were rst intro

duced to prove strong normalization for simply typ ed lamb da calculi with natural numb ers typ es

by W Tait in the s Highly sophisticated computability predicates for p olymorphically typ ed

systems like F and F were rst intro duced by Girard in his thesis Gi Gi For a partic

ularly clear presentation see his b o ok GLT These techniques were later revisited by Statman

and Mitchellusing more general logical relationsto also prove ChurchRosser and a host of other

syntactic and semantic results for such calculi see Mit

For general categorical treatments of logical relations see Mit MitSce MaRey and references

there Uses of logical relations in op erational semantics of typ ed lamb da calculi are covered in

AC Mit A categorical theory of logical relations applied to data renement is in KOPTT

Use of op erationallybased logical relations in programming language semantics is in Pib Pi

For techniques of categorical rewriting applied to lamb da calculus see JGh

Example ReductionFree Normalization

The op erational semantics of calculi have traditionally b een based on rewriting theory or pro of

theory eg normalization or cutelimination ChurchRosser etc More recently Berger and

Schwichtenb erg BS gave a mo deltheoretic extraction of normal formsa kind of inverting

of the canonical settheoretic interpretation used in Friedmans Completeness Theorem cf

b elow

In this section we sketch the use of categorical metho ds essentially from Yonedas Lemma

cf Theorem to obtain the BergerSchwichtenb erg analysis A rst version of this technique

was develop ed by Altenkirch Homan and Streicher AHS AHS The analysis given here

comes from the article CDS which also mentions intriguing analogues to the JoyalGordon

PowerStreet techniques for proving coherence in various structured bicategories The essential

idea common to these coherence theorems is to use a version of Yonedas lemma to emb ed into a

stricter presheaf category

To actually extract a normalization algorithm from these observations requires us to construct

ively reinterpret the categorical setting in P Set as explained b elow This leads to a nontrivial

example of program extraction from a structured pro of in a manner advo cated by MartinLof and

his scho ol ML Dy Ha CD The reader is referred to CDS for the ne details of the

pro of

In a certain sense the results sketched b elow are dual to Lamb eks original goal of categorical

pro of theory L L in which he used cutelimination to study categorical coherence problems

Here we use a metho d inspired from categorical coherence pro ofs to normalize simply typ ed lamb da

terms and thus intuitionistic pro ofs

Categorical Normal Forms

Let L b e a language T the set of Lterms and a congruence relation on T One way to decide

whether two terms are congruent is to nd an abstract normal form function ie a computable

function nf T T satisfying the following conditions for some ner congruence relation

NF nf f f

NF f g nf f nf g

NF

NF is decidable

From NF NF and NF we see that f g nf f nf g This clearly p ermits a decision

pro cedure to decide if two terms are related compute nf of each one and see if they are

related using NF The normal form function nf essentially reduces the decision problem of

to that of This view is inspired from CD

Here we let L b e typ ed lamb da calculus T the set of terms b e conversion and

b e congruence Let us see heuristically how category theory can b e used to give simply typ ed

calculus a normal form function nf

Recall that terms mo dulo conversion determine the free ccc F on the set of

X

2

sorts atoms X By the universal prop erty for any ccc C and any interpretation of the

atoms X in obC there is a unique up to iso cccfunctor F C freely extending this

X

op

F

X

interpretation Let C b e the presheaf category S et There are two obvious cccfunctors i the

op

F

X

Yoneda emb edding Y F S et cf and ii if we interpret the atoms by Yoneda there

X

We actually use the free ccc of sequences of terms as dened by Pitts Pi

op

F

X

is also the free extension to the cccfunctor F S et By the universal prop erty there

X

By the Yoneda lemma we shall invert the interpretation is a natural isomorphism q Y

on each homset according to the following commutative diagram

op

F

X

A B F A B S et

X

S

So

S

S

Y

1

o o

S q q

A A B

S

A

S

S

Sw

S

op

F

X

S et Y A Y B

1

f

q

q

B

A

A B That is for any f F A B we obtain natural transformations Y A Y B

X

1

q

f

A

AA

Then evaluating these transformations at A gives the Set functions F A A A A

X

q

B A

F A B Hence starting with F A A we can dene an nf function by B A

X A X

1

nf f q q f

def B A A A

AA

Clearly nf f F A B But alas by Yonedas Lemma nf f f Indeed this is just a

X

restatement of part of the Yoneda isomorphism But all is not lost recall NF says nf f f

This suggests we should reinterpret the entire categorical argument including the use of functor

categories and Yonedas Lemma in a setting where b ecomes a partial equivalence

relation Diagrams which previously commuted now should commute up to

This viewp oint has a long history in constructive mathematics where it is common to use sets

X equipp ed with explicit equivalence relations in place of quotients X b ecause of problems

with the Axiom of Choice Thus along with sp ecifying elements of a set one must also say what

it means for two elements to b e equal see Bee

P category theory and normalization algorithms

Motivated by enriched category theory K Bor this view leads to the setting of P category

theory in CDS In P category theory i homsets are P Sets ie sets equipp ed with a partial

equivalence relation p er ii all op erations on arrows are P Set maps ie preserve iii func

tors are versions of enriched functors iv P functor categories and P natural transformations

are versions of appropriate enriched structure etc One then proves a P version of Yonedas

lemma In essence P category theory is the development of ordinary enriched category theory in

a constructive setting where equality of arrows is systematically replaced by explicit p ers making

sure every op eration on arrows is a congruence with resp ect to the given p ers For an example see

Figure

Now consider the free ccc F as a P category where the arrows are actually sequences of

X

terms and the p er on arrows is equality Analogously to the ab ove freeness in the

P setting yields a unique P ccc functor

op

(F )

X

F P S et

X

where atoms X X are interpreted by P Yoneda ie as H om X Just as in the ordinary

(F )

X

Y case the P Yoneda functor Y is a P ccc functor so we have a P natural isomorphism q

of P ccc P functors

P Products

c c for any constant c f g

A 1 2

f f for any f f A

f g implies hf f i hg g i hf f i f where f g C A

i i 1 2 1 2 i 1 2 i i i i

h k k i k for k C A A

1 2 1 2

P Exponentials

ev

A

ev ev for B A B

A

h h implies h h C B where h h C A B

h h implies ev hh i h

1 2

A

l l implies ev hl i l C B

1 2

P Functor

f f implies F f F f for all f f

f f g g implies F g f F g F f for all comp osable f g and f g

F id id

A F (A)

= =

F A B F FA FB Sp ecied P isomorphisms

=

FA A

FB F B

Figure P cccs and P ccc Functors

op

1 (F )

X

q q F P Set

X

Y

In this setting nf as dened by Equation will b e a p erpreserving function on terms themselves

and not just on equivalence classes of terms recall that classically a free ccc has for its arrows

equivalence classes of terms mo dulo the appropriate equations Arguing as b efore but now using

the P Yoneda isomorphism it follows immediately that nf is an identity P function But this

means nf f f which is precisely the statement of NF Moreover the part of P category

theory that we use is constructive in the sense that all functions we construct are algorithms

Therefore nf is computable

It remains to prove NF f g nf f nf g This is the most subtle p oint Here to o the

op

C

P version of a general categorical fact will help us cf the P presheaf category P Set is a

P ccc for any P category C In particular let C b e the P category F of sequences of terms

X

up to change of b ound variable This is a trivially decidable equivalence relation on terms

called congruence in the literature and obviously Note that this P category has the

same ob jects and arrows as F but the p ers on arrows are dierent

X

By the freeness of F we have another interpretation P functor

X

op

(F )

X

F P Set

X

where we interpret atoms X X by the P presheaf H om X The key fact is that this

(F )

X

P functor That is has exactly the same settheoretic eect on ob jects and arrows as

one proves by induction

Lemma For al l objects C and arrows f in F j C j j C j and similarly j f j

X

j f j where j j means taking the underlying settheoretic structure

Hence we can conclude that f g implies f g here refers to the p er on arrows in

op

(F )

1 1

X

P S et We can show that q and q are natural in particular q and q preserve

B B A

B B A

It then follows that nf f nf g as desired

Remark

i The normal forms obtained by this metho d can b e shown to coincide with the socalled long

normal forms used in lamb da calculus CDS

ii The direct inductive pro ofs used ab ove corresp ond more naturally to a moreinvolved bicat

egorical denition of freeness CDS Remark

Finally in CDS it is shown how to apply the metho d to the word problem for typ ed

calculi with additional axioms and op erations ie to freelygenerated cccs mo dulo certain theories

This employs appropriate free P cccs over a P category a P cartesian category etc These are

generated by various notions of theory which are determined not only by a set of atomic typ es

but also by a set of basic typ ed constants as well as a set of equations b etween terms Although the

Yoneda metho ds always yield an algorithm nf it do es not necessarily satisfy NF the decidabili ty

of What is obtained in these cases is a reduction of the word problems for such free cccs to

those of the underlying generating categories

Example PCF

The language PCF due to Dana Scott in has deeply inuenced recent programming language

theory Much of this inuence arises from seminal work of Gordon Plotkin in the s on op era

tional and denotational semantics for PCF We shall briey outline the syntax and basic semantical

issues of the language following from Plotkins work We follow the treatment in AC Sie

although the original pap er Plo is highly recommended

PCF

The language PCF is an explicitly typ ed lamb da calculus with the following structure

Typ es Generated from nat bool e by

Lamb da Terms generated from typ ed variables using the following sp ecied constants

z er o nat bool e n nat for each n N

cond bool e nat nat nat T bool e

nat

cond bool e bool e bool e bool e F bool e

boole

succ nat nat

Y pr ed nat nat

Categorical Models

The standard model of PCF is dened in the ccc CPO as follows interpret the base typ es as

in Figure nat N bool e B function space in CPO

succ pr ed N at N at Interpret constants as follows for clarity we omit writing

2

z er o N at B cond bool e fnat bool eg cond is for conditional sometimes

called if then el se T t F f n n

xa ax z er o t

Y f f Y f z er on f

succn n cond tab a

pr edn n cond f ab b

f g f g

where u fsucc pr ed z er og

f a g a uf ug

f g

cond f ab cond g ab where cond xy z condxy z

Figure Op erational Semantics for PCF

y if p t

x if x

cond p y z z if p f succx

if x

if p

x if x

pr edx

denoting non is the least element of

else

termination or divergent

t if x

f if x

z er ox

Y is the least xed p oint op erator

W

n

if x

Y f ff j n g see Example

More generally a standard model of PCF is an CPOenriched ccc C in which each homset C A B

has a smallest element A B with the following prop erties i pairing and currying are

AB

o o

monotonic ii f f and ev h f i for all f of appropriate typ e iii there are ob jects nat

and bool e whose sets of global elements satisfy C nat N and C bool e B and in which

the constants are all interpreted in the internal language of C as in the standard mo del ab ove eg

o

interpreting succx by ev hsucc xi etc A mo del is orderextensional if is a generator and the

order on homsets coincides with the p ointwise ordering

The op erational semantics of PCF is given by a set of rewriting rules displayed in Figure

This is intended to describ e the dynamic evaluation of a term as a sequence of step transitions

The xedp oint combinator Y guarantees some computations may not terminate It is imp ortant

to emphasize that in op erational semantics with partiallydened ie p ossibly nonterminating

computations dierent orders of evaluation eg leftmost outermost vs innermost may lead to

nontermination in some cases and may also eect eciency etc See Mit Chapter We

have chosen a simple op erational semantics for PCF given by a deterministic evaluation relation

following AC

Adequacy

A PCF program is a closed term of base typ e ie either nat or bool e The observable behaviour

of a PCF program P nat is the set B ehP fn N j P ng and similarly for P bool e

The set B ehP is either empty if P diverges or a singleton if P converges to a necessarily unique

normal form The following theorem is proved using a logical relations argument

Theorem Computational Adequacy Let C be any standard model of PCF Then for al l

programs P nat and n N

P n i P n

and similarly for P bool e Hence P Q i their sets of behaviours are equal

We are interested in a notion of observational equivalence arising from the op erational se

mantics A program a closed term of base typ e can b e observed to converge to a sp ecic numeral

or b o olean value More generally what can we observe ab out arbitrary terms of any typ e The idea

is to plug them into arbitrary program co de and observe the b ehaviour More precisely a program

context C is a program with a hole in it the hole has a sp ecied typ e which is such that if

we formally plug a PCF term M into the hole of C we dont care ab out p ossible clashes of

b ound variables we obtain a program C M We are lo oking at the convergence b ehaviour with

resp ect to the op erational semantics of the resulting program cf AC

Denition Two PCF terms M N of the same typ e are observational ly equivalent denoted

M N i B ehC M B ehC N for every program context C

That is M N means that for all program contexts C C M c i C N c for

C M c either a b o olean value or a numeral Thus by the previous Theorem M N i

C N In order to prove observational equivalence of two PCF terms R Milner showed it suces

to pick applicative contexts ie

Lemma Milner Two closed PCF expressions M N nat

1 2 n

MP P NP P for al l closed P i n are observational ly equivalent i

1 n 1 n i i

Finally the main denition of the sub ject is

Denition Full Abstraction A mo del is called ful ly abstract if observational equivalence

coincides with denotational equality in the mo del ie for any two PCF terms M N

M N i M N

Which mo dels are fully abstract There are two main theorems due to Milner and Plotkin

First we intro duce the parallelor function por B B B on the standard mo del of PCF

t if a t or b t

por a b f if a b f

else

Theorem Plotkin

por is not denable in PCF

The standard model is not ful ly abstract

The standard model is ful ly abstract for the language PCF por

The pro of of the rst two parts of the theorem use logical relations Sie AC Gun In

R Milner proved the following Mil

Theorem Milner There is a unique up to isomorphism ful ly abstract orderextensional

model of PCF

Milners construction is syntactical so the question b ecame nd a more mathematical ie

not explicitly syntactical characterization of the unique fully abstract mo del This is related to the

Full Completeness Problems discussed in Section A satisfying solution to the Full Abstraction

Problem for PCF was recently given by S Abramsky R Jagadeesan and P Malacaria and also M

Hyland and L Ong who use various monoidal categories of games This has recently led to highly

active sub ject of games semantics for programming languages see Section and the articles

mentioned there

Parametricity

What is parametricity in p olymorphism We have already seen such notions as

Uniformity of algorithms across typ es

Passing typ es as parameters in programs

But the problem is that a typ e like when interpreted in a mo del as a large pro duct over

all typ es may contain in Stracheys words unintended ad hoc elements In addition to removing

some entities we may wish to include yet others For example should we consider closure of

parametric functions under isomorphism of typ es

We have already mentioned the idea of typ es b eing functors in Section Indeed this

suggests an obvious kind of mo delling

Typ es functors

Terms programs natural transformations

all dened over some ccc C This view of categorical program semantics has had a fruitful history

Reynolds Oles and later OHearn and Tennent have used functor categories to develop semantics of

lo cal variables blo ck structure noninterference etc in Algollike languages see OHT Ten

and references there

In the case of p olymorphism this is also not such a farfetched idea Imagine a term t

We know that for each typ e A tA A A Thus from our CurryHoward viewp oint we think

of this as an ob jectindexed family of arrows Combining this idea with the mild parametricity

condition of naturality then seems reasonable In the mid s Girard gave functor category

mo dels of System F Gi However to handle the functorial problem of cocontravariance in an

exression like or worse in which is not a functor at all he intro duced categories of

emb eddingpro jection pairs as in domain theory Section Below we shall consider dinaturality

a multivariant notion of naturality which takes into account such problems

ReynoldsRey also prop osed an analysis of parametricity using the notion of logical relations

a fundamental to ol in the theory of typ ed lamb da calculi The pap er BFSS studied the ab ove

two frameworks for parametricity Reynolds relational approach and the dinaturality approach

This work was extended and formalized in ACC BAC PlAb

Dinaturality

One attempt to understand parametric p olymorphism is to require certain naturality conditions on

families interpreting universal typ es In this view we b egin with some appropriate ccc C of values

and interpret p olymorphic typ e expressions A with typ e variables as certain kinds

1 n i

op n n

of multivariant denable functors F C C C Terms or programs t are then interpreted

as certain multivariant dinatural transformations b etween the interpretations of the typ es

We need to account for naturality not only in p ositive covariant p ositions but also in negative

contravariant ones As we shall see the diculty will b e comp ositionality

op n n

Denition Let C b e a category and F G C C C functors A dinatural transform

n

ation F G is a family of C morphisms f F AA GAA j A C g satisfying for any

A

n

ntuple f A B C

A

F AA GAA

GAf F f A

R

FBA GAB

R

F B f Gf B

B

GB B FBB

For a history of this notion see Mac Dinatural transformations include ordinary natural trans

formations as a sp ecial case eg construe covariant F G as bifunctors dummy in the contravariant

variable as well as transformations b etween co and contravariant functors The parametric asp ect

of naturality here is that may b e varied along an arbitrary map f A B in b oth the co and

A

contravariant p ositions

In the following examples K denotes the constant functor with value A where K f id

A A A

We use settheoretic notation but the examples make sense in any ccc eg using the internal

language We follow the treatment in BFSS

()

Example Polymorphic Identity Let F K let G Consider the family

1

2

A

A Denition reduces to the I fI A j A C g where I x A

A A x:A

following commuting square

A

A

A

f I

A

R

A

B

f

R

I

B

B

B

B

o o

which essentially says f id id f This equation is true in Set or in the internal language of

A B

any ccc since b oth sides equal f

()

Example Evaluation Fix an ob ject D C Let F D and G K The family

D

A

E v fev D A D j A C g F G is a dinatural transformation where ev is the usual

A A

evaluation in any ccc Denition reduces to the following commuting square for any f A B

A

D A

f

D A ev

A

R

B

D A D

B

R

ev

D f

B

B

D B

B

o o

This says for any g D a A ev g f a ev g f a More informally g f a g f a

A B

Again this is a truism in any ccc

A

Extending the ab ove example generalized evaluation EV fev A A A j A A C g

AA

determines a dinatural transformation b etween appropriate functors cf BFSS Dinaturality

B

o o

g f a for g A a A and any f A corresp onds to the true equation f g f a f

B f A B

() ()

Example Church Numerals Dene n to b e the family where n

A

A A n n

o o o

A A is given by mapping h h with h h h h h n times Dinaturality corresp onds

to the diagram for any f A B

n

A

A A

A A

f A

A f

R

B A

A B

R

f B

B f

n

B

B B

B B

n B n

o o o o

f f g f an instance of asso ciativity ie if g A f g

We shall see dinaturality again in Section Observe that each of the families f j A C g

A

ab ovewhich in essence arise from the syntax of cccshave uniform algorithms across all typ es

A

n

A

A For example n h uniform in each typ e A

A h:A

We end with an op erator which is fundamental to denotational semantics

Example Fixed Point Combinators In many cccs C used in programming language

semantics eg certain sub categories of CPO there is a dinatural xed p oint combinator

() A

Y That is we have a family fY A A j A C g making the follow

() A

ing diagram commute for any f A B

Y

A

A

A A

f

A f

R

B

A B

R

B

id

f

Y B

B

B

B B

B

o o

This says using informal settheoretic notation if g A f Y g f Y f g In particular

A B

setting B A and letting g id we have the xedp oint equation f Y f Y f

B A A

For example consider CPO the sub ccc of CPO whose ob jects A have a least element

A

but the morphisms need not preserve it It may b e shown that the family given by Y f the

A

W

n

least xedpoint of f ff j n g is dinatural see BFSS Mul Si

A

op n n

There is a calculus of multivariant functors F G C C C functors For example basic

typ e constructors may b e dened using pro ducts and exp onentials in C by setting

F GAB F AB GAB

F FBA

G AB GAB

Here A is the list of n contravariant and B the list of n covariant arguments Note the twist of

the arguments in the denition of exp onentiation Much of the structure of cartesian closedness

eg evaluation maps currying pro jections pairing etc exists within the world of dinatural

transformations and there is a kind of abstract functorial calculus cf BFSS App endix A

Fre

Unfortunately there is a serious problem in general dinaturals do not comp ose That is

A

A

given dinatural families fF AA H AA j A C g the comp osite GAA j A C g and fGAA

o

A A

H AA j A C g do es not always make the appropriate hexagon commute However fF AA

with resp ect to the original question of closure of parametric functions under isomorphisms of

typ es we note that families dinatural with resp ect to isomorphisms f do in fact comp ose But this

class is to o weak for a general mo delling Detailed studies of such phenomena have b een done in

BFSS FRRa FRRb

Remarkably there are certain categories C over which there are large classes of multivariant

functors and dinatural transformation which provide a comp ositional semantics

In BFSS it is shown that if C P er N that socalled realizable dinatural transformations

b etween realizable functors comp ose Realizable functors include almost any functors that

arise in practice eg those denable from the syntax of System F while realizable dinatural

transformations are families of p er morphisms whose action is given uniformly by a single

Turing machine This semantics also has a kind of universal quantier mo delling System

F see b elow

In GSS it is shown that the syntax of simply typ ed lamb da calculus with typ e variables

ie C a free cccadmits a comp ositional dinatural semantics b etween logically denable

functors and dinatural families This uses the cutelimination theorem from pro of theory

This work was extended to Linear Logic by R Blute Blu

In BS there is a comp ositional dinatural semantics for the multiplicative fragment of linear

logic generated by atoms Here C RT V E C a category of reexive top ological vector spaces

rst studied by Lefschetz Lef with functors b eing syntactically denable In BSb BS

this was extended to a comp ositional dinatural semantics for Yetters noncommutative cyclic

linear logic In b oth cases one demands certain uniformity conditions on dinatural families

involving equivariance wrt continuous group resp ectively Hopfalgebra actions induced

from actions on the atoms see also Section b elow For the noncyclic fragment this is

automatic

Asso ciated with dinaturality is a kind of parametric universal quantier rst describ ed by

Yoneda and which plays a fundamental role in mo dern category theory Mac

R

Denition An end of a multivariant functor G on a category C is an ob ject E GAA and

A

a dinatural transformation K G universal among all such dinatural transformations

E

R

X

GX X j X C g making GAA In more elementary terms there is a family of arrows f

A

f

the main square in the following diagram commute for any X Y and such that given any

o o

other family u fu j X C g such that GX f u Gf Y u there is a unique u making the

X X Y

appropriate triangles commute

D

Q

u

X

A

Q

Q

A

Q

u

Y

Qs

A

GX X

A

u

A

GX f

A

X

A

R

A

R

A GX Y GAA

A

A

A

A

Gf Y

AU

Y

R

GY Y

R

GAA as a subset of GAA note this is a large pro duct over all A C One may think of

A

A

so C must have appropriate limits for this to exist

Z

o o

GAA fg GAA j GX f Gf Y for all X Y f X Y C g

A X Y

A

In BFSS versions of such ends over P er N are discussed with resp ect to parametric mo delling

of System F

In a somewhat dierent direction coends dual to ends are a kind of sum or existential

quantier Their use in categorical computer science was strongly emphasized in early work of

Bainbridge B B on duality theory for machines in categories A useful observation is that we

may consider functors R C D S et and S D E S et as generalized relations with relational

R

D

comp osition b eing determined by the co end formula R S C E RC D S D E This

view has recently b een applied to relational semantics of dataow languages in HPW

We should mention that dinaturality is also intimately connected with categorical coherence

theorems and geometrical prop erties of pro ofs Blu So It also seems to b e hidden in deep er

asp ects of CutElimination GSS although here there is still much to understand We shall meet

dinaturality again in several places eg in Traced Monoidal Categories Section

Reynolds Parametricity

Reynolds Rey analyzed Stracheys notion of parametric p olymorphism using a relational mo del

of typ es Although his original idea of using a S etbased mo del was later shown by Reynolds

himself to b e untenable the framework has greatly inuenced subsequent studies As a concrete

illustration following BFSS we shall sketch a relational mo del over P er N Related results

in more general frameworks were obtained by Hasegawa RHas RHas and Ma and Reynolds

MaRey Although originally Reynolds work was semantical general logics for reasoning ab out

formal parametricity supp orted by such P er mo dels were develop ed in BAC PlAb ACC

Given p ers A A P er N a saturated relation R A A is a relation R dom dom

A A

satisfying R A R A where denotes relational comp osition For all p ers A A B B saturated

relations R A b dom b dom A and S B B and elements a dom a dom

B B A A

we dene a relational System F typ e structure as follows

A B given comp onentwise by a bR S a b i aRa and bS b R S A B

R S A B A B where f R S g i f g are co des of Turing computable

functions satisfying aRa implies f aS g a for any a a as ab ove

S B B is dened by a simultaneous inductive denition based

on the formation of typ e expression We shall omit the technical construction see

BFSS p but the key idea is to redene the P er interpretation of by triming

down the intersection A to only those elements invariant under all saturated relations

A

while S R S the intersection b eing over all p ers A A and saturated R

R

A A

The somewhat involved construction of ensures the typ e expressions act like functors

with resp ect to saturated relations More precisely Reynolds parametricity entails

A is a saturated relation then for any p olymorphic typ e R A A If R A

is a saturated relation

Identity Extension Lemma preserves identity relations ie id id as saturated

A (A)

A and similarly for id id relations A

A A

1 n

One obtains a Soundness Theorem essentially the interpretation of the free term mo del of

System F into the relational P er mo del ab ove For simplicity consider terms with only one

variable Let and denote p olymorphic typ es with free typ e

1 n 1 n

variables f g Let x t denote term t with free variable x Asso ciated

1 n

to every System F term t is a Turing computable numerical function f obtained by essentially

t

erasing all typ es and considering the result as an untyp ed lamb da term qua computable partial

function see BFSS App endix A

Theorem Soundness Let A A be pers and R A A saturated relations Then

i i i

i i

if m Rm then f m Rf m Also if t t in System F then f f as P er N maps

t t t t

A A

Thus terms programs b ecome relation transformers R R cf MitSce Fre of the

form

f

t

A A

R R

R R

A A

f

t

In particular this exemplies Reynolds interpretation of Stracheys parametricity if one instanti

ates an element of p olymorphic typ e at two related typ es then the two values obtained must b e

related themselves

Reynolds parametricity has the following interesting consequence BFSS RHas Recall

the category of T algebras for denable functors T cf the pro of of Theorem

Theorem Let T be a System F denable covariant functor Then in the parametric P er

model T is the initial T algebra

This prop erty b ecomes a general theorem in the formal logics of parametricity eg PlAb

RHas and hence would b e true in any appropriate parametric mo del Thus although the

syntax of secondorder logic in general only guarantees weakly initial data typ es as in GLT in

parametric mo dels of System F the usual denitions actually yield strong data typ es

The reader might rightly enquire do relational parametricity and dinaturality have anything in

common This is exactly the kind of question that requires a logic for reasoning about parametricity

Plotkin and Abadis logic PlAb extends the equational theory of System F with quantication

over functions and relations together with a schema expressing Reynolds relational parametricity

The dinaturality hexagon in Denition for denable functors and families is expressible as a

quantied equation in this logic

Prop osition In the formal system above relational parametricity implies dinaturality

Reynolds work on parametricity continues to inspire fundamental research directions in pro

gramming language theory even b eyond p olymorphism For example OHearn and Tennent

OHT OHT use relational parametricity to examine dicult problems in lo calvariable de

clarations in Algollike languages Their framework is particularly interesting They use cccs of

functor categories and natural transformations a la Oles and Reynolds but internal to the category

of reexive directed multigraphs The same framework somewhat generalized was then used by A

Pitts Pia in a general relational approach to reasoning ab out prop erties of recursively dened

domains Pitts work has led to new approaches to induction and coinduction etc see Section

The reader is referred to Pitts Pia p and OHearn and Tennent OHT for many

examples of these socalled relational structures over categories C

Linear Logic

Monoidal Categories

We briey recall the relevant denitions For details the reader is referred to Mac Bor

Denition A monoidal category is a category C equipp ed with a functor C C C an

ob ject I and sp ecied natural isomorphisms

a A B C A B C

AB C

I A A and r A I A

A A

satisfying coherence equations asso ciativity coherence Mac Lanes Pentagon and the unit coher

ence

A symmetric monoidal category is a monoidal category with a natural symmetry isomorphism

s A B B A satisfying s s id for all A B and omitting subscripts

AB BA AB AB

r s asa sas

Symmetric monoidal categories include cartesian categories with and co cartesian cat

egories with However in the two latter cases the structure is uniquely determined

up to isomorphismand similarly for the coherence isomorphismsby the universal prop erty of

pro ducts resp copro ducts This is not true in the general casethere may b e many symmetric

monoidal structures on the same category

We now intro duce the monoidal analog of cccs

Denition A symmetric monoidal closed category smcc C I is a symmetric mon

oidal category such that for each ob ject A C the functor A C C has a sp ecied right

adjoint A ie for each A there is an isomorphism natural in B C

H om C A B H om C A B

C C

ev

AB

As a consequence in any smcc there are evaluation and co evaluation maps A B A B

and C A C A determined by the adjointness We shall try to keep close to

our ccc notation Section In particular the analog of Currying arising from is denoted

f

op

C A B Moreover this data actually determines a bifunctor C C C

No sp ecial coherences have to b e supp osed for they follow from coherence for and

adjointness

For the purp oses of studying linear logic b elow we need among other things a notion of an

smcc equipp ed with an involutive negation or duality reminiscent of nite dimensional vector

spaces The general theory of such categories due to M Barr Barr was develop ed in the mid

s some ten years b efore linear logic

o

Consider an smcc C with a distinguished ob ject Consider the map ev s A A

A A

By this corresp onds to a map A A Let us write A for A

A

Thus we have a morphism A A Ob jects A for which is an isomorphism are called

A A

reexive or more precisely reexive with resp ect to

Denition A autonomous category C I is an smcc C with a distinguished ob ject

such that all ob jects are reexive ie the canonical map A A is an isomorphism for

A

all A C The ob ject is called the dualizing object

It may b e shown that a autonomous category C has a contravariant dualizing functor

op

C C dened on ob jects by A A There is a natural isomorphism H om A B

C

H om B A

C

In any autonomous category C there are isomorphisms

A B A B

I

The reader is referred to Barr for many examples Let us mention the obvious one

Example The category V ec of nite dimensional vector spaces over a eld k is

f d

autonomous Here A B LinA B the space of linear maps from A to B and the dualizing

ob ject k In particular A A is the usual dual space More generally within the smcc

category V ec of k vector spaces with k an ob ject is reexive i it is nite dimensional

B by de Morgan duality A In a autonomous category we may dene the cotensor

are identied A B The ab ove example V ec is somewhat degenerate since and

f d

see the Denition of compact category In a typical autonomous category this is not the

case indeed in linear logic one do es not want to identify tensor and cotensor

To obtain more general autonomous categories of vector spaces we add a top ological structure

due to Lefschetz Lef The following discussion is primarily based on work of M Barr Barr

following the treatment in Blute Blu

Denition Let V b e a vector space A top ology on V is linear if it satises the following

three prop erties

Addition and scalar multiplication are continuous when the eld k is given the discrete

top ology

is hausdor

V has a neighb orho o d basis of op en linear subspaces

Let TVEC denote the category whose ob jects are vector spaces equipp ed with linear top ologies

and whose morphisms are linear continuous maps

Barr showed that TVEC is a symmetric monoidal closed category when V W is dened to b e

the vector space of linear continuous maps top ologized with the top ology of p ointwise convergence

It is shown in Barr that the forgetful functor T V E C V E C is tensorpreserving Let V denote

V k Lefschetz proved that the emb edding V V is always a bijection but need not b e an

isomorphism This is analogous to Dana Scotts metho d of solving domain equations in denotational

semantics using the top ology to cut down the size of the function spaces

Theorem Barr RT V E C the ful l subcategory of reexive objects in TVEC is a complete

cocomplete autonomous category with I I k the dualizing object

Moreover in RT V E C and are not equated More generally other classes of autonomous

categories arise by taking a linear analog of Gsets namely categories of group representations

Denition Let G b e a group A continuous Gmodule is a linear action of G on a space V

in TVEC such that for all g G the induced map g V V is continuous Let T MO D G

denote the category of continuous Gmo dules and continuous equivariant maps Let RT MO D G

denote the full sub category of reexive ob jects

We have the following result which in fact holds in the more general context of Hopf algebras

see b elow

Theorem The category T MO D G is symmetric monoidal closed The category RT MO D G

is autonomous and a reective subcategory of T MO D G via the functor Furthermore

the forgetful functor j j RT MO D G RT V E C preserves the autonomous structure

Still more general classes of autonomous categories may b e obtained from categories of mo d

ules of co commutative Hopf algebras Given a Hopf algebra H a module over H is a linear action

H V V satisfying the appropriate diagrams analogous to the notion of Gmo dule Let

MO D H denote the category of H mo dules and equivariant maps Similarly T MO D H the

category of continuous H mo dules is the linearly top ologized version of MO D H where H is given

the discrete top ology and all vector spaces and maps are in TVEC

Prop osition If H is a cocommutative Hopf algebra MO D H and T MO D H are symmetric

monoidal categories

We then obtain precisely the same statement as Theorem by replacing the group G by a

co commutative Hopf algebra H Later we shall mention noncommutative Hopf algebra mo dels for

linear logic with resp ect to full completeness theorems Section

is an imp ortant class of monoidal categories The case where we do identify and

Denition A autonomous category is compact if A B A B ie equivalently

if A B A B

In addition to the obvious example of V ec there are compact categories of relations which

f d

have considerable imp ortance in computer science One such is

Example Rel is the category whose ob jects are sets and whose maps R X Y are

binary relations R X Y Comp osition is relational comp osition etc This is a compact

category with X Y X Y X Y the ordinary settheoretic cartesian pro duct Dene

def

op op

fg a oneelement set hence X X X On maps we have R R where y R x

i xRy

Given two smccs C and D we shall not distinguish the structure what are the morphisms

b etween them

Denition A symmetric monoidal functor is a functor F C D together with two natural

transformations m I F I and m F U F V F U V such that three coherence

I UV

diagrams commute In the case of the closed structure we can dene another natural transformation

o

m F U V FU FV by m F ev m A symmetric monoidal

UV UV U V U U V U

functor is strong resp strict if m and m are natural isomorphisms resp identities for all

I UV

U V A symmetric monoidal functor is strong closed resp strict closed if m and m are natural

I UV

isomorphisms resp identities for all U V Similarly one denes autonomous functors

Finally we need an appropriate notion of natural transformation for monoidal functors

Denition A natural transformation F G is monoidal if it is compatible with b oth

m and m for all U V in the sense that the following equations hold

I UV

o

m m

I I I

o o

m m

UV U V U V UV

Gentzens pro of theory

Gentzens pro of theory GLT esp ecially his sequent calculi and his fundamental theorem on Cut

Elimination have had a profound inuence not only in logic but in category theory and computer

science as well

In the case of category theory J Lamb ekL L intro duced Gentzens techniques to study

coherence theorems in various free monoidal and residuated categories This logical approach to

coherence for such categories was greatly extended by G Mints S Soloviev B Jay et al Min

So So J For a comparison of Mints work with more traditional KellyMac Lane coherence

theory see Mac More recently coherence for large classes of structured monoidal categories

arising in linear logic has b een established in a series of pap ers by Blute Co ckett Seely et al This is

based on Girards extensions of Gentzens metho ds see BCST BCS BCS CS CSb

Recent coherence theorems of GordonPowerStreet JoyalStreet et al GPS JS JS

have made extensive use of higher dimensional category theory techniques and Yoneda metho ds

rather than logical metho ds Related Yoneda techniques are now b eing intro duced in the reverse

direction into pro of theory as we outlined in Section ab ove

In computer science entire research areas pro of search AI Logic Programming op erational

semantics typ e inference algorithms logical frameworks etc are a testimonial to Gentzens work

Indeed Gentzens Natural Deduction and Sequent Calculi are fundamental metho dological as well

as mathematical to ols

A profound and exciting analysis of Gentzens work has arisen recently in the rapidly growing

area of Linear Logic LL develop ed by JY Girard in While classical logic is ab out

universal truth and intuitionistic logic is ab out constructive pro ofs LL is a logic of resources and

their management and reuse eg see Gi Gi GLR Sc Sc Tr

Gentzen sequents

Gentzens analysis of Hilb erts pro of theory b egins with a fundamental reformulation of the syntax

We follow the presentation in GLT

A sequent for a logical language L is an expression

A A A B B B

1 2 m 1 2 n

where A A A and B B B are nite lists p ossibly empty of formulas of L Sequents

1 2 m 1 2 n

are denoted for lists of formulas and Gentzen intro duced formal rules for generating

sequents the socalled derivable sequents Gentzens rules analyze the deep structure and implicit

symmetries hidden in logical syntax Computation in this setting arises from one of two metho ds

The traditional metho d is Gentzens CutElimination Algorithm which allows us to take a

formal sequent calculus pro of and reduce it to cutfree form This is closely related to b oth

normalization of lamb da terms cf Sections as well as the op erational semantics of such

programming languages as PROLOG

More recent is the pro of search paradigm which is the b ottomup goaldirected view of

building sequent pro ofs and is the basis of the disciplin e of Logic Programming MNPS

HM Mill

Categorically the cut elimination algorithm is at the heart of the pro oftheoretic approach to

coherence theorems previously mentioned On the other hand Logic Programming and the pro of

search paradigm have only recently attracted the attention of categorists cf FiFrL PK

Lamb ek p ointed out that Gentzens sequent calculus was analogous to Bourbakis metho d of

bilinear maps For example given sequences A A and B B B of R R bimo dules

1 m 1 2 n

of a given ring R there is a natural isomorphism

M ul tAB C M ul tA B D

b etween m n linear and m n linear maps Bourbaki derived many asp ects of tensor

pro ducts just from this universal prop erty Such a formal bijection is at the heart of Linear Logic

eg L

Traditional logicians think of sequent as saying the conjunction of the A entails the

i

disjunction of the B More generally following Lamb ek and Lawvere cf Section categorists

j

interpret such sequents mo dulo equivalence of pro ofs as arrows in appropriate categories For

example in the case of logics similar to linear logic CS the sequent determines an arrow

of the form

B B A A A B

2 n 1 2 m 1

in a symmetric monoidal category with a cotensor see Section b elow

Girards Analysis of the Structural Rules

Gentzen broke down the manipulations of logic into two classes of rules applied to sequents struc

tural rules and logical rules All rules come in pairs leftright applying to the left resp right

side of a sequent

Gentzens Structural Rules LeftRight

Permutation

p ermutations

B B A A

Contraction

A B

Weakening

A B

For simplicity consider intuitionistic sequents ie those of the form A A A B with

1 2 m

one conclusion So the right rules disapp ear and we consider the left rules ab ove We can give a

CurryHowardstyle analysis to Gentzens intuitionistic sequents cf Section assigning lamb da

terms qua functions to sequents eg x A x A tx B The structural rules say the

1 1 m m

following Permutation says that the class of functions is closed under p ermutations of arguments

Contraction says that the class of functions is closed under duplicating argumentsie setting two

input variables equal and Weakening says the class of functions is closed under adding dummy

arguments In the absence of such rules we obtain the so called linear lamb da terms terms where

all variables o ccur exactly once see GSS Abr L

3

By removing these traditional structural rules logic takes on a completely dierent character

see Figure Previously equivalent notions now split into subtle variants based on resource

allo cation For example the rules for Multiplicative connectives simply concatenate their input

hyp otheses and whereas the rules for Additive connectives merge two input hyp otheses

into one The situation is analogous for outputs and see Figure The resultant logical

connectives can represent linguistic distinctions related to resource use which are simply imp ossible

to formulate in traditional logic see Gi Abr Sc Sc

Remark First we should remark that on the controversial sub ject of notation in LL we have

chosen a reasonable categorical notation somewhere b etween Gi and See Observe that

in CLL twosided sequents can b e replaced by onesided sequents since is equivalent to

with the list A A where is A A

1 n

1 n

Thus the key asp ect of linear logic pro ofs is their resource sensitivity We think of a linear

entailment A A B not as an ordinary function but as an actiona kind of pro cess that in a

1 m

single step consumes the inputs A and pro duces output B For example this p ermits representing

i

in a natural manner the stepbystep b ehaviour of various abstract machines certain mo dels of

concurrency like Petri Nets etc Thus linear logic p ermits us to describ e the instantaneous state

of a system and its stepwise evolution intrinsically within the logic itself eg with no need for

explicit time parameters etc

But linear logic is not ab out simply removing Gentzens structural rules but rather mo dulating

their use To this end Girard intro duces a new connective A which indicates that contraction

and weakening may b e applied to formula A This yields the Exponential connectives in Figure

From a resource viewp oint an hyp othesis A is one which can b e reused arbitrarily Moreover this

p ermits decomp osing categorically the ccc function space into more basic notions

A B A B

? ? ?

Formulas of LL are generated from literals p q r p q r and constants I using binary op erations

? ? ? ? ? ?? ?

and unary Negation is dened inductively I I p p A B

? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

B A B A B A B A B A B A A A A Also B A A

?

B we dene A B A

p ermutations Perm

Structural

AxiomC ut Axiom A A

A A

Cut

A A

Negation

A A

A B

A B

M ul tipl icativ es T ensor A B A B

A B

A B

B P ar A A B

I

U nits I

A B

A B

Implication

A B A B

A B A B

Additiv es P r oduct A B A B A B

A B A B

Coproduct

A B A B A B

U nits

A A

Contraction

E xponential s W eak ening A A

A

A

Dereliction

S tor ag e A A

Figure Rules for Classical Prop ositional LL

Finally nothing is lost classical as well as intuitionistic logic can b e faithfully translated into

CLL Gi Tr

Fragments and Exotic Extensions

The richness of LL p ermits many natural subtheories cfGi Gia For a survey of the sur

prisingly intricate complexitytheoretic structure of many of the fragments of LL see Lincoln Li

These results often involve direct and natural simulation of various kinds of abstract computing

machines within the logic Sc Ka Of course there are sp ecic fragments corresp onding to

various sub categories of categorical mo dels in the next section There are also fragments directly

connected with classifying complexity classes in computing GSS Gi but these latter have not

b een the ob ject of categorical analysis

More exotic noncommutative fragments of LL are obtained by eliminating or mo difying the

p ermutation rule ie one no longer assumes is symmetric One such precursor to LL is the work

of J Lamb ek in the s on categorial grammars in mathematical linguistics for recent surveys

see L L Here the language b ecomes yet more involved since there are two implications

and and two negations A and A It has b een proven by Pentus Pen that Lamb ek grammars

are equivalent to contextfree grammars In L there is a formulation of Lamb ek grammars using

the notion of multicategory an idea currently of some interest in higherdimensional category theory

and higher dimensional rewriting theory HMP

DYetter Y considered cyclic linear logic a version of LL in which the p ermutation rule is

mo died to only allow cyclic p ermutations This will b e discussed briey b elow in Section

with resp ect to Full Completeness A prop osed classication of dierent fragments of LL including

braided versions based on Hopfalgebraic mo dels is in Blute Blu see also Section

Topology of Proofs

Let us briey mention one of the main novelties of linear logic Traditional Gentzen pro of theory

writes pro ofs as trees In order to give a CurryHoward isomorphism to arbitrary sequents

Girard intro duced multipleconclu sion graphical networks to interpret pro ofs These proof nets

use graph rewriting moves for their op erational semantics It is here that one sees the dynamic

asp ects of cutelimination In essense these networks are the lamb da terms of linear logic There

are known mathematical criteria to classify which among arbitrary networks arise from Gentzen

sequent pro ofs ie in a sense which of these parallel networks are sequentializable into a Gentzen

pro of tree Homological asp ects of pro of nets are studied in Met

The technology of pro of nets has grown into an intricate sub ject In addition to their uses in

linear logic pro of nets are now used in category theory as a technical to ol in graphical approaches

to coherence theorems for structured monoidal categories eg Blu BCST CS CSb

There are pro of net theories for numerous noncommutative cyclic and braided linear logics eg

Abru Blu Fl FR Y

The metho d of pro of nets has b een extended by Y Lafont Laf to a general graphical language

of computation his interaction nets These latter provide a simple mo del of parallel computation

with at the same time new insights into sequential computation

What is a categorical mo del of LL

General Models

As in Section we are interested in nding the categories appropriate to mo delling linear lo

gic pro ofs just as cartesian closed categories mo delled intuitionistic pro ofs The basic

equations we p ostulate arise from the op erational semanticsthat is normalization of proofs In

the case of sequent calculi this is Gentzens CutElimination pro cess GLT However there are

sometimes natural categorical equations which are not decided by traditional pro of theory The

problem is further comp ounded in linear logic and monoidal categories in that there may b e

several noncanonical candidates for appropriate monoidal structure

The rst categorical semantics of LL is in Seelys pap er See which is still p erhaps the

most readable account Subsequent development of appropriate term calculi Abr Bie BBPH

W BCST CS CSb have mo died and enlarged the scop e but not essentially changed the

original analysis for the case of classical linear logic CLL We imp ose the following equations

b etween CLL pro ofs in order to form a category C where sequents are interpreted as equivalence

classes of arrows according to formula based on the rules in Figure

C is a symmetric monoidal closed category with products coproducts and units from the

rules Axiom Cut Perm the Multiplicatives and the Additives

related by de Morgan duality C is autonomous from the Negation rule with and

C C is an endofunctor with asso ciated monoidal transformations id and

C

satisfying

forms a monoidal comonad on C

There are natural isomorphisms

I and AB A B

making C C I a symmetric monoidal functor

d e

A A

AA is a co commutative comonoid for all A in C and A In particular I

the coalgebra maps A A and A A are comonoid maps In fact these

A A

conditions are a consequence of but are required explicitly in weaker settings

For mo dications appropriate to more general situations eg various fragments of LL without

pro ducts linearly distributive categories etc see Bie BCS

The essence of Girards translation of intuitionistic logic into LL is the following easy result cf

See Bie

Prop osition If C is a categorical model of CLL as above then the Kleisli category K of the

C

comonad is a ccc Moreover nite products in K and C coincide while exponentials in K

C C

are given by A B A B

We should mention one formal rule MIX which app ears frequently in the literature To express

it we use onesided sequents

Mix

B This rule seems to b e valid in most Categorically MIX entails there is a map A B A

mo dels certainly so in ones based on RT V E C

Remark The categorical comonad approach to mo dels of linear logic has b een put to use by

Asp erti in clarifying optimal graph reduction techniques in the untyp ed lamb da calculus Asp see

also GAL

Concrete Models

There are by now many categorical mo dels of LL and its interesting fragments Let us just mention

a few Gia Tr

Posetal Models or Girards Phase semantics These are autonous p osets with additional

structure This gives an algebraic semantics analogous to Bo olean or Heyting algebras for

classical resp intuitionistic logic As categories they are trivial each hom set has at most

one element hence they do not mo del pro ofs but rather provability There is asso ciated a

traditional Tarski semantics with Soundness and Completeness Theorems Recently these

mo dels have b een applied in Linear Concurrent Constraint Programming for proving safety

prop erties of programs FRS

DomainTheoretic Models The category LIN coherent spaces and linear maps gave the

rst nontrivial mo del of LL pro ofs This mo del arose from Girards analysis of the ccc STAB

realizing that there were many other logical op erations available Indeed STAB is the Kleisli

category of an approproate comonad on LIN cf Prop osition In the mo del LIN

A is a minimal solution of the domain equation A I A AA indeed is a cofree

comonoid

Relational Models As discussed in Barr Barr many compact categories are complete

enough to interpret

A A E A E A

2 n

n

where E A is the equalizer of the n p ermutations of the nth tensor p ower A for n

n

For example Barr proves Rel has that prop erty More generally Barr Barr constructs

mo dels based on the Chuspace construction in Barr Chu spaces are themselves an

interesting class of mo dels of LL and have b een the sub ject of intensive investigation by M

Barr and by Vaughn Pratt Pra

Games Models Categories of Games now provide some of the most exciting new semantics

for LL and Programming Languages This socalled intensional semantics provides a ner

grained analysis of computation than traditional categorical mo dels taking into account

the dynamic or interactive asp ects of computation For example such games can b e used to

mo del interactions b etween a System and its Environment and provided the rst syntaxfree

fully abstract mo dels of PCF answering a longstanding op en problem Games categories

have b een extended to handle programming languages with many additional prop erties eg

control features callbyvalue languages languages with sideeects and store etc as well

as mo dern logics like LL System F etc For basic intro ductions see Abr Hy For a

small sample of more recent work see Mc AHMc AMc BDER

GoI and Functional Analytic Models The Geometry of Interaction Program see eg Gi

Gi Gib DR aims to mo del the dynamics of cutelimination by interpreting pro ofs

as ob jects of a certain C algebra with logical rules corresp onding to certain isomorphisms

The essence of Gentzens cutelimination theorem is summarized by the socalled execution

formula We shall lo ok at an abstract form of the GoI program in traced monoidal categories

in Section The GoI program itself has inuenced b oth game semantics and work on

optimal reduction

Finally as the name suggests linear logic is roughly inspired from linear algebra Thus A

is analogous to the Grassmann algebra Indeed in categories of Hilb ert or Banach spaces

one is reminded of the symmetric and antisymmetric Fo ck space construction Ge For a

noncategorical Banach space interpretation of LL see Girard Gi

Full Completeness

Representation Theorems

The most basic representation theorem of all is the Yoneda emb edding

op

A

Theorem Yoneda If A is local ly smal l the Yoneda functor Y A S et where Y A

H om A is a ful ly faithful embedding

A

Indeed Yoneda preserves limits as well as cartesian closedness

We seek mathematical mo dels which describ e the b ehaviour of programs From the viewp oint

of the CurryHoward isomorphism which identies pro ofs with programs we seek representation

theorems for pro ofsie mathematical mo dels which fully and faithfully represent pro ofs From

the viewp oint of a logician these are Completeness Theorems but now at the level of pro ofs rather

than provability

One of the rst such theorems was proved by H Friedman Frie Friedman showed com

pleteness of typ ed lamb da calculus with resp ect to ordinary settheoretic reasoning Consider the

pure typ ed lamb da calculus L whose typ es are generated from some base sorts by only We

interpret L settheoretically in a full typ e hierarchy A see Example

Theorem Friedman Let A be a ful l type hierarchy with base sorts interpreted as innite

sets Then for any pure typed lambda terms M N M N is true in A i M N is provable

using the rules of typed lambda calculus

Similar results but using instead full typ e hierarchies over CPO or Perbased mo dels have

b een given by Plotkin and by Mitchell using logical relations see Mit Friedmans original

Setbased theorem has b een extended by Cubric to the entire ccc language Cu to yield

the following

Theorem Cubric Let C be a free ccc generated by a graph Then there exists a faithful ccc

functor F C S et

Alas this representation is not full

Let B the free ccc with binary copro ducts generated by discrete graph G given syntactically

G

G

by typ es and terms of typ ed lamb da calculus For any group G the functor category S et is a

ccc with copro ducts So according to the universal prop erty if F is an initial assignment of Gsets

to atomic typ es then a pro of of formula qua closed term M qua F arrow M

G

corresp onds to a Gset equivariant map Such maps are xed p oints under M

F F

the action In particular letting G Z we obtain the easy half of the following theorem due to

Lauchli Lau

Theorem Lauchli A f gformula of intuitionistic propositional calculus is prov

Z

able if and only if for every interpretation F of the base types its S et interpretation has an

F

invariant element

Indeed Harnik and Makkai extend Lauchlis theorem to a representation theorem Recall a functor

is weakly ful l if H omA B implies H omA B

Theorem Harnik Makkai HM Let B be a countable free ccc with binary coproducts

Z

There is a weakly ful l representation of B into a countable power of S et If in addition the

Z

terminal object is indecomposable then there is a weakly ful l representation into S et

A weakly full representation of B corresp onds to completeness with respect to provability ie

Z

B implies H om B so B is provable We shall give stronger repres H om

B S et

entation theorems still based on the idea of invariant elements

Full Completeness Theorems

A recent topic of considerable interest is ful l completeness theorems Supp ose we have a free

category F We shall say that a mo del category M is ful ly complete for F or that we have ful l

completeness of F with respect to M if the unique free functor with resp ect to any interpretation

F M is full It is even b etter to demand that is a fully faithful of the generators

representation

For example supp ose F is a free ccc generated by the typ ed lamb da calculus cf Example

To say a ccc M is fully complete for F means the following given any interpretation of

the generators as ob jects of M any arrow A B M b etween denable ob jects is itself

denable ie of the form f for f A B If the representation is fully faithful f is unique

Thus by CurryHoward any morphism in the mo del b etween denable ob jects is itself the image

of a pro of or program indeed of a unique pro of if the representation is fully faithful Thus such

mo dels M while b eing semantical really capture asp ects of the syntax of the language

Such results are mainly of interest when the mo dels M are genuine mathematical mo dels not

apparently connected to the syntax In that case Full Completeness results are more surprising

op

C

is not and interesting For example an explicit use of the Yoneda emb edding Y C S et

op

C

what we want since S et dep ends to o much on C

Probably the rst full completeness results for free cccs were by Plotkin Plo using categories

of logical relations In the case of simply typ ed lamb da calculus generated from a xed base typ e

the free ccc on one ob ject Plotkin proved the following result Consider the Henkin mo del T

B

the ful l type hierarchy over a set B ie the full subccc of Sets generated by some set B The

Soundness Theorem for logical relations says that if a term f is lamb da denable it is invariant

under all logical relations We ask for the converse

The rank of a typ e is dened inductively rankb where b is a base typ e rank

max f rank rank g rank max f rank rank g The rank of an element

f B in T is the rank of the typ e

B

Theorem Plotkin Plo In the ful l type hierarchy T over an innite set B al l elements

B

f of rank satisfy if f is invariant under al l logical relations then f is lambda denable

This result has b een extended by Statman St but the same question for terms of arbitrary

rank is still op en However Plotkin Plo did prove the ab ove result for lamb da terms of arbit

rary rank by moving to Kripke Logical Relations rather than Setbased logical relations Kripke

op

P

relations o ccur essentially by replacing S et by a functor category S et P a p oset ie by lo ok

ing at P indexed families of sets and relations Extensions with new characterizations of lamb da

denability are in work of Jung and Tiuryn JT A clear categorical treatment of their work

and logicalrelationsbased full completeness theorems is in Alimohamed Ali cf also Mit

The name Full Completeness rst arose in Game Semantics where the fundamental pap er of

Abramsky and Jagadeesan AJb proved full completeness for multiplicative linear logic the

Mix rule using categories of games with historyfree winning strategies as morphisms It is shown

there that uniform historyfree winning strategies are the denotations of unique pro of nets A

more involved notion of game develop ed by Hyland and Ong see Hy p ermits eliminating the

Mix rule in pro ofs of full completeness for the multiplicatives These results paved the way for the

most sp ectacular application of these gametheoretic metho ds the solution of the Full Abstraction

problem for PCF by AbramskyJagadeesan and Malacaria and by Hyland and Ong referred to

in Section

In BS BSb BS Full Completeness for MLL Mix and for Yetters Cyclic Linear Logic

were also proved using dinaturality and a generalization of Lauc hli semantics Let us briey recall

that view

Linear Lauchli Semantics

Let C b e a autonomous category Given an M LL formula built from

1 n

with typ e variables we inductively dene its functorial interpretation

1 n 1 n

op n n

C C C as follows

B if

i 1 n i

AB

if A

1 n

i i

AB AB AB

1 2 1 2

AB BA AB

1 2 1 2

The last two clauses corresp ond to Equations and following Example in Section It

B From is readily veried that Also recall that in M LL A B is dened as A

now on let C RT V E C

The set D inatF G of dinatural transformations from F to G is a vector space under p ointwise

op erations We call it the space of proofs asso ciated to the sequent F G where we identify

formulas with denable functors If is a onesided sequent then D inat denotes the set of

dinaturals from k to In such sequents we sometimes abbreviate to

The following is proved in BS BS A binary sequent is one where each atom app ears

exactly twice with opp osite variances

Theorem Full Completeness for Binary Sequents Let F and G be formulas in multi

plicative linear logic interpreted as denable multivariant functors on RT V E C Given a binary

sequent F G then D inatF G is zero or dimensional depending on whether or not F G

is provable If it is provable every dinatural is a scalar multiple of the denotation of the unique

cutfree proof qua Girard proofnet

A diadditive dinatural transformation is one which is a linear combination of substitution in

stances of binary dinaturals Under the same hyp otheses as ab ove we obtain

Theorem Full Completeness The proof space D inatF G of diadditive dinatural trans

formations has as basis the denotations of cutfree proofs in the theory MLLMIX

Example The proof space of the sequent

1 1 2 2 3 n1 n n

has dimension generated by the evaluation dinatural

The pro ofs of the ab ove results actually yield a fully faithful representation theorem for a free

autonomous category with MIX canonically enriched over vector spaces BS

In BS a similar Full Completeness Theorem and fully faithful representation theorem is

given for Yetters Cyclic Linear Logic In this case one employs the category RT MO D H for a

Hopf algebra H This is based on the following observation Blu

2

Prop osition If H is a hopf algebra with an involutive antipode ie S id then

RT MO D H is a cyclic autonomous category ie a model of Yetters cyclic linear logic

The particular Hopf algebra used is the shue Hopf algebra describ ed in Ben Haz BS Once

again we consider formulas as multivariant functors on RT V E C but restrict the dinaturals to so

called uniform dinaturals ie those which are equivariant with resp ect to the H action

jV jjV j

1 n

induced from the atoms for H mo dules V RT MO D H This is completely analogous to the

i

techniques used in logical relations

4

Related results using Chu spaces are in Pra

Feedback and Trace

Traced Monoidal Categories

This new class of categories intro duced by Joyal Street and Verity JSV have shown surprising

connections to mo dels of computation and iteration The original versions were very general in

cluding braided and tortile categories arising in several branches of mathematics At the moment

most of the applications to computing omit any braided structure But even at the abstract level

of JSV the authors illustrate a computational geometric calculus somewhat akin to Girards

pro of nets in linear logic Gi and indeed some precise connections have b een made BCS

Moreover the main construction in JSV has b een shown by Abramsky Abr to have fascin

ating connections with Girards GoI program as already hinted by Joyal Street and Verity

We now give a version of traced symmetric monoidal categories For ease of readability and

without loss of generality we consider strict monoidal categories recall from Mac Lanes coherence

theorem that every monoidal category is equivalent to a strict one

Denition A traced symmetric monoidal category tmc is a symmetric monoidal category

C I s where s X Y Y X is the symmetry morphism with a family of functions

XY

U

T r C X U Y U C X Y called a trace sub ject to the following conditions

XY

U U

Natural in X T r f g T r f g where

U

XY X Y

f X U Y U g X X

U U

g f where Natural in Y g T r f T r

U

XY XY

f X U Y U g Y Y

U U

f g Dinatural in U T r g f T r

X Y

XY XY

where f X U Y U g U U

I U V U V

Vanishing T r f f and T r g T r T r g

XY XY XY X UY U

for f X I Y I and g X U V Y U V

U U

f T r g f Sup erp osing g T r

XY W XZ Y

U

Yanking T r s

UU U

UU

Added in pro of there has b een recent progress on the ab ove work Masahiro Hamano JAIST has managed to

eliminate the use of dinaturals in the full completeness pro ofs in b oth of the BluteScott pap ers BS BS giving

a direct denotational interpretation of MLL Mixfull completeness in the categories of reexive top ological vector

spaces ab ove This pap er will app ear in Ann Pure and Applied Logic Also Hamano has proved MLL without

Mix full completeness in Barrs category of Reexive Top ological Ab elian Groups using Pontrjagin duality and the

dinatural framework ab ove This will app ear in Math Struc in Computer Science in a volume dedicated to the

th birthday of J Lamb ek

Y X

f

U U

U

Figure The trace T r f

XY

U

From a computer science viewp oint the essential feature is to think of T r f as feedback

XY

along U as in Figure The axiomatization given here diers slightly from those in Abr JSV

although it can b e shown to b e equivalent We shall leave it to the reader to draw the diagrams for

the trace axioms We note however that Vanishing expresses trace along a tensor U V in terms

of iterated traces along U and V This is related to the socalled Bekic Lemma in Domain Theory

The ab ove notion is really a parametrized trace The usual notion from linear algebra is when

X Y I see Example b elow

Example Rel The ob jects are sets cartesian pro duct and maps are binary

U

relations Comp osition means comp osition of relations and x T r y i there exists a u such that

XY

x uRy u

P

k j

U

y where Example V ec Given f X U Y U dene T r f x

k f d i

ij

jk XY

P

k m

f x u y u where u x y are bases for U X Y resp In the case that

i j k m i j k

ij

k m

X Y are onedimensional this reduces to the usual trace of a linear map f U U ie the usual

trace determines a function T r H omU U H omI I where I k

U

U

Example More generally any compact category has a canonical trace T r f X X

XY

id f id

idev

o

I X U U Y U U Y I Y where ev ev s

Example CPO with I fg In this case the dinatural leastxedp oint

()

combinator Y induces a trace given as follows using informal lamb da calculus

U

o

notation for any f X U Y U T r f x f x Y uf x u where f f

1 U 2 1 1

XY

U

o

f x f x u where u is the smallest X U Y f f X U U Hence T r

1 2 2

XY

element of U such that f x u u A generalization of this idea to traced cartesian categories is

2

in MHas and mentioned in Remark in the next Section

Unfortunately these examples do not really illustrate the notion of feedback as data ow the

movement of tokens through a network More natural examples of traced monoidal categories in

the next section given by partially additive and similar iterative categories more fully illustrate

this asp ect

Example Bicategories of Processes The pap er of Katis Sabadini and Walters KSW

develops a general theory of pro cesses with feedback circuits in symmetric monoidal bicategories

They prove their bicategories C ir cC have a parametrized trace op erator A small dierence

with the ab ove treatment is that their feedback is given by a family of partial lydened functors

U

C ir cC X U Y U C ir cC X Y f b

XY

Remark The pap er ABP develops a general theory of traced ideals in tensored categories

The category HI LB the tensored category of Hilb ert spaces and b ounded linear maps illustrates

the diculty In passing from the nite dimensional case cf Example ab ove to the innite

dimensional one not all endomorphisms have a trace for example the identity on an innite

dimensional space However T r may b e dened on traced ideals of maps and this extends to

U

parametrized traces See ABP for many examples

X Y Z

f

f g

X Y Z

R

g

Y Z

Figure Generalized Yanking

An amusing folklore ab out traced monoidal categories is that general comp osition is actually

denable using traces of simple comp ositions

Prop osition Generalized Yanking Let C be a traced symmetric monoidal category with

Y

o o

arrows f X Y and g Y Z Then g f T r s f g

Y Z

XZ

Although a fairly short algebraic pro of is p ossible the reader may wish to stare at the diagram

in Figure and do a stringpulling argument cf JSV Similar calculations are in KSW

Mil

Denition Let C and D b e traced symmetric monoidal categories A strong monoidal functor

F C D is traced if it is symmetric and satises

FU 1 U

T r F f F T r f

AU

F AF B B U AB

1

F f f

AU B U

F A U F B U where A U B U and FA FU FB F U In the case of

strict monoidal functors they are traced if they preserve the trace on the nose

We dene TraMon and TraMon to b e the categories whose cells are traced monoidal

st

categories resp strict traced monoidal categories whose cells are traced monoidal functors

resp strict traced monoidal functors and whose cells are monoidal natural transformations

Partially Additive Categories

We shall b e interested in sp ecial kinds of traced monoidal categories those whose homsets are

enriched with certain partiallydened innite sums which p ermits canonical calculation of iteration

and traces see formulas and b elow A useful example is Manes and Arbibs partial ly additive

categories which rst arose in their categorical analysis of iterative and owchart schema MA

Categories with similar additive structure on the homsets had already b een considered in the

s by Kuros Ku with regards to categorical KrullSchmidtRemak theorems

Denition A partial ly additive monoid is a pair M where M is a nonempty set and

is a partial function which maps countable families in M to elements of M we say that x ji I

i

5

is summable if x ji I is dened sub ject to the following

i

PartitionAssociativity Axiom If x ji I is a countable family and if I j j J is a

i j

countable partition of I then x j i I is summable if and only if x j i I is summable

i i j

for every j J and x j i I j j J is summable In that case

i j

x j i I x j i I j j J

i i j

We sometimes abbreviate x j i I by x Throughout countable means nite or denumerable All

i i2I i

index sets are countable A partition fI j j J g of I satises I I I I if i j and fI j j J g I

j j i j j

But we also allow I for countably many j

j

Unary Sum Axiom Any family x j i I in which I is a singleton is summable and x j i

i i

I x if I fj g

j

Limit Axiom If x j i I is a countable family and if x j i F is summable for every

i i

nite subset F of I then x j i I is summable

i

We observe the following facts ab out partially additive monoids

i Axioms and imply that the empty family is summable We denote x j i by

i

which is an additive identity for summation

ii Axiom implies the obvious equations of commutativity and asso ciativity for the sum when

dened

iii Although Manes and Arbib use the Limit Axiom to prove existence of Elgotstyle iteration

see b elow Kuros did not have it And for many asp ects of the theory b elow it is not

needed

Denition The category of partial ly additive monoids PAMon is dened as follows Its

f

ob jects are partially additive monoids M Its arrows M N are maps from

M N

M to N which preserve the sum in the sense that f x j i I f x j i I for all

M i N i

summable families x j i I in M Comp osition and identities are inherited from Sets

i

A PAMoncategory C is a category enriched in PAMon This means the homsets carry a

PAMonstructure compatible with comp osition In particular in each homset H om X Y there

C

is a zero morphism X Y the sum of the empty family

XY

Remark In a PAMoncategory C

The family of zero morphisms f g satises g f for any f W X

XY XY C WZ WY XY

and g Z Y

If f then all summands f in H om X Y

iI i XY i XY C

L

X For any j I Denition Let C b e a PAMoncategory with countable copro ducts

i

iI

L

we dene quasi projections PR X X as follows

j i j

iI

id if k j

X

j

PR in

j k

else

X X

k j

Denition A partial ly additive category pac C is a PAMoncategory with countable cop

ro ducts which satises the following axioms

Compatible Sum Axiom If f j i I C X Y is a countable family and there exists

i

P

f X I Y such that PR f f we say the f are compatible then f exists

i i i i

Untying Axiom If f g X Y exists then so do es in f in g X Y Y

1 2

The following facts ab out partially additive categories follow from Manes and Arbib MA

L L

Y there is a unique family X Matrix Representation of maps For any map f

j i

j J iI

P

ff X Y g with f in f PR and PR f in f Notation we write

ij i j iI j J j ij i j i ij

iI j J

f for f

X Y ij

i j

Y

X

g

X

y

Figure Elgot Dagger g

y

P

g g

y n

Elgot Iteration Given X Y X there exists X Y where g g g satisfying

XY

n=0 XX

the xedpoint identity

y y

g g g

Y

This corresp onds to the owchart scheme in Figure The pro of of Elgot iteration

MAp uses the Limit Axiom

Prop osition A partial ly additive category C is traced monoidal with coproduct and if

f X U Y U

X

U y n

T r f f f f f f f

Y 1 XY UY XU

XY 2 UU

n=0

where f f f with f X Y U f U Y U

1 2 1 2

Formula corresp onds to the data ow interpretation of traceasfeedback in Figure see

the examples b elow We should also remark that Formula corresp onds closely to Girards

Execution Formula Abr and is related to a construction of Geometry of Interaction categories

in the next section

Remark Conversely a traced monoidal category where is copro duct has an Elgot iterator

y X

g g where g X Y X An axiomatization of the opposite of such categories g T r

XY

which corresp ond to categories with a parametrized Y combinator is considered in Hasegawa

MHas More generally Hasegawa considers traced monoidal categories built over cartesian

categories and it is shown how various typ ed lamb da calculi with cyclic sharing are Sound and

Complete for such categorical mo dels

Finally we should mention the general notion of Iteration Theories These general categorical

theories of feedback and iteration their axiomatization and equational logics have b een studied

in detail by SL Blo om and Z Esik in their b o ok BE A more recent categorical study of

iteration theories is in BELM

We shall now give a few imp ortant examples of pacs

Example Rel the category of sets and relations Ob jects are sets and maps are binary

+

relations Comp osition means relational comp osition The identity is the identity relation and

L

the zero morphism is the empty set X Y Copro ducts X are as in Set ie

XY i

iI

disjoint union Al l countable families are summable where R R Finally let R b e

iI i iI i

the reexive transitive closure of a relation R Supp ose

R X U Y U Then formula b ecomes

U n

o o

T r R R R R R

XY UY XU

XY UU

n0

o o

R R R R

XY UY XU

UU

Example Pfn the category of sets and partial functions The ob jects are sets the maps

are partial functions Comp osition means the usual comp osition of partial functions The zero map

is the empty partial function A family ff j i I g is said to b e summable i i j I i

XY i

j D omf D omf f is the partial function with domain D omf and where

i j iI i i i

f x if x D omf

j j

f x

iI i

undened else

The following example comes from Giry Giry inspired from Lawvere and is mentioned in

Abr The fact that this is a pac follows from work of P Panangaden and E Haghverdi

Example SRel the category of Sto chastic Relations Ob jects are measurable spaces X

X

where X is a set and is a algebra of subsets of X An arrow f X Y is a

X X Y

transition probability ie f X such that f B X is a measurable

Y

function for xed B and f x is a subprobability measure ie a additive

Y Y

set function satisfying f x and f x Y The identity morphism id X

X X

X is a map id X with id x A x A where for A xed x A is

X X X X

the characteristic function of A and for x xed x A is the Dirac distribution

Comp osition is dened as follows given f X Y and g Y Z

X Y Y Z

R

o o

g y C dff x g where the notation dff x g means g f X Z is g f x C

X Z

Y

that we are xing x and using f x as the measure for the integration the function b eing integrated

is the measurable function g C

Given X and Y the zero morphism X Y is given by

X Y XY X Y

x B for all x X and B

XY Y

The partially additive structure on the homsets of SRel is as follows we say an I indexed family

P

f x Y Since we are dealing of morphisms ff ji I g is summable if for all x X we have

i i

iI

with b ounded p ositive measures it is easy to verify that the sum so dened is a subprobability

measure Note that we would have only trivial additive structure only singleton families summable

if we had used probability distributions rather than subprobability distributions

L

Finally let fX ji I g b e a countable family of ob jects We dene the copro duct X as

i i

iI

follows We take the disjoint union of the sets X equipp ed with the evident algebra Thus a

i

measurable subset will lo ok like the disjoint union of measurable subsets of each of the X say

i

A of course some of the A may b e empty and a p oint will b e a pair x i where i I and

i i

L

x X The canonical injections in X X are in x A x A Given Y and

i j j i j i j

iI

L

X Y by the formula i I arrows h X Y we obtain the mediating morphism h

i i i

iI

hx j B h x B The verications are all routine

j

The next example while not a pac is essentially similar

Example P inj the category of sets and injective partial functions This is a fundamental

example that arises in Girards Geometry of Interaction program Although this category is traced

monoidal with an iterative trace formula given in Abramsky Abr it do es not have copro ducts

However its paclike asp ects may b e captured in a Kurosstyle presentation via a generalization

of partially additive categories in which countable copro ducts are replaced by countable tensors

and in which suitable axioms guarantee analogously to pacs a matrix representation of maps

N N

Y and a trace formula as in X

j i

j J iI

GoI Categories

Girards Geometry of Interaction GoI program intro duces some profound new twists into com

putation theory In particular the idea that pro ofs are like dynamical systems interacting lo cally

The dynamics of information ow in comp osition via cutelimination is then related to tracing

out paths in certain algebraic structures Girard originally used op erator algebras but the results

can b e expressed without them Gib The connection of Girards functional analytic metho ds

in GoI with lamb da calculus and pro of nets is further explored in DR MaRe

Starting with a traced monoidal category C we now describ e a compact closed category G C

called I ntC in JSV which captures in abstract form many of the features of Girards Geo

metry of Interaction program as well as the general ideas b ehind game semantics We follow the

evo cative treatment in Abramsky Abr The idea is to create a category whose comp osition is

given by an iterative feedback formula using the trace

Denition The Geometry of Interaction construction Given a traced monoidal category C

we dene a compact closed category G C as follows JSV Abr

+ +

Ob jects Pairs of ob jects A A where A and A are ob jects of C

+ + + +

Arrows An arrow f A A B B in G C is an arrow f A B A B

in C

+ +

Identity

(A A ) A A

+ +

Comp osition given by symmetric feedback Arrows f A A B B and g

+ + + +

o

B B C C have comp osite g f A A C C given by

+

B B

o

f g g f T r

+ +

A C A C

+ + + + + +

where and

A B C B A C B B A C B B A

o

+ + + +

An informal picture displaying g f is given b elow

B C B A B B C

+

+

A

B

B B C

B

B

B

B

f g

B

B

B

B

+

+

B

A B C

B

+ + + + + +

Tensor A A B B A B A B and for A A B B and

+ +

+ + + +

g C C D D f g f g

A B C D A C B D

Unit I I

+ + +

Duality The dual of A A is given by A A A A where the unit I I

+ + + +

+

and counit A A A A I I A A A A

def def A A

+

A A

+ + + + + +

Internal Homs As usual A A B B A A B B A B A

B

Remark We have used a sp ecic denition for and ab ove however any other p ermuta

= =

+ + + + + + + +

tions A C B B A B B C and A B B C A C B B

o

for and resp ectively will yield the same result for g f due to coherence

Translating the work of JSV in our setting we obtain that G C is a kind of free compact

closure of C

Prop osition Let C be a traced symmetric monoidal category

G C dened as in Denition is a compact closed category Moreover F C G C

C

dened by F A A I and F f f is a ful l and faithful embedding

C C

The inclusion of categories C ompC l T r aM on has a left biadjoint with unit having

component at C given by F

C

+

Following Abramsky Abr we interpret the ob jects of G C in a gametheoretic manner A

is the typ e of moves by Player the System and A is the typ e of moves by Opp onent the

Environment The comp osition of morphisms in G C is the key to Girards Execution formula

esp ecially for paclike traces In Abr it is p ointed out that G P inj is essentially the original

6

Girard formalism while G CPO is the dataow mo del of GoI given in AJa

Literature Notes

In the ab ove we have merely touched on the large and varied literature The journals Mathematical

Structures in Computer Science Camb Univ Press and Theoretical Computer Science Elsevier

are standard venues for categorical computer science Two recent graduate texts emphasizing

categorical asp ects are J Mitchell Mit and R Amadio and PL Curien AC Mitchells

b o ok has an encyclop edic coverage of the ma jor areas and recent results in programming language

theory The AmadioCurien b o ok covers many recent topics in domain theory and lamb da calculi

including full abstraction results foundations of linear logic and sequentiality

We regret that there are many imp ortant topics in categorical computer science which we

barely mentioned We particularly recommend the comp endia PD FJP AGM Let us give a

few p ointers with sample pap ers

Operational and Denotational Semantics See the surveys in the Handb o okAGM The clas

sical pap er on solutions of domain equations is SP For some recent directions in domain

theory see FiPl ReSt For recent categorical asp ects of Op erational Semantics see

Pi TP Higherdimensional category theory has also generated considerable theoretical

interest eg Ba HMP Coalgebraic and coinductive metho ds are a fundamental tech

nique and have considerable inuence eg see AbJu Pia Mul CSp JR Mil

Fibrations and Indexed Category Models This imp ortant area arising from categorical logic

is fundamental in treating dep endent typ es System F F mo dels and general variable

binding quantierlike op erations for example hiding in certain pro cess calculi For bred

category mo dels of dep endent typ etheories see the survey by M Hofmann Ha cf also

PowTh HJ Pi See Indexed category mo dels for Concurrent Constraint Logic

Programming are given in PSSS MPSS see also FRS for connections of this latter

paradigm to LL

Added in pro of recent progress on these matters has b een achieved in the PhD thesis of Esfan Haghverdi Dept

of Mathematics U Ottawa Feb and in a pap er to app ear in the Lamb ek Festschrift Math Structures in

Computer Science See also httpaixuottawacaehaghver

Computational Monads E Moggi greatly inuenced programming language semantics and

asso ciated logics using the categorists notion of monads and comonads Mac Moggis ap

proach p ermits a mo dular treatment of such imp ortant programming features as exceptions

sideeects nondeterminism resumptions dynamic allo cation etc as well as their asso ci

ated logics Mo Mo Practical uses of monads in functional programming languages are

discussed in P Wadler W More recently E Manes M showed how to use monads

to implement collection classes in Ob jectOriented languages Alternative categorytheoretic

p ersp ectives on Moggis work are in Power and Robinsons PowRob

Concurrency Theory and Categorical Bisimulation This large and imp ortant area is surveyed

in Winskel and NielsenWN In particular the fundamental notion of bisimulation via op en

maps is intro duced in Joyal Nielsen and Winskel JNW Presheaf mo dels for Milners

calculus Mila and other concurrent calculi are in CaWi CSW For categorical work

on Milners recent Action Calculi see GaHa Milb Mil P

Complexity Theory Characterizing feasible eg p olynomialtime computation is a ma jor

area of theoretical computer science eg Cob Co ok Typ ed lamb da calculi for feasible

higherorder computation have recently b een the sub jects of intense work eg CoKa CU

Versions of linear logic have b een develop ed to analyze the ne structure of feasible computa

tion GSS Gi Although there are some known mo dels KOS general categorical

treatments for these versions of LL are not yet known Recently M Hofmann eg Ha

has analyzed the work of Co ok and Urquhart CU as well as giving higherorder extensions

of work of Bellantoni and Co ok using presheaf and sheaf categories

Three volumes MR FJP GS are conferences sp ecializin g in applications of categories

in computer science note MR is the th Biennial such meeting Similarly see the biennial

meetings of MFPS Mathematical Foundations of Programming Semanticspublished in either

Springer Lecture Notes in Computer Science or the journal Theoretical Computer Science There

is currently an electronic website of categorical logic and computer science HYPATIA

Other b o oks covering categorical asp ects of computer science andor some of the topics covered

here includeAL BW Cu DiCo Gun MA Tay In categorical logic and pro of

theory we should mention our own b o ok with J Lamb ek LS which b ecame p opular in theoretical

computer science The category theory b o ok of Freyd and Scedrov FrSc is a source b o ok for

Representation Theorems and categories of relations

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