Order Sorted Computer Algebra
and Co ercions
submitted by
Nicolas James Doye
for the degree of PhD
of the
University of Bath
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Nicolas James Doye
Abstract
Computer algebra systems are large collections of routines for solving mathematical
problems algorithmically e ciently and ab ove all symb olically The more advanced
and rigorous computer algebra systems for example Axiom use the concept of strong
typ es based on order sorted algebra and category theory to ensure that op erations are
only applied to expressions when they make sense
In cases where Axiom uses notions whichare not covered by current mathematics we
shall present new mathematics which will allow us to prove that all such cases are
reducible to cases covered by the current theory On the other hand we shall also
p oint out all the cases where Axiom deviates undesirably from the mathematical ideal
Furthermore we shall prop ose solutions to these deviations
Strongly typ ed systems esp ecially of mathematics b ecome unusable unless the system
can change the typ e in a way a user exp ects We wish any typ e change exp ected by
a user to be automated natural and unique Co ercions are normally viewed as
natural typ e changing maps this thesis shall rigorously de ne the word co ercion
in the context of computer algebra systems
We shall list some assumptions so that wemayprove new results so that all co ercions
are unique this concept is called coherence
We shall give an algorithm for automatically creating all co ercions in typ e system
which adheres to a set of assumptions We shall prove that this is an algorithm and
that it always returns a co ercion when one exists Finally we present a demonstration
implementation of this automated co ercion algorithm in Axiom
Contents
Intro duction
Intro duction
Computer algebra
Strong typ es categories varieties and theories
Abstract datatyp es in general
The problem
Examples of how Axiom co erces
Mathematical solution overview
Constructing co ercions algorithmically
Background of Axiom
OBJ
Aims of the thesis
Typ es in computer algebra
Intro duction
Mathematica
Reduce
Axiom
Economyofe ort
Interest
Contents
Functoriality
Newsp eak
OBJ
Ob jects
Theories
Views
Comparison
Category theory
Intro duction
Category theory
Categories and Axiom
Functors and Axiom
Co ercion and category theory
Conclusion
Order sorted algebra
Intro duction
Term Algebras
Order sorted algebras
Extension of signatures
The equational calculus
Signatures theories varieties and Axiom
Conclusion
Extending order sorted algebra
Intro duction
Contents
Partial Functions
Conditional varieties
A Category theory approach
Co ercion
Conclusion
Coherence
Intro duction
Web er s work I de nitions
Web er s work I I assumptions and a conjecture
The coherence theorem
Extending the coherence theorem
Conclusion
The automated co ercion algorithm
Intro duction
Finitely generated algebras
Constructibility
The algorithm
Existence of the co ercion
Proving co erciveness
Conclusion
Implementation details
Intro duction
Bo ot and Axiom
The top level
Lab elling op erators
Getting information from domains
Contents
Checking information from domains
Flaws in the implementation
Conclusion
Making Axiom algebraically correct
Intro duction
Explicitly de ned theories
Op erator symb ols and names
Moving certain op erators
Retyping certain sorts
Sorts and their order
Altering Axiom s databases
Conclusion
Conclusions
Intro duction
Summary of work done
Formalising strongly typ ed algebra systems
Representation and syntax issues
On coherence
The automated co ercion algorithm
Future work and extensions
A Extra category theory
A Intro duction
A Extra de nitions
B Set theory
B Intro duction
B Basics
Contents
B Inductive class approximation
B Inductive classes and category theory
B Inductive classes categories and Axiom
Bibliography
Structure of the Thesis
Structure of the Thesis
The thesis can b e viewed as decomp osable into the following topics
Background and existing theory
Chapter intro duces computer algebra systems in general strong typing and abstract
datatyping We intro duce some of the problems that can occur in a strongly typ ed
language which uses abstract datatyp es We present an analogy of the solution to one
of the problems in such a system
In chapter we discuss and compare the typ e systems in several mo dern computer
algebra systems We also mention OBJ a computer language which implements a typ e
system based on abstract datatyp es
Both category theory chapter and universal algebra chapter may be used as
the basis for systems of abstract datatyp es In these chapters we state all the current
theory whichwe shall require in the later chapters
W e shall also compare how many of these theoretical notions are represented in the
computer algebra system Axiom
New theory
Chapter details some extra mostly new mathematics which allows for certain ex
tensions to the theory of abstract datatyp es as currently implemented in Axiom to
b e formalised in the language of order sorted algebra
We brie y lo ok at the corresp ondence b etween the twoapproaches to abstract datatyp
ing This allows us to use either terminology when one may b e more appropriate than
the other
Finally we formalise the meaning of the word co ercion which is used widely in Axiom
without any strict de nition
In chapter we lo ok at the work of Web er in the area of coherence We adjust and
extend his work so that w e may sensibly restrict the typ e system of Axiom and still
have unique co ercions
Chapter intro duces new de nitions and assumptions on the typ e system whichallow
us to create co ercions automatically Weprovide a function to do so and provethatit
is an algorithm which creates a unique co ercion if and only if one exists
Structure of the Thesis
Demonstration implementation
In chapter we detail how the automated co ercion algorithm may b e implemented in
Axiom We mention where this prototyp e implementation disagrees with the theoretical
algorithm
Conclusions and future extensions
In chapter we list the di erences b etween Axiom s currenttyp e system and the formal
theoretical typ e system wehavespeci ed
We draw conclusions from this work in chapter We also state future directions
research may take towards the aim of creating the p erfect computer algebra system
based on abstract datatyp es
Acknowledgements
Acknowledgements
None of this work would have b een p ossible without the original input and ideas of my
sup ervisor Professor James H Davenp ort
The implementation would have b een imp ossible without the help from the following
Lisp and or Axiom internals exp erts Professor James H Davenp ort again Mike
Dewar Russell Bradford Pete Broadb erry
Mathematical issues were discussed at length with Professor James H Davenp ort yet
again Tracy Gardner Russell Bradford Mark Beanz Hindess
My thanks go out to both Professor James H Davenp ort and Russell Bradford for
pro of reading various versions of this thesis
In addition to all the intellectual supp ort there are two p eople without whom none of
this would have been p ossible I refer of course to my wife Deb orah and my father
Mr Geo rey Doye
thank Professor John tch for giving me this chance in the rst place I d I d liketo
also b etter thank all the p ostgrads in the computing group in the last few years for
putting up with me
My thanks go to NAG for supplying the Axiom Bo ot source co de Also to the follow
ing computing services departments for the use of their systems University of Bath
Department of Mathematical Sciences computing services BUCS Universityof Bath
Computing Services NISS National Information Services and Systems
This research was funded by the EPSRC previously known as SERC Much needed
p ersonal nancial assistance came from my father and part time employment from
b oth BUCS and the Department of Mathematical Sciences In the overrun I have
b een employed by both the Department of Mathematical Sciences and more recently
by NISS
Finally I ought to thank the authors of XEmacs for making editing everything just so
A
easy and the authors of L T X for the layout of this thesis Diagrams were pro duced
E
using Paul Taylor s diagrams package
Dedication
For Deborah
Chapter
Intro duction
Intro duction
This chapter intro duces the reader to the main ideas b ehind this thesis Wegive some
background information explain the context of the work and say why the work is a
go o d idea
This work centres on typ es in computer algebra
Computer algebra
Computer algebra DST is an all encompassing term that covers p erforming anysort
of mathematics on a computer Numerical analysis mathematics involving oating
p ointnumb ers and a certain degree of error is often excluded from this term though
most mo dern computer algebra systems are at the very least comp etent in this area
Computer algebra systems may excel in various areas of mathematics Some are masters
of discrete mathematics groups rings monoids etc the more common systems deal
with more mainstream concerns such as inde nite integration solving di erential
equations matrices etc
The imp ortant feature of all computer algebra systems is their abilitytosolve equations
like di erential equations symbolical ly In other words they maysolve equations for
the general case they need not substitute values for the variables
Mo dern computer algebra systems are large collections of routines and algorithms
These algebra systems often come with their own language s for user input The user
Chapter Intro duction Computer algebra
is usually supplied with an interpreter which can usually read both input from the
keyb oard and from les The more advanced computer algebra systems come with a
compiler to o
Common computer algebra systems are Axiom JS Derive Der GAP GAP
Macsyma Mac Magma Pla Maple CGG Mathematica Wol and Re
duce Hea We discuss the relevant details of some of these systems in chapter
The user of a computer algebra system is usually someone who wishes to solve a math
ematical problem Mathematics is a formal sub ject and as such there is a need for
rigour Hence computer algebra systems need a rigorously de ned language to avoid
confusion and incorrect results
this Many of the more popular commercial computer algebra systems do not provide
rigour Instead they trade rigour and strictness for ease of use and hence mass market
app eal This is not to say that they consistently supply incorrect results merely that
if the user has not given all the input that they can or are allowed to by constraints
of the system then the system may misinterpret their input
Being able to restrict the domain over which a variable may range or from where a
function may take its arguments can reduce the amount of confusion and ambiguityin
any mathematics a user mayenter
For example if a user enters a then a computer algebra system has to judge whether
this is the from the integers a constant p olynomial the identity of some group a
from the integers mo dulo some n the distance which is not zero in a discrete metric
space etc
Restricting variables to various sets of values is a useful concept and is called typing
For example if one says that x is an integer and then declares x to b e then there is
no confusion as to what x is We are free to add x to x safe in the knowledge that the
answer is the integer If we did not know that x were an integer then the answer
could havebeen in Z in a group where is the identity and the binary op erator
is or any other value in some contrived example
Such restrictions on variables are usually called typ es For example declaring x to b e
an integer is the same as saying that the typ e of x is an integer
Mathematics written in b o oks is usually stated formally with all the context and
meaning of certain symb ols stated in the text When mathematicians talk to one
another they may and in practice do drop some of this formality since they can infer
text from extra mathematical factors such the meaning of certain symb ols and the con
Chapter Intro duction Strong typ es categories varieties and theories
as with whom they are holding a conversation previous conversations they mayhave
had etc They mayeven correct eachother
Computer algebra systems are not p eople and can fail to infer the typ e and meaning
of terms successfully They also can not understand paragraphs of English or other
natural languages to gain the context from there They are highly unlikely to correct
the user This is why their languages should b e strictly de ned and yet fully expressive
Guessing the typ e intelligently is called typ e inference All the untyp ed systems
e ectively guess the typ e internally whenever a function is applied to any data
Entering will result in simply this is b ecause this is the most common case
etyp e inference in Axiom is to allowthe user to say A more intelligent approach lik
which function they require Axiom will then infer the typ es of the two s from the
fact it knows whichtyp es that is exp ecting Alternatively the user may state which
stheyhave inputted and what typ e they exp ect in return Axiom will then use the
function uniquely determined by this information
Strong typ es categories varieties and theories
In this thesis we are mainly concerned with the typing mechanisms in mo dern computer
algebra systems However this work is also relevant to typ e systems in general Many
languages principally the so called ob ject oriented languages use a typ e system which
may o ccasionally b e viewed in a mathematical fashion
Axiom and Magma are two of the b est computer algebra systems and what sets them
apart from their p eers is the concept of categories chapter or varieties sp eci ed by
theories section Some other algebra systems do possess the concept of typ es
and Reduce Hea will allownewtyp es to b e intro duced section GAP GAP
also claims to have a concept of categories though they are not so deeply ingrained
in the typ e system as they are in Axiom section and Magma section GAP
will also allow new typ es to b e intro duced
Certain forms of inheritance found in ob ject oriented languages follow the paradigms
used in Axiom and Magma However most ob ject oriented languages C Str
Ei el Eif Java Gra youto o Kin etc are more exible and allow many
di erent ways for typ es to re use bits of co de from each other However automated
co ercion in any sense is not a feature of these languages OBJ GWM is not a
computer algebra system however this work is wholly suited to it since it uses a similar
typ e paradigm to Axiom and Magma We discuss OBJ further in sections and
Chapter Intro duction Abstract datatyp es in general
The category or variety mechanism found in Axiom and Magma is a way of abstracting
a further level away from mere typ es The categories are used to typ e the typ es and
provide a standard interface and or implementation for eachtyp e in a category
Magma also abstracts one further level away by allowing the existence of varieties of
categories in a similar way
The concept of strong typing is that every thing must haveatyp e Axiom and Magma
are both strongly typ ed Most of the other systems are not strongly typ ed Indeed
some do not even have any notion of typ es at all
e The categories or varieties of Axiom Magma and OBJ are abstract datatyp es W
shall detail this more in the next section
Abstract datatyp es in general
Abstract datatyp es provide a way of sp ecifying a typ e As a useful side e ect an
abstract datatyp e can express how a whole collection of typ es mayact
An incredibly sp eci c sp eci cation will only sp ecify typ es which are all the same
isomorphic A less exacting sp eci cation may result in typ es which are similar but
not the same
This side e ect of using a less exacting sp eci cation to collect similar typ es together
forms the basis of strict categorical typ e systems likethat in Axiom and many ob ject
oriented languages
For example we may say which op erations are available on a certain family of typ es
We may also state certain facts ab out the structure of all typ es in a certain family
Most imp ortantly of all we can enforce relationships b etween various typ es of di erent
families how they interact and howtheymay dep end up on each other or not
See section for howwemay sp ecify typ es and what op erations are available to them
Lo ok at section for the relationships between di erent typ es Section details
howwemay restrict the structure of certain typ es using sets of equations
As a simple example a p olynomial ring dep ends for its sp eci cation on the underlying
ring from which the co e cients are taken
As amore complicated example a p olynomial ring could also dep end up on the typ e
from which it takes its variables the typ e of the exp onents a b o olean algebra an
often a ordered free monoid with one generator the natural numb ers adjoin f g is
Chapter Intro duction The problem
go o d choice and mayb e a few others
Another factor is the actual relationship between the various typ es In our ab ove
example of the p olynomial ring weknowthat
there is a ring monomorphism from the underlying ring to the p olynomial ring
there is an injection from the variable typ e to the p olynomial ring In fact an
ordered set monomorphism
Other typ es often have more complicated lattices of relationships These relationships
are often able to be abstracted out to apply all the instance typ es of a particular
abstract data typ e
The problem
A commonly encountered di culty in languages which utilise typ es is the following
Given thing of typ e Type can we change the typ e of thing to be that of
Type More accurately canwe create an elementof Type which corresp onds
in some natural waytothing of typ e Type
If so how do we go ab out p erforming such an op eration Can we p erform this
task algorithmically
Do es there exist a way of abstracting this question or must every Type Type
pair be considered Moreover can we abstract such an algorithm as mo oted
in ab ove out to cover all cases
Such typ e changes are often called coercions conversions or castings with all three
words having slightly di erent meanings
versions to mean any typ e change In this work we shall consider castings and con
regardless of mathematical rigour We shall call a typ e change a co ercion if it is in
some way natural We will de ne this more rigorously later in de nition
The typ e from which all the variables are taken In Axiom the variables are usually elements such
as X Y and Z which are from the typ e Symbol
This is an injective function which is b oth a set homomorphism which is just a total function
and preserves order So if a b then a b Mathematicians often utilise the order on the
variables and extend it to an order on p olynomials
Chapter Intro duction The problem
Computer algebraists often need to use co ercions since informally most mathematicians
alter the domain of computation without saying phrases like in the monomorphic
image of A in B or forgetting that x is a T and viewing it as a U
For example we often view the integers Z as p olynomials in Z x rather than
considering the constant p olynomials of Z x as b eing a monomorphic copyofZ This
is the co ercion Z Z x
Other simple examples of this sort of co ercion are
Z Q Z x Z y x Z Q x y S S Z x Z x
where S n is the symmetric group on a set of n symb ols
Co ercions are useful when creating elements of quotient structures suchas Z Z
n
Z nZ where n N Other epimorphic surjective examples include
Z Z G G G
where G is any group and G is the commutator subgroup of G
Under certain circumstances co ercions can be lifted into other constructors We
refer to these as structural co ercions later in this work As examples consider
List Z List Q List Z List Z Z x Z x
Z M Z x M
Examples of typ e conversions which are not co ercions include
Q Float List N S n Z Z Z Z Z x Z y
Natural typ e changes co ercions can b e created in the Axiom interpreter at the moment
using a series of ad hoc measures All of this o ccurs transparently to the user and works
for most common typ es However it is not algorithmic One of the aims of this thesis
is to provide an algorithm algorithm to create unique theorems
co ercions
Chapter Intro duction Examples of how Axiom co erces
Examples of how Axiom co erces
Axiom knows or b elieves that all functions called coerce are co ercions and will use
them to build other co ercions using certain rules and sp ecial cases de ned in the
interpreter Here we give examples of how Axiom co erces between certain common
typ es
Simple examples
Example Z Q
In Axiom Z is the typ e Integer and Q is the typ e Fraction Integer The typ e
constructor Fraction R Ring exp orts a function
coerce R
which is the natural monomorphism including the ring R in its fractional eld
Fraction R
Example Z Z X
In Axiom the typ e UnivariatePolynomial X Integer typically represents Z X
This is constructed bythetyp e constructor
UnivariatePolynomial S Symbol R Ring PolynomialCategory R
NonNegativeInteger Symbol
UnivariatePolynomial exp orts a function
coerce R
which is the natural monomorphism including the ring R in the p olynomial ring as the
constant p olynomials
Example Z X Z X Y where Z X Y Z X Y
This is the same as the previous example Y is acting as a morphism Z X Z X Y
Actually de ned in a Category to which Fraction b elongs
Chapter Intro duction Examples of how Axiom co erces
Example Z X Z Y X whereZ Y X Z Y X
Axiom knows that Z Z Y is a co ercion and that the map X Z Y Z Y X
ie UnivariatePolynomial lifts this co ercion
Example Z Q X Y
This is built by the chain of co ercions
Z Q Q X Q X Y
Example S S
These do not exist as domains in Axiom
Example Z X Z X
the co ercion from Fraction discussed ab ove since Z X is represented in This uses
Axiom by Fraction UnivariatePolynomial X Integer
Example Z Z
This uses the function
coerce Z
de ned in IntegerMod n PositiveInteger
Example Z Z
Axiom cannot co erce from IntgerMod to IntegerMod
Structural co ercions
The structural co ercions are lifts of other co ercions In the case of Lists one may
p erform a structural co ercion as follows
coerce x List A List B
map coerce x List B
map f List A List B x List A List B
nullp x nil List B
cons f car x map f cdr x
Chapter Intro duction Mathematical solution overview
In the case of Polynomials a similar map function exists which uses instead of cons
In the case of Matrix a similar map function can be used though since matrices are
of xed sizes unlike sets lists and p olynomials a default valued matrix has to b e
created then all the values substituted into added to it
Mathematical solution overview
The solution to the questions raised in of section rely on abstract datatyp es
These collect large numbers of typ es into collections of similar typ es We shall be
considering abstract datatyp es in the language of universal algebra sp eci cally order
sorted algebra section
Universal algebra has close links with the ideas of category theory See chapter for
category theory and section for the links b etween the two theories Indeed Axiom
algebraic design Thus we shall uses the language of category theory to describ e its
also discuss the fundamentals of category theory in relation to this work
Using the order sorted algebra framework we shall show that there is indeed a strict
mathematical de nition de nition which we may use to tell us which typ e
changes are natural and therefore in our de nition co ercions
Moreover we shall show that this approach allows us to create such co ercions algo
rithmically algorithm
In section we shall show an analogy with moving buildings to the solution of chang
ing the typ e of ie co ercing something
Constructing co ercions algorithmically
Consider the following analogy
Supp ose you have been given the job of moving London Bridge from one lo cation to
another The bridge is to o large to move in one piece
You will merely be sup ervising the work and will not have to carry any of the bricks
yourself hence e ciency is not your concern
You are also planning to move other di erent bridges in the future and once you have
a metho d for moving one bridge you would like to b e able to apply that metho d to the
next This means that next time a bridge needs moving you won t have to do much
Chapter Intro duction Constructing co ercions algorithmically
work at all
Luckily for you someone has already worked out a way of moving any of the bricks
at the very b ottom of the bridge to their new lo cation regardless of any obstruction
whichwould normally leave civil engineers crying This p erson was clearly a magician
So here is your algorithm for moving London bridge and hence any future bridge
If the original bridge has no bricks left then finished
Else
take a brick from the top of the original bridge
If this brick is a bottom brick then
use the magical bottom brick mover to place
the brick in the corresponding location of
the new bridge
Else
the brick in place in the corresponding hold
location in the new bridge
Endif
Endif
Repeat
Indeed you realize that this metho d will work for other brick based constructions or
indeed anything that is built Obviously there are other considerations is the new
lo cation for the bridge a natural one For example Is it long or tall enough
The analogue of the bridge is a thing or item in our computer algebra system The
original lo cation of the bridge is the original typ e of the item and the new lo cation
for the bridge is the analogue of the typ e to whichwe are co ercing the item
Thus what weneedtoknoware
what are the foundations of a bridge
Analogue what are the constants of our typ e
what is the length width strength of a bridge
Analogue what are the parameters of the typ e
howdoyou alter the bricks to b e of a new shap e material
Analogue we need to be able to recursively call the co ercion algorithm on
typ es whichare recursively required for co ercion cf De nition That
is typ es whic h form the bits of the item once wehavechopp ed it up
Chapter Intro duction Background of Axiom
This is our analogy to algorithm and we may consider many data typ es to
be constructed or built similarly For example lists in the Lisp sense are always
constructed either by b eing the empty list or by b eing the cons of something on
to the front of another list
As a more complicated example p olynomials are constructed via one of the following
mechanisms
by being a member of the included monomorphic copy of the underlying ring
a constant p olynomial
by b eing a symbol exp onentiated to some p ositive power a univariate monic
monomial
y b eing the pro duct of a monic monomial and a univariate monic monomial a b
monic monomial
by b eing the pro duct of a monic monomial and a memb er of the underlying ring
a monomial
by adding a monomial to a p olynomial
So we see that at least two of the most basic typ es in computer algebra are constructed
in this way
Background of Axiom
Computer algebra is a sub ject which dates back to and has b een describ ed as
constructive algebra plus e cient algorithms Bra Towards the late s large
collections of routines or programs were starting to b ecome what one might think of
as the rst computer algebra systems
Scratchpad I GJ was b orn in ab out the principal architect b eing R D Jenks
JS Scratchpad I to ok the best ideas from Reduce Hea and Mathlab and
with new ideas and a new language b ecame IBM s research computer algebra system
In Jenks initiated the Scratchpad II pro ject Over the next couple of years
he along with J H Davenp ort and Barton designed and implemented prototyp es of
a su ciently clever algebra system one which used the concept of categories
An idea which apart from research pro jects suchas Newsp eak Fod up until the
Chapter Intro duction OBJ
release of Magma Pla November made Scratchpad II unique amongst
computer algebra systems
The category inheritance mechanism is based on an original abstract datatyp e design
by Jenks and B M Trager The algebraic category hierarchy has subsequently b een
redesigned byDavenp ort with Trager
The implementation of Scratchpad I I commenced in and in supp ort for the
system transfered to NAG and the name of the system changed to Axiom
OBJ
The sub ject of data abstraction has not been monop olised by the computer algebra
community Indeed much of the ma jor work has been done by those who are almost
completely disjoint from the computer algebra eld
Akey player in the area has b een Joseph A Goguen In Goguen designed OBJ as
a language for dealing with algebraic abstract data typ es errors and partial functions
in a simple and more imp ortantly uniform way See GWM for more details
The current version of OBJ is OBJ Release and is implemented in Common Lisp
OBJ is based on order sorted equational logic whichprovides a notion of subsort
OBJ also provides parameterised programming Both of these notions are found
in Axiom and Magma but OBJ s idea of subsort is far more rigorous that that of
Axiom We shall discuss OBJ in more detail in section
Aims of the thesis
We shall lay down a strict mathematical foundation for a typ e system in a computer
algebra system like Axiom We state the mathematics which inspired Axiom s typ e
system chapter
Axiom s typ e system uses ideas which are not explained using this basic mathematics
we explain some current mathematics which mo dels how we shall be In section
viewing Axiom s partial functions In we state some new mathematics to deal
with Axiom s conditional categories In section we formalise what is meant by a
co ercion In chapter we state the di erences b etween Axiom and the mathematical
theory
However the greater part of this work is dedicated to stating sections and and
Chapter Intro duction Aims of the thesis
creating sections and chapter enough mathematics to state an algorithm
for automatically creating mathematically correct co ercions
In Axiom co ercions are de ned by
b eing a rare sp ecial case hand co ded in the interpreter
b eing an explicitly written coerce function from a Category Domain or o cca
sionally a Package or
b eing created via sp ecial interpreter tricks whichcombine the ab oveinanad hoc
manner
These sp ecial interpreter tricks cover most cases but not all Users can not guarantee
that co ercions entered in the interpreter will exist
It is also imp ortant to note that these sp ecial interpreter tricks are not available in the
Axiom compilers for Spad and Aldor
We shall present an algorithm which builds co ercions for a very large sub class of typ es
which should exist from a p otentially but not necessarily minimal set This algorithm
is implementable chapter in the Axiom interpreter Sub ject to a set of conditions
b eing met bythetyp e system these co ercions are unique
We draw some conclusions ab out this work in chapter
The interpreter is the interactiveinterface to the Axiom system
Sp ecial hand written pieces of co de which execute b ehind the scenes in the interpreter
Chapter
Typ es in computer algebra
Intro duction
In this chapter we shall go into more detail ab out typ es in various computer algebra
systems We shall discuss how the di erent systems view the concept of typing in
formation from the simplest non typ ed languages in the so called M M systems
through Reduce s typ e system to the full blown mathematical b eauty of systems such
as Magma and Axiom
Mathematica
Mathematica is one of the most popular computer algebra systems on the market
to day Priced very reasonably and with an attractive user interface it is easy to see
why ordinary PC owners are attracted to this pro duct
In chapter of Mathematica a System for Doing Mathematics by Computer Wol
Wolfram states
At a fundamental level there are no data typ es in Mathematica Every
ob ject you use is an expression
Typ es may be created in Mathematica but they are merely tags on which one can
p erform switches For example the tag List This is an obvious approach since
Mathematica uses rule based programming
In JT Mathematica and Maple are referred to as M M
The ability to extend the typ e system of Reduce was inspired by Axiom then called Scratchpad
Chapter Typ es in computer algebra Maple
Abstract data typ es for Mathematica are discussed in Mae and inheritance for such
a system in Har However this approach is only marginally more sophisticated than
that used in Reduce describ ed in BHPS There is some form of abstraction in
this Mathematica abstract data typ e mo del but function metho d selection is just as
primitive as in the Reduce mo del
Maple
Maple CGG is a small fast p opular computer algebra system and is well re
sp ected both by users and exp erts in the area of symb olic computation Maple ver
sion V release do es have some concept of typ e but as you can see this is not
particularly well founded
x array a b c d
x array
a
b
c
d
m linalg matrix x y z a b c
x y z
m
a b c
eval m
array
a
b
c
d
Maple V release is nowavailable
Chapter Typ es in computer algebra Maple
type m matrix
true
Notice that m is now a matrix in Maple s view whose th element is an
array of symbols Three of these symb ols are also in the second rowof m The array
typ e do es not form a ring and moreover the elements of m are from di erenttyp es
It is ridiculous that one may create matrices over anytyp e whichdoesnotformaring
One learns at scho ol that given two matrices A and B with the same number of rows
p and the same numb er of columns q onemay calculate
the p q additive identity matrix
if p q I thep p multiplicative identity matrix
p
the negation of a matrix A
their sum A B
T T
the pro duct AB where B represents the transp ose of T
However should one create matrices over typ es which do not form rings then one of
the ab ovemaynotbewell de ned A in case and A B in cases and This is
the situation in Maple it lets one create ob jects of the typ e matrix which are not
matrices in the mathematical sense
Maple will now also let one assume certain prop erties ab out or typ es for elements
but these are entirely at the user s discretion This mechanism is based on the alge
bra of prop erties as describ ed in WG WG One can think of this assumption
mechanism as an advanced way of declaring the typ e of a variable
The algebra of prop erties is a fascinating and extremely clever way of assuming certain
facts ab out a given symbol and works very well in the context of saying We assume
that is real and greater than zero and then deducing various facts ab out f
Admittedly Axiom do es not always get these things p erfectly correct Martin Mar p oints out
that Axiom allows Matrix Float as a typ e yet the Float typ e do es not really form a ring
Chapter Typ es in computer algebra Reduce
However the algebra of prop erties do es not work particularly well for typing sym
b ols data Typ e inference for example is problematic given real and greater
p
than zero what are the typ es of and log
Inserting new elements in the prop erty lattice is inherently di cult and in a dynamic
system means that binding op erations to implementations is imp ossible at compilation
time
It could be argued that this metho d could run along side of Axiom s more advanced
notion of typ es Part of the problem with the Maple system is a confusion between
di erent classes of unknowns There is no distinction between the imp ortance of
b eing a real numb er and that of b eing greater than zero Surely b eing a real should
b e far more imp ortant than some statement regarding a symb ol s p ossible values
Much work has been done on this sub ject with sp ecial emphasis on Axiom and the
de nitivework is DF which de nes the di erent classes of unknowns and provides
details of an implementation for Axiom
Reduce
Reduce current release is Str Neu is a well established Lisp based
computer algebra system and although the user interface is archaic until recently its
algebra system was still one of the very b est around
Reduce has a well de ned concept of typ e and indeed new datatyp es may b e added
and included at will BHPS
Reduce prototyp es have also been develop ed HS which take an order sorted ap
proach to algebra This implementation seems to take some of the Axiom ideas and
some of the algebra of prop erties ideas It app ears to be far cleaner and clearer than
the mo del used in Maple
However in this prototyp e it is still stated in HS that
Mathematical structures can have several isomorphic representations A
go o d example is the representation of p olynomials in a distributed or re
cursive form For the simultaneous use of multiple representations
Being greater than zero is not just a prop osition restricted to real numb ers or subsets thereof
Polynomial typ es are often ordered and zero is a constant p olynomial
The commercial success of some of the other systems has forced them and gained them the capital
to correct many of the bizarre bugs that used to ab ound in earlier releases Thus the commercial systems
nowwork as well as Reduce have a friendly GUI and maywell have consigned Reduce to history
Chapter Typ es in computer algebra Axiom
explicitly de ned co ercions are necessary My emphasis
This is not currently the case in Axiom where co ercions can b e built in the interpreter
by clever tricks in the underlying co de Furthermore it is the aim of this thesis to
demonstrate that using an order sorted approach to algebra one can build manyof
these co ercion functions without explicitly de ning the co ercions themselves
Axiom
Axiom JS n ee Scratchpad II is a Lisp based general computer algebra system
Axiom s main view of things is that everyobjecthasatyp e and that there are osten
sibly four layers
items Domains Categories Category
For example
Integer Ring Category
and the top layer is the unique distinguished symbol Category In general this top
layer is never referred to and the three lower layers are all that are ever considered
Strictly sp eaking if one considers Axiom s Categories to b e categories and Axiom s
Category symb ol to signify the category of all the Categories then one should write
Integer Integer Ob j Ring Ring Ob j Category
where Integer means is an element of the carrier of the principal sort of the
signature to which Integer b elongs
Users may extend the Axiom system by writing new Domains and Categories in the
Axiom extension language called Aldor WBD a They may also be written in
an older Axiom language called Spad This is the language describ ed in JS
Aldor can b e complied into many languages for use inside Axiom to Lisp currently
AKCL AKC or CCL to C WBD b This makes Axiom far more than
Category is not one of the Categories in Axiom
This language was sometimes referred to as A in internal do cuments and has often b een referred
to as Axiom XL
Chapter Typ es in computer algebra Axiom
just an algebra system Aldor is also a rst class language for writing fast e cient
stand alone applications
A category is a collection of domains all of which are similar In DT the authors
state that they designed the Scratchpad system of categories or abstract algebras
qv chapter for the following reasons
economy of e ort section
interest section
functoriality section
Economy of e ort
The most obvious reason for this is the view of inheritance A category in Axiom is
said to extend another if it inherits all the other s functions attributes and equations
This allows for a signi cantamount of re use
A more imp ortant issue in an algebra system is that if one can prove a theorem in some
generality ie for all the ob jects in a category then one do es not have to go around
proving the theorem for each ob ject of the category
Interest
Some categories are interesting and some are not For example Ring and Field are
particularly interesting Many theorems can b e proved for al l Fields
However to borrow an example from DT the category of all rings which when
viewed as Ab elian groups have an involution with precisely one xed point is not
particularly interesting
One may think of the designers concept of interest as congruent to useful usually
to the user but o ccasionally to the designers
Functoriality
This is called higher order Polymorphism in Cro There are op erators which
t given ob jects of a category can create new ob jects of a given p otentially di eren
Axiom s Categories can also be thought of as order sorted algebras but the order on sorts is
never explicitly de ned and there are some di culties in the de nition of the op erator symbols This
is discussed in more detail in section
Chapter Typ es in computer algebra Newsp eak
category For example List is a functor from SetCategory to ListAggregate
List SetCategory ListAggregate
S List S
Indeed this target category mayeven b e the distinguished symbol Category
In Axiom the Category constructors often called functors are functors from
Y
Category Category
n N
For example the Ring functor takes no arguments and returns the Category of Rings
Ring Category
Ring
The sp ecial Category creation op eration Join is also a functor Join is used to declare
a new Category to be a sub category of the intersection of two or more Categories
Here we illustrate the case of Join acting on two Categories
Join Category Category Category
A B A B
More commonly in Axiom and Axiom s own use of the word functor covers cases
as follows This is similar to the case of Ring only less trivial ListAggregate is the
Category of all typ es of nite lists
ListAggregate SetCategory Category
S ListAggregate S
Here we see the sub category ListAggregate S of the Category ListAggregate
When using Axiom one usually thinks of one s typ e as b elonging to the smaller con
crete sub Category ListAggregate S Mathematically one usually thinks of one s
typ e as b eing in the larger more abstract ListAggregate
Newsp eak
As an aside from the main computer algebra systems listed in this chapter a research
system called Newsp eak Fod or more correctly Newspeak will now be dis
cussed
Chapter Typ es in computer algebra Newsp eak
Newsp eak was a language written at the University of California at Berkeley in the
early s
It di ered from Axiom then in the early stages of b eing Scratchpad I I otherwise called
Newspad by discarding the category mechanism and making a list of improvements
most of which are now in Axiom
The di erences are listed in Fod and are summarised below We shall also p oint
out whether the di erences still hold or not
It should be pointed out that Barton of the Scratchpad II design team also had
an implementation of Scratchpad II called Andante Andante grew in parallel with
Scratchpad I I but this chapter will refer mainly to Scratchpad I I
of Newsp eak over Axiom which have not yet b een incorp orated Sp eci c advantages
into Axiom are
Axiom s implementation language is not itself Newsp eak however was written
in Newsp eak This allowed the authors of Newsp eak to write dedicated garbage
collection and fast function selection routines
Function lifting If one writes a Category in Axiom which has parameters from
avery generic Category suchasRing then for certain functions to b e available
in the typ e one has to write co de such as the following from the de nition of
FiniteAbelianMonoidRing R Ring E OrderedAbelianMonoid
if R has IntegralDomain then
exquo R Union failed
exquo p r returns the exact quotient of
polynomial p by r or failed
if there is none exists
if R has GcdDomain then
content R
content p gives the gcd of the
coefficients of polynomial p
primitivePart
primitivePart p returns the unit
normalized form of polynomial p
divided by the content of p
Newsp eak s parameterised representationless typ es equivalent to Axiom s
Chapter Typ es in computer algebra Magma
Categories get around this problem by extending or in their doublethink word
ing restricting the Polynomial constructor Poly to PolyId Polyno
mials over Integral domains etc
To this author this do es not app ear to solve the problem of having to know the
category hierarchy of one s parameters It merely abstracts it and moves the
problem to a less satisfactory place
Problems Scratchpad I I Axiom then had whichhave since b een xed
A domain could only b elong to one Category This was not true even then for
Andante
Functions declared in Categories wereonlyabletohave their sources and targets
from a sp eci c domain an arbitrary typ e of this Category the value of a
parameter of this Category
Although Newsp eak allowed functions between categories which it called pa
rameterised representationless typ es which cannot b e written in Axiom s Spad
or Aldor languages one could do this in Axiom s Bo ot language but this is not
a desirable option
Axiom s items did not know their own typeorthattyp e s p osition in the hierarchy
In conclusion Newsp eak was a brave attempt at rewriting Scratchpad I I Axiom with
out explicit categories and solved many of the problems that the language then had
Fortunately mayb e thanks to Newsp eak these problems have since been redressed
and Axiom is all the more exible and easier to use for it
Magma
Magma the system to replace Cayley isanewandpowerful computer algebra system
As with Cayley the main purp ose of Magma is to deal with discrete algebraic ob jects
While Cayley was useful as a to ol for investigating groups Magma can deal with many
di ering discrete algebraic ob jects
Magma users are p eople more interested in branches of mathematics such as Semigroup
theory Group theory Galois theory or other areas of mo dern algebra Magma do es
It should b e noted that Axiom has some comp etence in all these areas but for discrete mathematics
it can not match the pure sp eed of Magma
Chapter Typ es in computer algebra Magma
have some ability to deal with the mainstream areas of computer algebra p olynomials
etc but this is not Magma s raison d etre
The design philosophy of Magma BCM takes its cues from Axiom but go es further
supp orting algebraic structures as rst class ob jects This makes Magma far more
powerful than Axiom when dealing with algebraic structures Indeed manipulating
algebraic structures is as easy as dealing with their elements in Magma In Axiom this
is simply not the case
A fundamental design di erence b etween Axiom and Magma is the following Axiom s
designers to ok their cue from order sorted algebra Thus their de nitions of typ es and
their metho d of collecting them Categories fo cus on the op erations whicharevalid
on those typ es
The Magma metho dology is more representation centred A category in Magma consists
of all ob jects with a similar representation the category of all p ermutation groups
for example However all the categories of di erent representations of an algebraic
idea are collected together in a variety thevariety of groups for example
This is a subtle di erence b etween Axiom and Magma since Axiom views varieties as
categories At some future date should Axiom ever supp ort algebraic structures as
rst class ob jects then the noticeable di erence would b e minimal
Magma also supp orts the idea of order sorted algebra more concretely than Axiom
do es From the Magma WWW page Pla
The primary concept in the design of the Magma system is magma Fol
lowing Bourbaki a magma can b e de ned as a set with a law of comp osition
Thus typ es corresp ond to magmas a collection of magmas sharing a
common representation forms a category e g the category of p ermuta
tion groups a collection of categories satisfying the same set of identical
relations forms a variety e g the variety of groups Functors may be
used to movebetween categories and the variety op erations substructure
ailable as uniform con homomorphic image and Cartesian pro duct are av
structors across all categories
While every value in Magma b elongs to a unique parent magma the system
provides a mechanism for automatic and forced co ercion to move a value
from one magma to another To take a simple example when an integer
is added to a rational Magma automatically co erces the integer into the
rational eld so that the addition op eration can be carried out on two
Chapter Typ es in computer algebra OBJ
values with the same parent
This automatic co ercion also implemented in Axiom is from the order on the sorts
R Ring Fraction R in the theory of fractional quotient elds as we shall
de ne in section roughly means that there is some sort of natural way of mapping
every elementofR to one in Fraction R
OBJ
In this section we shall discuss how OBJ implements the ideas of order sorted algebra
in a more concrete way than Axiom or Magma Much of the information in this section
can b e found in GWM
It should b e noted that OBJ is not marketed as a computer algebra system but as a
comprehensive language with a strong mathematical foundation
OBJ provides a strongly typ ed language based more closely on order sorted algebra
than any other
At OBJ s top level there exist three typ es of entity
ob jects These are describ ed in the literature as an encapsulation of executable
co de or an algebra Though it should be noted that an OBJ algebra need not
be a mo del of any theory Note OBJ do cumentation sometimes refers to the
ob jects as Mo dules
theories In the literature theories are describ ed as varieties but in our language
OBJ s theories are indeed theories
views Views showhow ob jects algebras satisfy or mo del certain theories
In the next few sections we shall investigate these entities in more detail so that we
may compare them with Axiom s top level forms
Ob jects
An OBJ ob ject is an executable algebraic sp eci cation for an abstract data typ e That
is to say an OBJ ob ject de nes a sorted signature or indeed a theory since equations
are allowed and provides the initial ob ject of that theory
The initial ob ject of a theory is the term algebra factored out by the equations This is equivalent
to the category theory notion of initiality
Chapter Typ es in computer algebra Comparison
The parameters of an OBJ ob ject can b e typ ed and forced to b e mo dels of particular
theories Indeed OBJ ob jects can b e viewed as algebras mo delling particular theories
themselves as we shall see in section
Theories
OBJ theories are indeed order sorted signatures with sets of equations Semantically
OBJ ob jects and OBJ theories are very similar However one imp ortant di erence
is that the equations presented in a theory do not have to have to satisfy the same
restrictions as those in an OBJ ob ject
OBJ ob jects must have equations whichcanbeconverted into rewrite rules however
this restriction do es not apply to OBJ theory equations This is b ecause the OBJ
ob jects are executable whereas OBJ theories are not Remember that OBJ is
based up on a rewriting system
Views
Views are the most exciting feature of OBJ A view in OBJ is a metho d for stating
that a particular OBJ ob ject mo dels a particular OBJ theory
What makes OBJ unique is not the fact that the carriers mayhave di erent names to
the sorts but that the op erator names may b e di erent to the op erator symb ols
Comparison
So howdo our de nitions of higher order entities OBJ s and the other systems com
pare
The current implementation of OBJ is called OBJ
Chapter Typ es in computer algebra Comparison
Universal Category OBJ Magma Axiom
Algebra Theory abstract concrete
Variety Variety
Theory Category Theory Theory
de ned
in an
ob ject
Signature Category Category
Algebra Ob ject View Ob ject Magma Domain
In the Universal Algebra column wehave written variety in a di erentrow to theory
This is not quite correct yet the two are di erent A theory is a sp eci cation of a typ e
and a variety is the collection of all typ es with satisfy or mo del that sp eci cation
There is a di erence b etween theories and signatures since signatures contain no infor
mation ab out equations However the di erence is small a theory with no equations
is a signature
We have slotted categories in as the equivalent of universal algebra s theories Yet
since everything may b e thoughtofasacategory this decision is merely arbitrary The
sp eci cation of a category could be the same as a theory indeed this is the case in
Axiom but the collection of all ob jects of that category would form a variety
In the OBJ abstract column we refer to genuine OBJ theories declared to b e theories
as opp osed to those mentioned in the OBJ concrete column OBJ s theories are
most certainly theories and the collection of views of a particular theory forms a
variety A view of an OBJ theory is most certainly a mo del satisfying that theory
and hence an algebra of the signature of the theory by de nition
In the OBJ concrete column the theories are those which exist by inference from an
ob ject de nition OBJ ob jects de ne a singleton variety de ned by the inferred theory
which the ob ject mo dels This inferred theory is not an OBJ theory However it is still
a mathematical theory
Magma has a view of varieties and categories matching in with the mathematical variety
and signature de nition It should be noted that Magma s categories neither have
an explicit notion of sortedness nor one for equations Though certain assertions
assumptions and statements maybemade
Chapter Typ es in computer algebra Comparison
Axiom s categories written Categories elsewhere in this thesis are discussed at length
in other chapters of this work
Chapter
Category theory
Intro duction
As we have already hinted at category theory is one of the main foundations of b oth
Magma and Axiom In this chapter we shall give the necessary de nitions to investigate
this claim We shall also investigate the claim in itself
The de nitive text on Category Theory is Mac Lane ML The amount of theory
we require is relatively small We de ne categories without worrying ab out Russell s
Paradox Men In app endix B wehave a brief lo ok at the complications which arise
when one do es
Category theory
De nition A category C consists of two col lections The rst col lection is
cal ledtheob jects of the category or Ob j C The second col lection is cal led the arrows
of the category or Arr C
Also for each arrow f there exists two special objects with which it is associated The
rst is the source of f cal led source f The second is cal led the target of f cal led
target f
There also exists a law of comp osition for arrows
g f arrows source g target f
source f target g f target g g f arrow source g f
Chapter Category theory Category theory
Next for each object c there exists a unique arrow cal led the identity arrowonc or
id
c
Final ly the fol lowing two axioms must hold
k g f arrows g f k g arrows
k g f k g f arrows k g f k g f
f arrows id f f f id f
target f source f
Here endeth the de nition
Tointro duce the concept here are some simple nite categories
Example is the empty category it has no ob jects and no arrows
Example is the category with one ob ject and one arrow
Example is the category with two ob jects a b and one non identity arrow
f a b
Example is the category with two ob jects a b and two non identity arrows
f g a b
Now let is consider some more useful categories The collection of ob jects in the
following examples do not always form a set ML Ber Dev BHFL
Example Set is the category which has as ob jects all sets and has as arrows
all total functions b etween sets
Example Grp is the category which has as ob jects all groups and has as
arrows all group homomorphisms b etween them
Example Ring is the category which has as ob jects all rings and has as arrows
all ring homomorphisms b etween them
Example Poly is the following category An ob ject of Poly is a set of all
p olynomials with variables from a xed ordered set V co e cients from a xed ring
R and the exp onents of the variables from an ordered free monoid with one generator
E
Chapter Category theory Category theory
An arrow of Poly is a Polynomial homomorphism That is a Ring homomorphism
which also acts homomorphically on a certain set of functions which act on p olynomials
Some texts call the arrows of a category the morphisms of a category Examples
show us that for some naturally o ccurring categories the arrows or morphisms
are just the homomorphisms of the category Indeed an arrowofSet a total function
is really just a homomorphism b etween sets
So we see that the arrows preserve a certain set of prop erties for each element of the
ob ject
We also see that every ob ject of Grp is an ob ject of Set every ob ject of Ring is an
ob ject of Grp and every ob ject of Poly is an ob ject of Ring
Now similarly replace the word ob ject with the word arrow in the previous para
graph and it still holds true
algebra system there is a certain amount of inheritance Hence we see that in an
amongst the categories The categories are the so called abstract datatypes since they
typ e the usual datatyp es
We can also see how the categories can collect together all similar typ es This function
ality can be used for the three design goals of Axiom economy of e ort the co de
for many similar typ es need only be written once interest collecting similar typ es
together gives the user an identical interface to similar typ es and functoriality which
is b est left to b e discussed after the following de nition
De nition Let B C be categories then a functor T C B consists of two
functions
C Ob j B the ob ject function T Obj
T Arr C Arr B the arrow function
These must obey the fol lowing
Which has an entirely di erent yet identical meaning in Java Gra Java do es not allowmultiple
inheritance from classes due to the usual problems Towork around this Javaprovides interfaces
An interface is just like a class although declared using di erent syntax except that it can not
provide a metho d for any op eration It may declare many signatures of op erations available to any
class which implements the interface
This is similar to abstract typ es classes with all functions pure virtual and no data memb ers in
C Str
Chapter Category theory Category theory
f arrow T source f source T f
f arrow T target f target T f
c ob ject T id id
c
T c
g f arrows source g target f T g f T g T f
This is a very useful de nition It shows us that when we have two ob jects and an
arrowbetween them in one category then a sensible map of these ob jects to another
category will induce the obvious map b etween these image ob jects
fact one can see that if one were to de ne a category which had as ob jects all In
categories and as arrows all functors then wewould set theoretical concerns aside
haveawell de ned category
Categories just like many other mathematical constructs may form pro ducts
De nition For two categories B and C we may construct a new category de
noted B C cal led the pro duct of B and C
Anobject of B C is a pair hb ci where b is an object of B and c an object of C
An arr ow of B C is a pair hf gi where f b b is an arrow of B g c c is an
arrow of C the source of hf gi is hb ci and the target of hf gi is hb c i
Composition of arrows hf g i hb c i hb c i and hf gi hb ci hb c i is de ned
via
hf g i hf gi hf f g g i
Axiom makes use of such pro ducts implicitly in its functor de nitions For example
see the discussion of PolynomialCategory b elow
Natural transformations are another imp ortant part of category theory They are to
functors as functors are to categories Here is a more formal de nition
De nition For two functors S T C B a natural transformation S T
is a function which assigns to every c Ob j C an arrow c Sc Tc of B such
c
that
c Ob j C f c c Arr C Tf c c Sf
Nowwemay de ne the following interesting category
C
De nition For two categories B C the functor category B is the category
Chapter Category theory Category theory
with objects the functors T C B and arrows the natural transformations between
two such functors
As another fairly abstract de nition consider the following
De nition Let S D C be a functor and c Ob j C An universal arrow
c S is a pair hr ui Ob j D Arr C where u c Sr such that
hd f i Ob j D Arr C wheref c Sd f r d Arr D Sf u f
Functoriality in a computer algebra system allows us to view ob jects of one category
as ob ject of another
As wehave already seen an ob ject of Poly is an ob ject of Ring and hence an ob ject
of Grp and thus an ob ject of Set Wehave also seen this is true for their arrows This
relationship is called the sub category relationship de ned formally as follows
De nition A category C is a sub category of a category B i every object of
C is an object of B and every arrow of C is an arrow of B
Functors from sub categories to the categories of which they are sub categories are often
called forgetful functors This is more often true when the target of the functor is
Set
Axiom s designers also use functors to create instances of abstract datatyp es
PolynomialCategory Axiom s equivalentof Poly is in Axiom s view a func
tor
Ring OrderedAbelianMonoid OrderedSet PolynomialCategory
R E V PolynomialCategory R E V
This functorialityprovides the glue for Axiom s typ e mechanism
Now trivially for categories A B if F A B a forgetful functor then B is a sub
category of A Equally trivially a concrete instance of PolynomialCategory R E V
is a sub category of PolynomialCategory R which is a sub category of Poly
inheritance mecha It is this rst form of sub category relation that provides Axiom s
nism
Chapter Category theory Categories and Axiom
Categories and Axiom
Notation To distinguish between Axiom s internal structures and those com
monly used in mathematics things which belong to Axiom wil l be written in this
font Speci cal ly
Category wil l always refer to Axiom s distinguished symbol to which al l Axiom s
Categories belong
Hence a Category is an Axiom object declared to be such an object eg Ring
PolynomialCategory R E V
A Domain is an Axiom object declared to bea member of a particular Category
eg Integer Polynomial Integer
An item is an element of a Domain eg x
As stated in section Axiom s main view of things is that every ob ject has a typ e
and that there are four layers
items Domains Categories Category
Categories also may inherit from or extend other Categories forming an inheritance
lattice Thanks to the higher order p olymorphism available in Axiom Categories
may also b e parameterised by items or Domains
This parameterisation may cause some confusion For example
List S ListAggregate S
declares for each and every S List S is in the category ListAggregate S For exam
ple List Integer is an ob ject of ListAggregate Integer However
ListAggregate Integer is a mere sub category of the category of all ob jects which
are domains of linked lists This is ListAggregate S
So ListAggregate S contains for example both List Integer and
List Fraction Integer as ob jects but what are the arrows of this category This
One may view a domain as a category whose ob jects are the items and whose arrows are either triv
ial ie solely the identityarrows or some other natural o ccurring meaning eg In PositiveInteger
one may think that there is a natural map since j This would then mean that this map
could b e lifted to IntegerMod IntegerMod
Chapter Category theory Functors and Axiom
is something which is not made explicit in the Axiom literature and is indeed the core
of this thesis
We shall discuss what it means to b e a natural map and how this relates to categorical
arrows section
Functors and Axiom
Axiom describ es its domain constructors as functors and this is true After all ne
glecting di culties with constructors which take domain elements for arguments these
constructors are maps from a cross pro duct of categories to another category
The di culties with constructors whichtake domain elements for arguments disapp ear
when considering the argument used in fo otnote of section
Co ercion and category theory
If coerce A B thenwe are going to have that A and B are ob jects of the same
category ACat andthat coerce is an arrow of that category See de nition
Also if T is a functor from ACat to TCat whichinAxiomwould lo ok like
T A ACat TCat
then T lifts the arrows of ACat to TCat and in particular
T coerce A B coerce TCat A TCat B
In manytyp es and some other languages T is acting like the familiar map op erator
on the coerce function
Conclusion
We have seen in this chapter how category theory and abstract datatyping esp ecially
with resp ect to Axiom s Category mechanism are ideologically similar
We have also shown how Axiom s functors interact with co ercions from a category
theoretical p ersp ective
Chapter
Order sorted algebra
Intro duction
In this chapter we shall intro duce the concepts of universal algebra We shall lo ok at
the unsorted case to start with and then move on to the sorted case
We then follow up with the equational calculus which allows us to consider sets of
equations whichmust hold in a concrete instance of an algebra
All the work in this chapter is taken from Dav except example the Perl
example in section and section
Universal Algebra
Universal algebra will provide us with a natural way of representing categories of typ es
which p ossess certain functions In the following lengthy section of de nitions keep
in mind that what we shall de ne as a signature will b e equivalent in some sense
to our notion of a category or abstract datatyp e An algebra will b e the ideological
equivalent of an ob ject or typ e
De nition A sort list S is a nite sequenceofsymbols cal led sorts normal ly
denoted s s
m
De nition Given a sort list of size m a set A of S carriers is an ordered m
tuple of sets A indexed by S
s
i
Chapter Order sorted algebra Universal Algebra
Example One may consider the sort list of a vector space to be K V where
K is to be identi ed with the underlying eld and V is to be identi ed with the set of
al l points in the vector space Then considering C to be a vector space over R the
K V carriers of C are R C
So we see that the carriers of a particular typ e are a list of typ es which have something
in question The sort list is a list of the same length where the to do with the typ e
ith element is a symb ol corresp onding to a particular abstract datatyp e to which the
ith carrier b elongs
De nition Given a sort list S and n N f g an S arity of rank n is an
ordered n tuple of elements of S
An arity is merely a list of elements of the sort list This will b e useful when we wish
to typ e p olymorphic or abstract functions
De nition Given a sort list S n N f g q q q an S arity of rank
n
n and a set A of S carriers we de ne
Y
q
A A
q
i
i f ng
This is the map of an arity to the list of carriers
q
De nition An S op erator of arity q is a function from A to one of the elements
of A
De nition An S op erator set or S sorted signature is a set of sets
n q s
S
is a indexed by n N f g q an S arity of rank n and s S such that
n q s
n q s
is cal led operator symbol subset of some alphabet An element of some
n q s
The usual notation for such a signature is h Si
So this de nes us a set of sets of p olymorphic functions for a particular sort list A
signature is like a category in that it is an abstract datatyp e
A signature collects together all typ es which share a similar family of op erators
Notice that the sort list is de ned rst and that the carriers dep end on the sort list Onedoesnot
de ne a list of carriers and then x a list of abstract datatyp es post facto
Functions without metho ds
Chapter Order sorted algebra Universal Algebra
Example Monoids N f g is an additive monoid whereas N is a multiplicative
monoid
The signature for monoids could beviewed as being
h Ui h M B e id ep b i
where M is the monoid sort B is the sort of boolean logic types The operator symbol
e is a member of and corresponds to the function which always returns the
M
identity constant of the monoid
id is the operator symbol in which corresponds to the identity function
M M
ep is the operator symbol in which corresponds to the is this the identity
M B
element function
Last in our example but by no means least b is the monoid s binary operator from
M M M
So in N f g e id ep b correspond to id respectively Whereas in N e id ep b
correspond to id respectively
De nition A multi sorted total algebra is an ordered pair hA i where A
is an S carrier set and
f jn N f g q an arity of rank n s S g
n q s
q
A g f A
s n q s n q s
n q s
So if h Ui is the monoidal signature then
h N Boolean id i
is an algebra
Notation Let hA i be a algebra For an operator symbol of the sig
n q s
nature h Si the function associated with this symbol in hA i is represented by either
or
n q s
n q s
Constants are often represented by if not compiled in the same way as functions in programming
languages such as Axiom s Spad language This gives a homogeneous interface for providing constants
Chapter Order sorted algebra Term Algebras
This second form of notation is useful for when we refer to op erator symb ols by names
other than those in the form For example if then
n q s n q s
n q s
n q s
De nition Given hA i hB i both algebras then a homomorphism
hA i hB i is an S indexed family of functions A B for each s in S
s s s
such that
q
s S n N f g q arities of rank n a a A
n q s n
we have
a a a a
s n q s n n q s q q n
n
As an example of a homomorphism let us consider our monoidal case once more The
n
map from N f g N which maps n is a homomorphism
Term Algebras
Term algebras provide us with at least one example of an algebra for each signature
In some sense it is the freest algebra of the signature and all other algebras are
isomorphic to quotients of the term algebra
Notice that given h Ui the de nition of the monoidal signature from example
the term algebra is not the free monoid since we have not added in any laws or
equations to the algebra Thus we do not have asso ciativity in the term algebra
whereas we do in the free monoid
Thus we see that as yet universal algebra do es not mo del real mathematics p erfectly
However this situation will b e remedied somewhat in section
Notation We shal l de ne the set to be the set containing threespecial symbols
f g f g f g
is a just a piece of notation that will make the following de nition less verb ose
Wehaven t provided an exp onentiation function in our monoidal algebra but adding an extra sort
and a function it is p ossible Otherwise think of the map as taking to to to to
and so on
Chapter Order sorted algebra Order sorted algebras
De nition Let X bean S indexed family of sets disjoint from each other from
S
the set and from We de ne T X to be the S indexed family of sets
n q s
n q s
S S
of strings of symbols from X each set as smal l as possible
s n q s
s S n q s
satisfying these conditions
s S T X
s
s
S X T X s
s s
i f ng t T X t t T X
n q s i q n s
i
We make T X into a algebra by de ning operators on T X for each
T
via
n q s
If n then Guarante ed to be in T X by
T s
Else de ne t t to be the string t t
T n n
T X is cal led the term algebra and an element of T X is cal led a term
Order sorted algebras
Order sorted algebras extend the concept of of universal algebras by imp osing an order
on the elements of the sort list This can b e useful if weknow that all algebras of our
signature h Si have the carrier of S as a subset of S say
The ordering of sorts imp oses a subtyp e lattice on the sorts and hence their carriers
This can then be used in any algebra of the signature to either restrict a function
already de ned on one typ e to a subtyp e of that typ e or extend a function on a typ e
to a partial function on a sup ertyp e Partial functions and order sorted algebra are
discussed in section
Firstly we had b etter de ne what we mean by an order
De nition A strict partial order on a set S is a relation on S which is
transitive antisymmetric and irre exive
A weak partial order on a set S is a relation such that a b a b a b
Such a relation is transitive
Chapter Order sorted algebra Order sorted algebras
Notation Let S be a set of sort symbols such that there is a partial order
de ned on S and a top element u of S such that s u for al l s in S
u provides us with a universe see app endix B in which to work Equivalently we
could set u to be any class which is big enough The term class is de ned in
app endix B
De nition Extend from S to the S arities of r ank n by de ning s s
n
t t i i f ng s t
n i i
De nition An order sorted total algebra is an ordered triple hA fA s
s
S g where A is a class known as the universe fA s S g is an S indexed family
s
of subsets of A known as the carriers of the algebra and is a set of sets of functions
S
q
A g such that f A
n q s s n q s
n q s
A A
u
If s s in S then A A
s
s
If with s s and q q then j
q
n q s n q s
n q s n q s
A
Strictly sp eaking that last condition should b e
j
q
A A n q s
n q s
s
A
s
where is the inclusion op erator A A since otherwise the target of the
A A s
s s
s
s s
left hand side would b e A and of the right hand side would b e A
This is typical of the sort of abuse of notation that computer systems often have to
implement
In a language like C KR there is not much scop e for such abuse The compiler will
complain if we attempt to change the typ e of anyvariable
However in a language likePerl WCS where typ e changes are p erformed automat
ically abuse is everything For example to convert the string to a number one
simply adds to it
The following command line session shows the abuse Perl allows
Chapter Order sorted algebra Order sorted algebras
nic pangur ban Tmp perl e
a string
a number
print a string
print n
print a number
print n
print a string a number
print n
a string text
print a string
print n
text
print n means print carriage return linefeed as in C See how a string is
treated as a number when added to a number and a string when we app end using
the op erator some text to it
Notice that the opp osite can b e p erformed to o Wemay treat numb ers as strings We
have already seen this with the line print a number but as a more explicit example
e can app end text to a number w
nic pangur ban Tmp perl e
a number
a number text
print a number
print n
text
Perl is incredibly go o d at all such built in typ e changes However Perl is not a computer
algebra system and do es not have the vast number of typ es that Axiom has
Perl also has a xed universe see app endix B and de nition and knows how
to retract any variable back to the universe typ e From this it can see how to co erce
Chapter Order sorted algebra Order sorted algebras
anytwo ob jects to typ es over which an op erator suchasprint or isvalid
The de nition of order sortedness ensures some con uence amongst op erators on sub
typ es However it do es not provide enough for most sensible applications The follow
ing de nition ensures a more con uent system
De nition The order sorted signature is regular if whenever q is an arity and
with q q there is a least pair q s such that and q q and
n q s
n q s
s s
Now we shall extend the de nition of term algebras to the order sorted case
om each other from De nition Let X bean S indexed family of sets disjoint fr
S
We de ne T X to be the S indexed family of sets the set and from
n q s
n q s
S S
each set as smal l as possible X of strings of symbols from
n q s s
n q s s S
satisfying these conditions
s S T X
s
s
s S X T X
s s
s s S s s T X T X
s
s
i f ng t T X t t T X
n q s i q n s
i
We make T X into a algebra by letting the rst component be T X and de ning
u
operators on T X for each via
T n q s
If n then Guaranteed to be in T X by
T s
Else de ne t t to be the string t t
T n n
T X is cal led the term algebra and an element of T X is cal led a term
The following theorem proves the freeness of term algebras That is to say all algebras
are isomorphic to a quotient of the term algebra In this waywe see that all algebras
ted as a term algebra and a set of rewrite rules are easily implementable once represen
in a rewrite system suchasOBJ
Perhapsthisiswhy OBJ is a rewriting system
With the usual caveats on b eing able to check whether two elements are equal or not and other
unsolvable problems
Chapter Order sorted algebra Extension of signatures
S
It also says something ab out the constructibilityoftyp es That is if is nite
n q s
n q s
then clearly only a nite number of functions in each and every algebra construct
the whole algebra
Moreover supp ose in every real algebra whichwe wish to study a certain set of equa
tions hold see section Then all our algebras are isomorphic to factors of the term
algebra factored out by that set of equations
Then if every element of this freest factor algebra is equal to one constructed by a
S
p otentially very small nite subset of a p otentially in nite we may
n q s
n q s
utilise this to construct elements of our algebra
In the automated co ercion algorithm section we utilise a small but not necessarily
minimal set to construct all or some of the elements of one of the sorts of an algebra
any algebra any Theorem First universality theorem Let hA i be
map S indexed family of maps from X into A Then there exists a unique
homomorphism from T X to A such that s S x X
s
x x
s
s
The pro of may b e found in Dav
Extension of signatures
In this section we shall formalise what wemean for one algebra to be an extension of
another or more imp ortantly from our p oint of view for one algebratobea p ortion
of another
More formally we are saying how the abstract datatyp es or categories or signatures
may inherit from each other This sometimes corresp onds to algebras dep ending on
each other
First a piece of notation
Notation If S and T are two sets of sets both indexed by the same set I we
say that S T i i I S T
i i
This extends the de nition of subsets to n tuples of sets
De nition Let S S betwo sort lists such that as sets of symbols S S and
Chapter Order sorted algebra The equational calculus
that s t S s t s t Let be an S sorted signature and T be a
S
S
S sorted signature If T we say that is a sub signature of T and that T is a
sup er signature of
Thus we have the obvious de nition of sub signature As an example the monoidal
signature is a sub signature of the signature for groups The notion of sub signature
corresp onds directly with that of sub category
De nition With S S T as in the previous de nition let hA i be an S
sorted T algebra De ne hA i j cal led hA i restricted to to be the S sorted
algebra with carriers those carriers of hA i which are indexed by sort symbols from
S and operators n ary operators of arity q and result sort s for every
n q s
This is applying the forgetful functor from the category corresp onding to T to that
corresp onding to to hA i
The equational calculus
Again we borrow heavily from Davenp ort s lecture notes Dav Throughout this
section we assume that S fs s g is our indexing family of sort symb ols and
n
f g is a S sorted signature
n q s
The equational calculus presented here applies to multi sorted algebra The reader will
see that it clearly may b e extended to the order sorted case
The equational calculus allows us to add equations to our signatures and hence
algebras This will allow us to assert facts ab out all the algebra in a signature For
example we may wish note that one of the binary op erators is always asso ciative
Other more complicated expressions are available also
The equational calculus do es not allow us to de ne everything that weneed only those
things that are easily de nable as equations
De nition A S indexed family of relations R fR R g on a algebra
s s
n
hA i is cal led a congruence if it satis es the fol lowing four families of conditions
a A aR a R
s s
i i
aR b bR a S
s s
i i
aR b and bR c aR c T
s s s
i i i
b a R b a a R b b C a R
n q n n s n n q s q
n
Chapter Order sorted algebra The equational calculus
The rst three conditions imply that R is an equivalencerelation on the set A while
s s
i i
the conditions C explain how R relates to the various op erators of The op erators
of are well de ned on the equivalence classes
n q s
De nition Let hA i be a algebra and bea congruence on hA i De
ne hA i to be the algebra hB i where the carrier set B of B is the set of
s
i
and is the operator equivalence classes of A under the equivalence relation
s s
i i
de ned by
a a a a a a
n n
for every operator symbol in every
n q s
This is merely the quotient algebra and to b e sure the following is a theorem
Theorem The operators in the above de nition are wel l de ned
The following de nition actually turns out to b e very imp ortant
De nition We say that a sort s is void in the signature if T
s
Basically having a void sort s in the signature means that there are no constants of
that typ e and that one of the following is true
s
no op erators have s as a return typ e q n where q is an arity of
n q s
rank n
every op erator withs as a return typ e has an argument whose sort is either void
or s q n i f ng q is a void sort or s where q is
n q s i
an arity of rank n
Lemma If s is not void in the signature theninevery algebrahA i A
s
Goguen Meseguer suggest the following rules of deduction for a sound multi sorted
equational calculus
Notation Let X be a S indexed family of sets of variable symbols such that
each X is disjoint from al l the others from the operator symbols of and from any
s
i
symbols in any particular algebras we may be reasoning over
Chapter Order sorted algebra The equational calculus
We wil l bereasoning frequently with S indexed families of sets and wishing to perform
operations on them Let X hX X i and Y hY Y i be two S indexed
n n
families of sets and de ne X Y to be the S indexed family hX Y X Y i
n n
Similarly we wil l write X Y to indicate that each component of X is a subset of the
corresponding element of Y
A nal piece of notation is that the indexed family of empty sets wil l bedenoted by
De nition A multi sorted equation for the signature consists of a triple
Y t t where Y X t and t are terms from the same carrier set of T X or
i and every T X A if we are dealing with equations in a particular algebra hA
variable occurring in t and t occurs in the appropriate member of Y t t T Y
Such equations are written Y t t
If t and t come from the same carrier set of T X we say that the equation is
generic
One should read the symbol Y as meaning for all values of all the variables of Y in
the appropriate sorts and there had b etter b e some values in those sorts It is this
interpretation that will solve the paradoxmentioned earlier
system for the signature is a set of equations for De nition An equational
T X or T X A if we are dealing with equations in a particular algebra hA i
We will tend to write e f for an equation from an equational system meaning
Y e f where Y is the S indexed family of sets of variables consisting precisely of
those variables o ccurring in e and f
Notation x t means substitute the variable x with the term t We cal l this
i i i i
a substitution instance
De nition A pro of in the multi sorted equational calculus of the equation
Y e f from the equational system E is a nite set of equations Y e f such
i i i
that each equation is justi ed by one of the seven fol lowing rules of inference
Y e x t x t f x t x t E
n n n n
where X e f is an equation of E thet are terms of the appropriate sort of T X
i
and Y is the S indexed family of sets whose s th component is the set of al l variables
Chapter Order sorted algebra The equational calculus
of sort s in al l the t and those variables of X which have not been substituted for i e
i s
which are not one of the x
i
Y e e R
where Y is the S indexed family of sets whose s th component is the set of al l variables
of sort s in e
Y f e S Y e f
Y Y e g T Y e f Y f g
Y e f Y e f Y Y e e f f
n n n n n n
C
where is any symbol of and each e is a term of sort q
n q s i i
Y e f A Y e f
where Y Y
Y e f hY Y Y nfy g Y Y i e f Q
i i i n
where y does not occur in e or f and the sort i is non void for
We use the notation Y e f or Y e f if we wish to make clear which
E
equational system is b eing considered to mean that e f is provable in the equational
system using the ab ove rules of inference
De nition If Y e f is an S sorted equation c al l it E and hA i is
a algebra then we say that hA i satis es E or that hA iis a mo del for E if for
al l S sorted maps from Y to A e f where is the map from T X A
A
to hA i whose existence is guaranteed by theorem We extend the notation to
sets of equation E by insisting that hA i bea model for each equation in E
Theorem The Soundness Theorem If hA i is a model for E and
E
Y e f then hA i is a model for E f Y e f g
The pro of of the soundness theorem may b e found in GM
De nition Let hA i be a algebra and let E be an equational system for
d by A B if hA i The congruence induced by E denoted on hA i is de ne
E E
and only if Y a b where Y is the S indexed family of empty sets
E
Chapter Order sorted algebra The equational calculus
The following two de nitions are very imp ortant A theory sp eci es a typ e and we
de ne a varietyto b e the collection of all typ es which mo del that theory
De nition We de ne a theory to be the ordered pair hh Si Si where is
an S sorted signature and S is a set of S sorted equations We say that a algebra
mo dels or satis es the theory i it is a model for S
De nition The col lection of al l models of a particular theory is cal led a vari
ety
Clearly since a signature h Si may b e viewed as the theory hh Si i the collection
of all algebras forms a variety
When referring to the variety of all mo dels of a particular signature for example
hh Si Si we usually say the variety de ned by or sp eci ed by the signature
hh Si Si
Theorem Second universality theorem Let X beaS sorted set of vari
ables an S sorted op erator set E a set of equations for hA i a algebra which
is a model for E and a S sorted mapping from X to A Then there is a unique
homomorphism from T X to A such that
E
x x
for al l x X where is the map from X into T X de ned by x x
E
Theorem The Completeness Theorem If every algebra which satis
es the equation E also satis es the equation Y e f then Y e f
E
Again the pro of of this may b e found in GM
Finally we state the de nition of an extension and a protecting extension Extensions
are self explanatory
De nition Suppose that and T are signatures where is S sorted and
T is S sorted where S S
e an extension of Then T is said to b
A protecting extension is an extension which preserves the equational system for a the
ory Combining the second universality theorem with the de nition of protecting
Chapter Order sorted algebra Signatures theories varieties and Axiom
extension allows us to view a mo del of a theory as a mo del of a protecting extension
of that theory
More imp ortantly ifwehave an algebra which is a mo del for a protecting extension of
a theory then wemay view the algebra as a mo del of that theory unextended
For example if the Ring theory is de ned to be a protecting extension of the Group
theory then wemay view anyring Ring algebra as a group Group algebra
De nition With T S and S as in de nition
Let E bean equational system on Also let E F bean equational system on T
cal led a protecting extension if Such an extension is
T X j
E F
T
is isomorphic to T X isomorphism meaning isomorphism as algebras
E
Notice that the de nition of protecting extension is equivalent to the notion of enrich
ment given in Pad although this do es not allow for the existence of S Thus a
theory is a protecting extension of another i is complete and consistent
with resp ect to
Signatures theories varieties and Axiom
Axiom s typ e system uses the terminology from category theory yetitsdesignisbased
on order sorted signatures
A Category de nition in Axiom ie the source co de that de nes the Category is
equivalent to a signature or theory b eing a sp eci cation of a typeortyp es
The Category viewed as a collection of ob jects Domains is the variety sp eci ed by
the theory which de nes the Category in the Category de nition
The sorts are for the principal sort see de nition and the argument and
return typ es of all the op erator symb ols of the Category
For example PolynomialCategory R E V is a Category is the principal sort
Other sorts include R E V a sort each for the Boolean and PositiveInteger typ es
The equations of a theory in an Axiom Category de nition are either de ned in the
Unless they are concrete typ es Domains See section
Chapter Order sorted algebra Conclusion
comments or as certain attributes of op erators For example commutative means
that is commutative and the traditional commutativity equation for the op erator
is an equation of the theory There is currently no metho d for enforcing the equations
to hold in any mo del
An algebra or mo del for a theory in Axiom is a Domain Declaring a Domain to be
in a Category is equivalenttosaying that is an algebra of that signature or mo del of
that theory Or in other words a memb er of the variety de ned by the theory which
sp eci es the Category
For example Polynomial Integer is a mo del for PolynomialCategory R E V
An op erator symbol in Axiom is a function declaration in a Category An op erator
name in Axiom always corresp onds to the op erator symbol
the For example is an op erator symbol in Ring In Integer a mo del for Ring
op erator name of is and has to b e
A carrier is the concrete typ e substituted for a parameter in any instantiation of a
Domain
In Polynomial Integer Polynomial Integer is the carrier Whereas Integer
NonNegativeInteger and Symbol are the carriers of R E and V resp ectively No sort
is given for the Boolean and PositiveInteger typ es
All of Axiom s extensions are protecting extensions That is if a Category is declared
to extend another then it is always a protecting extension of that Category
In chapter we shall discuss how Axiom di ers from order sorted algebra and howwe
may remedy this situation
Conclusion
In this chapter wehaveintro duced all the basics of order sorted algebra and the equa
tional calculus Wehave also shown how these notions are represented in Axiom
Wehave demonstrated that order sorted algebra combined with the equational calculus
provides a sound basis for a computer algebraic typ e system
Amulti sorted signature is a sp eci cation for a typ e and hence is an abstract datat yp e
Adding an order on the sorts to obtain an order sorted signature enforces relations
between the sorts It also guarantees sensible interaction between these carriers of
these sorts and op erators thereon in any algebra of the signature
Chapter Order sorted algebra Conclusion
Combining signatures with the equational calculus yields theories Theories further
enhance the usefulness of signatures by ensuring certain equations will hold in any
mo del algebra of the theory signature
Chapter
Extending order sorted algebra
Intro duction
In this chapter we shall discuss how partial functions and conditional signatures may
tie in with traditional order sorted algebra This will mean that any facts that we
mayprove for or use from traditional order sorted algebra will also apply when partial
functions and or conditional signatures are considered to o provided certain extra facts
hold
Next we shall lo ok at the similarities b etween order sorted algebra and category theory
This will demonstrate why computer algebra systems such as Axiom use category theory
terminology whereas most of this thesis uses order sorted algebra as its framework
Lastly we de ne what we mean by a co ercion This de nition is fundamental to the
rest of this thesis
All the work in this chapter is the author s own except for section which is based
almost entirely on Bro after a suggestion by Ric and Mar
Partial Functions
In this section we ask the question How do partial functions interact with classical
universal algebra We need this in case some of the functions used to create our
co ercions later are only partial This can happ en see de nition For example the
division op erator in any quotient eld is partial but could b e viewed as a constructor
function
Chapter Extending order sorted algebra Partial Functions
In our previous chapter on universal algebra we de ned an S op erator of arity q to
q
be a total function from some A to some element of A Where A is a set of S
carriers Is it p ossible to rede ne this so that the S op erators are partial Indeed is
this necessary One of the original reasons for the invention of order sorted algebra
de nition rather than multi sorted algebra de nition was so that all
functions could b e considered total See for example GM
i A A were a partial function in an algebra hA For example if
P T
P T
A A were a total then we could insert a new sort N P such that
N T
N T
function
In the example of a quotient eld one would intro duce a subsort of the integral domain
whichwould represent all the non zero elements of the domain
Wemay pro ceed by either attempting to rede ne all of universal algebra using partial
functions or through some di erentmechanism The following mechanism which uses
virtual sorts turns out to b e awed Virtual sorts are an on the y way of generating
sorts to representthingslike the non zero elements of an integral domain or non empty
stacks
De nition Let h Si de ne an order sorted signature If we de ne h Li to be
the signature where
L S f u u g a set of symbols distinct from al l those in S
m
i f mg s S u s
u i L u
i i
n
n N f g s S q S
n q s n q s
n n
e than n N q L n S s S Note that there may be mor
n q s
one such triple hn q si
then we say that the u are virtual sorts of h Si and n N q
i n q s n q s
n n
L n S s S a virtual op erator symbol of h Si
Unfortunately virtual sorts virtual op erators and homomorphism do not interact in a
satisfactory manner The intro duction of a virtual sort in the source of a homomor
phism may b e meaningless in the target or vice versa
For traditional examples like the non empty stack they are ne since this has some
meaning in all stack algebras But for typ es like all the elements which do not map
Chapter Extending order sorted algebra Partial Functions
to under a particular co ercion then it may be meaningless to create a virtual sort
which attempts to representthis
Broy Bro de nes algebras as having either partial or total op erators A partial
homomorphism is then de ned as follows
De nition Partial homomorphism If hA i and hB i are h Si al
gebras and is a family of partial maps A B then is a partial
s s s
homomorphism hA i hB i i the fol low two conditions are ful l led
Firstly
n N f g q arities of rank n s S
n q s
q
a a A
n
if b oth a a and a a are de ned then
s n q q n
n q s n q s n
a a a a
s n q q n
n q s n q s n
Second ly
n N f g q arities of rank n s S
n q s
a a A a a A
q n q
n n
a a
q i strong q
i
i i
i f ng
a a a a
s n strong s
n q s n q s
n
Where is the strong equality de ned via
strong
de ned a de ned a b a b a de ned b
strong
Broy s de nition of homomorphism de nition is as he states rather lib eral A
sp ecial factor whichhe notes is that the comp osition of partial homomorphisms need
not b e a homomorphism again
The following two de nitions turn out to b e useful in distinguishing b etween typ es of
partial homomorphism
De nition Let h S i bea sub signatureof h Si A algebra hA i is said
to be a subalgebra of the algebra hB i i
with partial op erators in our terminology
Chapter Extending order sorted algebra Partial Functions
s S A B
s s
q
n N f g q arities of rank n s S j
n q s A
De nition Let h S i bea sub signatureof h Si A algebra hA i is said
to be a weak subalgebra of the algebra hB i i
A B s S
s s
q
n N f g q arities of rank n s S j
n q s A
where is de ned via
f g x f x is not de ned f x g x
Broy notes that any partial homomorphism may be decomp osed using the following
eak subalgebra A of A metho dology A partial homomorphism A B de nes a w
de ned via
s S A fa A a de nedg
s s
a a a a if i with the functions taking the meanings
n n
n q s
n q s
a a is de ned Otherwise f ng a are de ned and
n i
n q s
a a is not de ned
n
n q s
By we denote the weak partial identity function A A Then we de ne A B
to b e the total homomorphism which is the restriction of to A
Being total induces a weak subalgebra B of B de ned via
s S B fb B a A a b g
s s s
with the functions de ned via a a a a where b
n n i
n q s
n q s
a and a a is de ned
i n
n q s
This then induces a total surjective homomorphism A B Also by con
struction B is a weak subalgebra of B Therefore there also exists a total injective
homomorphism B B which is the natural inclusion op erator
As examples consider the following co ercions We are only lo oking at the carrier of
the principal sort de nition in these examples
Example Q Z
Chapter Extending order sorted algebra Partial Functions
Fraction Integer PrimeField
A Q nfq j q Q nf gg
B B
Example Q x Z x Q x
Polynomial Fraction Integer Fraction Polynomial Integer
A A
B Q x fp j p Q x q Q x p q denom q Q g
Example Q x Z x
Polynomial Fraction Integer Fraction Polynomial PrimeField
i
A Q x g nf q x Q x j i N f g s Q nf g q s
i i
i N f g
B Z x fp j p Z x q Z x p q denom q Z g
These show that real examples of co ercions may indeed cause A and A to di er as well
as B and B
Notice that in example ab ove we are considering a eld homomorphism and
that A is a weak sub eld of A For example is unde ned on the pair in A
Clearly B is a sub eld of B
In example it is an integral domain homomorphism that concerns us So although
B is only a weak sub eld of B this do es not matter as it is sub integral domain of B
Similarly in example we are again considering an integral domain homomorphism
and b oth A and B are sub integral domains of A and B respectively
As Broy notes this de nition of partial homomorphism allows the everywhere unde ned
function to be a partial homomorphism Thus this de nition is to o weak for our
purp oses
De nition Apartial homomorphism is cal led strict i
q
n N f g q arities of rank n s S a a A
n q s n
Chapter Extending order sorted algebra Conditional varieties
a a a a
n strong n
Broy notes that a strict homomorphism ensures that B is a subalgebra of B and
not just a weak subalgebra is always strict and if is strict then B is indeed
a subalgebra of B
Our requirements for homomorphism are that it act strictly on a certain set of partial
functions the constructors de nition In other words we shall require that our
C
co ercion b e a strict partial homomorphism again see de nition
In the rest of this work we shall not in general concern ourselves with the partiality
of homomorphisms op erators or strictness Since provided our homomorphisms act
ely required for construc strictly on the constructors of the typ e and anytyp e recursiv
tion de nition then the strong equality in the de nition of strictness puts us
in go o d shap e
Conditional varieties
Axiom contains the notion of conditional Categories or in the language of universal
algebra conditional varieties In this section we de ne and reconcile the notions of
conditional and non conditional varieties This means that in future sections we shall
b e able to ignore the existence of conditional varieties in Axiom
Notice that our choice of the word conditional here is similar to that used in the
phrase conditional algebraic sp eci cations The di erence b eing that our conditions
are predicates evaluated over arities not terms Ours is a higher order notion and
should not b e confused with the ordinary lower order work
We extend the de nition of signature to conditional signature via the following de ni
tion
De nition A S sorted conditional signature is a set C of sets C indexed
n q s
by n N f g q an S arity of rank n ands S Each element of C is of the form
if P w then
n q s
where P is a wel l formed proposition in some language and w is an arity of nite
Unconditional may have been a better choice of word from an English language point of view
However we are not saying that an algebra is unconditionally an algebra of a variety we are saying
that it is an algebra of a variety which has no conditions
Chapter Extending order sorted algebra Conditional varieties
rank if it is a tautology it is always represented by T The are the conditional
n q s
op erator symbols
Firstly we need to de ne a conditional term algebra
De nition Let X bean S indexed family of sets disjoint from each other from
S
the set the set fif P theng and from f j arity w if P w then g
n q s
n q s
We de ne T X to be the S indexed family of sets of strings of symbols from
C
S S
fif P theng X each set as smal l as possible satisfying these
s n q s
s S n q s
conditions
s S T X
C s
s
s S X T X
s C s
if P w then C i f ng t T X if P w then
n q s i C q
i
t t T X
n C s
We make T X into a conditional C algebra by de ning conditional operators if
C
P w then on T X for each if P w then C via
T C n q s
If n then if P w then if P w then Guaranteed to be in
T
T X by
C s
Else de ne if P w then t t tobethe string if P w then t t
T n n
T X is cal led the conditional term algebra and an element of T X is cal led a
C C
conditional term
Asso ciated with the notion of conditional signature are the notions of conditional the
require a new de nition for equations and ory and conditional variety However we
equational systems
De nition A conditional multi sorted equation for the conditional signature
C consists of a quadruple P w Y t t where P is a wel l formed proposition in
some language and w is an arity of nite rank if it is a tautology it is always
representedbyT where Y X t and t are terms from the same carrier set of T X
C
or T X A if we aredealing with equations in a particular algebra hA i and every
C
Chapter Extending order sorted algebra Conditional varieties
variable occurring in t and t occurs in the appropriate member of Y t t T Y
C
Such equations are more usual ly written if P w Y t t
If t and t come from the same carrier set of T X we say that the equation is
C
C generic
Note that this de nition of conditional equation should not be confused with that
found in work such as KR In that context a conditional equation is a clause such
as if P t Y t t where t is a subterm of t for example
De nition An conditional equational system for the signature is a set of
conditional equations for T X or T X A if we aredealing with equations in a
C C
particular algebra hA i
Asso ciated with de nition are two sp ecial signatures and The former is the
extension of all the C s the latter is extended byallC s More formally
De nition Using the de nitions of de nition we de ne to be the S
sorte d signature where n q s n N f g q an S arity of rank n and s S
f j if P w then C g
n q s n q s n q s n q s
Now similarly we de ne to be the S sorted signature where n q s n N f g q
an S arity of rank n and s S
T then C g f j if
n q s n q s n q s n q s
where S is the largest subset of S such that has no void sorts
The algebras of conditional signatures are the same as ordinary algebras except that
w
the prop osition P is evaluated over A and i this is true there is an op erator
n q s
in that algebra More formally
De nition An order sorted total conditional C algebra is an ordered triple
hA fA s S g i where A is a class known as the universe fA s S g is an
s s
S indexed family of subsets of A known as the carriers of the algebra and is a set
S
q w
of sets of functions f A A j P A g such that
n q s n q s s
n q s
A A
u
Chapter Extending order sorted algebra Conditional varieties
If s s in S then A A
s
s
If if P w then C and if P w then C with s s and
n q s
n q s
w w
q q and also P A and P A then j
q
n q s
n q s
A
Clearly if all the P are tautologies then a conditional signature C is identical to its
non conditional partner Therefore all conditional C algebras are non conditional
algebras in this case
In any case a C algebra is a algebra Thus wecanseethat is a minimal signature
for C
De nition We de ne a conditional theory to be the ordered pair hhC Si Si
where C is an S sortedconditional signatureandS is a set of S sortedconditional
equations We say that a C algebra hA i mo dels or satis es the conditional theory
i it is a model for Sj where Sj is de ned as fol lows
hA i hA i
w
Sj f Y t t j P w Y t t S P A g
hA i
A theoretical interpretation of a conditional theory hhC Si Si is a theory hh Si S i
equivalent to hhC Si Si with al l the propositions P w evaluated in some consistent
manner
Final ly we demand that for al l possible pairs of theoretical interpretations hh Si S i
hh Si S i where and S S that hh Si S i is a protecting extension of
hh Si S i
De nition The col lection of al l models of a particular conditional theory is
cal led a conditional variety
In fact the conditional variety de ned by hhC Si Si is equivalent to the variety de ned
by hh S i Si
Also the variety sp eci ed by any theoretical interpretation of the conditional theory
hhC Si Si forms a sub class of the the conditional variety Moreover for a pair of theo
retical interpretations hh Si S i hh Si S i where and S S thevariety
sp eci ed by hh Si S i forms a sub class of the variety sp eci ed by hh Si S i
Minimal in the sense that it has fewest op erator symbols and sorts The variety it sp eci es is
maximal if you consider the numb er of algebras that mo del it
Chapter Extending order sorted algebra Conditional varieties
If we consider two conditional algebras from the same conditional variety thenwe wish
to know whether there is a homomorphism from one to the other For this we need to
de ne conditional homomorphism
Atechnical de nition which will allow us to de ne homomorphisms more easily
De nition Let hA i and hB i be C algebras For al l n q s we de ne
n q s n q s n q s
to be the map
n q s n q s
Similarly de ne for hB i
n q s
The family of maps as we show in notation map an algebra hA i to a
non conditional signature which it mo dels
De nition hA i hB i is a conditional C homomorphism i it is a
AB
non conditional homomorphism where
AB
ff j g f j gj
n q s n q s n q s n q s n q s n q s n q s
n q s
n N f g q an S arity of rank n s S g
AB AB
is a S sorted signature where S is the largest subset of S such that has no
void sorts
In fact if we use the following piece of notation
NC A
Notation If hA i is a conditional C algebra we de ne to be the
non conditional signature
ff j gjn N f g q an S arity of rank n s S g
n q s n q s n q s n q s
NC A NC A
is a S sorted signaturewhere S is the largest subset of S such that has
no void sorts
As an aside notice that for each C algebra hA i there is an asso ciated non conditional
T X This term algebra is the equivalentofevaluating each P w term algebra
NC A
over A in T X
C
Chapter Extending order sorted algebra A Category theory approach
NC A
Then hA i is a non conditional algebra and
AB NC A NC B
f j n N f g q an S arityof rank n s S g
n q s n q s
So for any two conditional algebras of the same conditional signature there exists
a xed maximal non conditional signature of which they are both non conditional
algebras Wemayhave to forget some op erators and even some sorts whichmay only
exist in conditions
Our de nition of conditional homomorphism is that of the non conditional homomor
phism from this xed non conditional signature
one for non Hence the concept of conditional signatures can always be reduced to
conditional signatures and therefore any results we prove or use need only b e for non
conditional signatures
As we have seen in de nition this concept may also be extended to theories In
addition to adding new op erators when certain prop ositions hold we also allow for new
equations to be added conditionally However we demand that any such extension
is a protecting extension of the minimal non conditional theory the theory over the
signature h S i
More explicitly let us de ne S to be all those equations from S which only involve
NC A
arities from as well as all equations not involving arities Similarly de ne
S to be all those equations from S which only involve arities from as well as all
equations not involving arities
NC A
If hA i is a conditional C mo del then hh S i Si must b e a protecting exten
sion of hh S i Si
A Category theory approach
This section is an aside from the rest of the chapter We are not extending the theory
of order sorted algebra nor category theory merely explaining whywecover b oth
There is a great deal of corresp ondence b etween category theory sp eci cally categor
ical typ e theory and universal algebra There is not enough ro om here to cover this
huge topic but the reader is pointed to Cro whichcovers categorical typ e theory
in great depth
Alternatively de ne S to b e those equations from S for which hA i is a mo del since it maynot
mo del all the equations in the de nition of S in the main text
Chapter Extending order sorted algebra Co ercion
As usual Mac Lane ML also contains a great deal of information on the corresp on
dence between universal algebra and adjoint functors in the chapter on Monads and
Algebras
All the ideas from multi sorted algebra theory may be represented in the categorical
typ e theory framework although an equivalent for order sorted algebra app ears not to
be covered However the author b elieves that the notion could b e intro duced
One area which we shall discuss here is the concept of algebraic homomorphism and
the arrows morphisms of a given category
The corresp ondence between a categorical typ e theory C and its asso ciated algebraic
typ e theory hh Si Si is the following
Ob j C all the algebras for a giv en which mo del S
Arr C all or a xed sub collection of the homomorphisms
The category of all algebras of a given signature h Si has as an initial ob ject the term
algebra T X
The category of all algebras which mo del a given theory hh Si Si has as an initial
ob ject the free algebra T X This is why the second universality theorem
E
holds
So we see that both categories and theories collect together similar typ es This ab
straction of information forms the basis of abstract datatyping
Co ercion
In this work we are interested in co ercions whichso far have been explained as being
natural typ e changing maps Wemay de ne them in a far more strict fashion
The following w ould seem to be a go o d de nition for co ercion However in practice
this de nition is not well de ned enough since it do es not state which theory a co ercion
should come from
De nition Let hA i and hB i be algebras from the variety speci ed by some
theory hh Si Si The map hA i hB i is a coercion if it is a homomorphism
of that theory
Using the category theory corresp ondence ab ove we see that a function b etween two
Chapter Extending order sorted algebra Co ercion
typ es of the category of all algebras which mo del a particular theory is a co ercion i
it is an arrow of the category
The only problem with this de nition of co ercion is that it do es not sp ecify from which
theory or equivalently category from which we should demand the homomorphism
be taken In general we would like this to be the most sp eci c or smallest category
to which b oth algebras b elong
If we just used the de nition of co ercion ab ove de nition then any total map
between typ es would b e a co ercion Any small category may b e forgotten backto Set
via the forgetful functor and any total function is an arrow of Set Most typ es in
computer algebra system are ob jects of Set which is equivalent to SetCategory in
Axiom the second most basic Category in Axiom
Richardson Ric notes that we also require a xed framework in which we de ne
our co ercions If we were to attempt to de ne a co ercion hA i to hB i to be any
map which is a homomorphism for all theories which both algebras mo del then the
de nition would not b e well de ned Weneedto state a context
We are attempting to re ect the situation that app ears in a system like Axiom Th us
wemust state that wehavebeengiven a priori a xed collection of theories and that
given anytyp e we know precisely which theories it mo dels
This precludes a user from adding another theory which the algebras mo del which could
rede ne what co ercion means for those algebras
In the category theory corresp ondence we say that Axiom s Category is a xed col
lection of categories and given anytyp e weknow of which categories it is an ob ject
De nition Co ercion Let bea xed col lection of theories
Let hA i be a model for theories for i in some indexing set I where i I
i i
is a theory from
hB i be a model for theories for j in some indexing set J where Similarly let
j
j J is a theory from
j
Then we cal l a map hA i to hB i a co ercion i it is a homomorphism for al l the
theories in
f j i I g f j j J g
i j
In Axiom Domains are only memb ers of a xed set of Categories
A user could re implement one of these Categories and ruin everything
Chapter Extending order sorted algebra Conclusion
In fact in Axiom we are in a slightly b etter p osition If a Domain is an ob ject of two
Categories then there must exist a Category which extends p otentially trivially
b oth these Categories of whichthe Domain is an ob ject
Thus in Axiom de nition reduces to
A co ercion in Axiom from a typ e A toatyp e B is a homomorphism of the
most restrictive Category to which b oth A and B b elong
It would be useful to have a name for maps which are co ercions in the sense of
de nition These maps are in a sense natural and may b e the next most natural
map b etween twotyp es However as wehave shown in the example using Set the map
need not particularly natural from a realistic p ointofview
with our current terminology maps which satisfy de nition which As it stands
are not co ercions are merely homomorphisms which are not co ercions
If no co ercion existed b etween twotyp es but a homomorphism which is not a co ercion
existed and a user required that homomorphism then either the user only required a
conversion or the theory lattice for the algebra system has not b een designed correctly
Conclusion
In this chapter we have shown how classical order sorted algebra may b e extended to
encompass the notion of partial functions and conditional signatures We have also
stated how these notions interact with the equational calculus These are imp ortant
extensions due to the fact that these notions are used extensively in Axiom
Wehave also mentioned some of the correlation b etween category theory and universal
algebra Finally wehav e made the imp ortant de nition of a co ercion and demonstrated
why the de nition is necessarily strict
Chapter
Coherence
Intro duction
In this chapter we will lo ok at a conjecture of Web er which is imp ortant to our work
We shall state his assumptions and pro of which is incorrect
We shall then add in some extra assumptions and truly prove the theorem Finally we
shall relax one of Web er s assumptions and prove that the theorem still holds
The assumptions made byWeb er provide a strict formal setting for typ es in an algebra
system The theorem in itself proves con uence for co ercions in this setting
All the work in section except the explanation of n ary typ e constructor which
is by this author and section is taken from Web b Web a Web All the
rest of the work is the author s own
Web er s work I de nitions
This section contains the de nitions from Web er s thesis Web b Web a Web
required for the statement and Web er s pro of of Web er s co ercion conjecture
His assumptions are in the next section
These statements will also b e used when we correctly prove the coherence theorem
in section
It should b e noted that Web er s co ercion conjecture only makes up part of one chapter
of his thesis Web b which also covers various areas of typ e classes typ e inference
and co ercion in great depth and detail
Chapter Coherence Web er s work I de nitions
Web er uses the phrase typ e class where we would use the terms category or va
riety
De nition A base typ e is any type which is not a parametrical ly de ned type
i e a ary operator in the term algebra of type classes
So for example the Integer and Boolean typ es are base typ es
De nition A ground typ e is any type within the system which is either a base
type or a parametrical ly de ned type with al l the parameters present Any non ground
e is cal led a p olymorphic typ e typ
As examples we have Integer and Polynomial Fraction Integer As a non
example wehave Polynomial R Ring
De nition If there exists a coercion from t to t we say that t t
This de nition places an order on the ground typ es
Web er uses the phrase n ary typ e constructor to mean a functor from the pro duct
of n categories to a sp eci c category
Equivalently it is a function from the pro duct of n varieties sp eci ed resp ectively
by the theories to a variety sp eci ed by the theory hh Si Si This
n
is a function which for all i maps the carrier of the principal sort and p otentially
the carriers of some of the non principal sorts of a mo del of to one for each sort
i
mapp ed of the non principal sorts of a mo del of hh Si Si
If the mo del returned is hA i this function must map one and only one sort carrier
to each and every member of A nfA g where S is the principal sort
S
we de ne com t to be if t is a base type De nition For a ground type t
or if t f t t an n ary type constructor then com t is de ned to be
n
max fcom t ji f ngg
i
This de nes the complexity of a ground typ e to b e how far up the typ e lattice it is
De nition Coherence A type system is coherent if the fol lowing condition
is satis ed
Chapter Coherence Web er s work I de nitions
ground types t t t t coercions
This guarantees that there only exist one co ercion from one ground typ e to another
This is a highly desirable feature of anytyp e system The main results of this chapter
the coherence theorem and the extended coherence theorem are that we
may be able to guarantee that our typ e system is coherent providing some sensible
assumptions hold true
In the following de nitions all the and are typ e classes
type t is an object of the type class Notation t means that the
De nition The n ary typeconstructor f n N induces a structural co ercion
if thereare sets A f ng and M f ng such that the fol lowing condition
f f
is satis ed
If f and f and i f ng
n
n
i A M t t and there exist ground types t and t
i i i
f f
i i i
coercions
if i M t t
i i
f
i
t if i A t
i i
f
i
id id if i A M
i t
f f
t
i
i
then there exists a uniquely de ned coercion
F t t t t f t t f t t
n n n
f
n n
The type constructor f is covariant or monotonic in its i th argument if i M f
f
is contravariant or antimonotonic in its i th argument if i A
f
t Note that if i A M then t
i
f f
i
As an example of covariance the list constructor in Axiom List a functor Set
ListAggregate takes one argument in whichit is covariant Given typ es A and B
h that there exists a co ercion A B then suc
F A B List A List B
List
Chapter Coherence Web er s work I de nitions
Since Axiom s typ e constructors are functors then category theory states this more
simply as
F A B List
List
Contravariance is a rarer case However Axiom s Mapping functor is contravariant in
its rst argument and covariant in its second Mapping takes two typ es A and B and
returns the typ e of all mappings from A to B
As a concrete example for Mapping supp ose we wish to nd the uniquely de ned
co ercion
Mapping Fraction Integer Fraction Integer
Mapping Integer Fraction Integer
There exists a co ercion
Integer Fraction Integer
the inclusion op erator There also exists a co ercion
id Fraction Integer Fraction Integer
which is the identity op eration The uniquely de ned co ercion is as follows
F Fraction Integer Fraction Integer Integer Fraction Integer id
Mapping
Mapping Fraction Integer Fraction Integer
Mapping Integer Fraction Integer
which sends f id f
The following de nitions shows that there is a homomorphic image of a parameter in
the created typ e For example there is a homomorphic image of the underlying ring in
any p olynomial ring
De nition Let f be an n ary type constructor If i
n
f ng for some ground types t such that there exists a coercion
i i
i
t f t t
i n
f t t
n
then we say that f has a direct emb edding at its i th p osition
Chapter Coherence Web er s work I I assumptions and a conjecture
Moreover let
D fijf has a direct embedding at its i th position g
f
be the set of direct emb edding p ositions of f
The de nition of D is a technical de nition of Web er s needed for one of his assump
f
tions
Web er s work II assumptions and a conjecture
This section provides all the assumptions and results whichWeb er uses in his pro of
of Web er s coherence conjecture which we shall state and at the end of this
section The assumptions and results are also required in our pro of of the coherence
theorem
Assumption For any ground type t the identity on t wil l be a coercion The
wel l de ned composition of two coercions is also a coercion
This is clearly a sensible if not often implemented statement Since our co ercions are
always to b e arrows of a category then the ab ove assumption must hold
Lemma If assumption holds then the set of ground types as objects to
gether with their coercions as arrows form a category
Proof Immediate
The following assumption will provide us with the basis for a coherent typ e system
y ensuring con uence amongst di erent paths leading to the Our coherence is built b
same co ercion If we do not have coherence at the base typ es then we shall not have
coherence amongst the general typ es
Assumption The subcategory of base types and coercions between base types
forms a preorder i e if t t are base types and t t are coercions then
The following condition states that F is a functor o ver the category of all f s
f
Chapter Coherence Web er s work I I assumptions and a conjecture
Assumption Let f bean n ary typeconstructor which induces a structural co
ercion and let f t t f t t f t t beground types Assume that the
n
n n
fol lowing are coercions
i M t t t t
i i
f
i i i i
t i A t t t
i i
f
i i i i
i A M t t t and id
i i t
f f
i i i
i
Then the fol lowing conditions are satis ed
id F t t t t id id
n n t t
f
f t t
n
n
F t t t t
n n
f
n n
F t t t t F t t t t
n n
f f
n n n n
This is a condition which stops direct emb eddings b ecoming confused Firstly Web er
declares that anytyp e constructor can only have one direct emb edding We shall show
how to relax this condition in a later section Secondly he states that direct
emb eddings where they exist are unique
Assumption Let f beann ary typeconstructor Then the fol lowing conditions
hold
jD j
f
i
Direct embedding coercions are unique i e if t f t t and
i n
f t t
n
i
t f t t then
i n
f t t
n
i i
f t t f t t
n n
The following assumption is highly technical and shows how direct emb eddings interact
with structural co ercions Basically they commute
Assumption Let f beann ary typeconstructor which induces a structural coer
cion and has a direct embedding at its r th position Assume that f
n
and f and i f ng t and t If there are
i i
n
i i
r r
coercions t t if the coercions and are de ned and if f is
r r
r
f t t
f t t
n
n
th argument then the fol lowing holds covariant in its r
r r
F t t t t
r n n
f
n f t t
f t t
n
n
Chapter Coherence Web er s work I I assumptions and a conjecture
or equivalently the fol lowing diagram commutes
r
t t
r
r
r r
f t t
f t t
n
n
F t t t t
n n
f
n
f t f t t t
n
n
However if f is contravariant in its r th argument then
r r
F t t t t
n n r
f
n f t t
f t t
n
n
or equivalently the fol lowing diagram commutes
r
t t
r
r
r r
f t t
f t t
n
n
t F t t t
n n
f
n
f t t t f t
n
n
We are now in a p osition to state Web er s coherence conjecture and his pro of This
attempts to show that when the aforementioned assumptions hold true then wehavea
coherenttyp e system We shall give more assumptions and a prop er pro of in section
Conjecture Web er s coherence conjecture Assume that al l coercions be
tween ground types are only built by one of the fol lowing mechanisms
coercions between base types
coercions induced by structural coercions
direct embeddings in a type constructor
composition of coercions
coercions identity function on ground types as
If assumptions and are satis ed then the set of ground
types as objects and the coercions between them as arrows form a category which is a
preorder
Chapter Coherence The coherence theorem
This is the pro of of this conjecture given in Web b Web a Web
Weber s Proof By assumption and lemma the set of ground typ es
as ob jects and the co ercions b etween them form a category
For anytwogroundtyp es t and t we will proveby induction on max com t com t
that if t t are co ercions then
If max com t com t then the claim follows by assumption Now assume
that the inductivehyp othesis holds for k and let max com t com t k Assume
t t are co ercions w l o g that t t and that
Now t t com t com t So we may assume that t f u u for some
n
n ary typ e constructor f
By assumption and the induction hyp othesis we can assume that there are ground
typ es s s and unique co ercions t s and t s such that either
F t s
f
or
i
f s
Similarly either
t s F
f
or
j
f s
If is of form and is of form then by assumption and the
uniqueness of F
f
If is of form and is of form then by assumption
Analogously if is of form and is of form
If is of form and is of form then assumption implies that i j and
s s Because of the induction hyp othesis wehave and hence again
by assumption
The coherence theorem
In the ab ove pro of there seem to b e some irregularities
Chapter Coherence The coherence theorem
On co ercibility and complexity
Web er states in the pro of that
t t com t com t
Web er do es not prove this and indeed it is not true
Supp ose that f is an n ary typ e constructor and that it is contravariant in its i th
p osition If
F s s t t f s s f t t
n n n n n
f
and t s with
i i i
com s max com s com s com s com s
i i i n
In other words com s istheunique maximum memb er of the set
i
fcom s j j f ngg
j
Then if com t com s and for all j in f ngnfig we have that s t we
i i j j
haveacounterexample Web er s claim that
com f s s com f t t
n n
Thus Web er s assertion is invalid
Structural co ercions in the pro of
In equation and similarly equation is given as a function
f t f s
however is supp osed to b e a function from t
Chapter Coherence The coherence theorem
Structural co ercions syntax
Web er calls the structural co ercion function from f s s to f t t
n n
F s s t t
n n n
f
where is from s to t or t to s dep ending on whether f is covariantorcontravariant
i i i i i
in its ith argument resp ectively
However this is merely the functorial action of f on the maps and could b e
n
represented more compactly as
f
n
The source and target of each and knowledge of the sets A and M uniquely de
i
f f
termine the source and target of f This also demonstrates the uniqueness
n
of f and guarantees that assumption holds
n
Identity co ercions
Web er states in assumption that the identity function is a co ercion However
he never proves this to b e unique Indeed automorphisms are p erfectly natural maps
t t
In a computer algebra system like Axiom many automorphisms are not automorphisms
of the smallest category to whichatyp e b elongs
For example the ring automorphism Z X Y Z Y X isnotaPolynomialCategory
homomorphism since for instance the leadingMonomial function is not preserved
under the map
For some categories like the category of groups this may not b e so easy to implement
We add the following sensible assumption
Assumption The only co ercion from a type to itself is the identity function
We shall of course allowanynumb er of functions from a typ e to itself including con
versions It is merely the numb er of co ercions whichwe are restricting
Chapter Coherence The coherence theorem
Comp osition of co ercions
It is p ossibly symptomatic of the previous errors that Web er has neglected to cover all
the p ossible cases of and
All our co ercions are comp ositions of co ercions or just a single co ercion from the list
co ercions b etween base typ es
co ercions induced by structural co ercions
direct emb eddings in a typ e constructor
identity function on ground typ es as co ercions
Supp ose that and that where and are single co ercions and
and are also comp ositions of co ercions may b e the identity co ercion as may
be
For a pro of by induction on com weneedtocover all the cases of pairs Thus
the list whichwe need to consider is
is a co ercion b etween base typ es is a co ercion b etween base typ es
is a co ercion b etween base typ es is a structural co ercion
is a co ercion b etween base typ es is a direct emb edding
is a co ercion b etween base typ es is an identity function
is a structural co ercion is a co ercion b etween base typ es
is a structural co ercion is a structural co ercion
is a structural co ercion is a direct emb edding
is a structural co ercion is an identity function
is a direct embedding is a co ercion b etween base typ es
is a direct embedding is a structural co ercion
is a direct embedding is a direct emb edding
is a direct embedding is an identity function
is an identity function is a co ercion b etween base typ es
Chapter Coherence The coherence theorem
is an identity function is a structural co ercion
is an identity function is a direct emb edding
is an identity function is an identity function
Firstly notice we are only interested in the pairs as unordered entities Thus some of
these cases are duplicates
Indeed case is case case is case case is case case is case case is
case case is case So wemay discard cases and
Now notice that a base typ e can not haveadirect emb edding neither can it induce a
structural co ercion If either or is co ercion b etween base typ es then the target of
and is a base typ e Hence the other can not be a direct emb edding nor can it
i
b e a structural co ercion
tly Thus wemay All the cases of this form are equivalently and equivalen
ignore these to o
Our remaining cases are and
To really prove the coherence theorem we are going to require some more assumptions
Only one of them is di cult to justify
We shall replace this rst assumption which uses Web er s de nition of a direct
emb edding presently with our own de nition In our de nition assumption
shall always hold trivially
Assumption If a type constructor f has a direct embedding at its i th position
i
and f t t exists then t f t t
n i n
f t t
n
Web er de nes a f to hav e a direct emb edding de nition at a particular p osi
tion if there exists some n tuple of typ es t t for which t directly emb eds in
n i
f t t
n
This seems sensible if we consider a typ e constructor to be a function which returns
the carrier of the principal sort of the algebra Wearesaying that direct emb edding of
atyp e in a constructor is equivalenttosaying that a sort the principal sort
It is true that an Axiom typ e constructor returns the carrier of the principal sort of
the algebra but Axiom is more sp eci c than that The typ e constructor furnished with
full complement of arguments is the principal sort of the algebra
Chapter Coherence The coherence theorem
In fact it would be b etter to replace de nition and assumption with the
following de nition
De nition Let f be an n ary type constructor If i
n
f ng ground types t there exists a coercion
i i
i
t f t t
i n
f t t
n
then we say that f has a direct emb edding at its i th p osition
eover let Mor
D fijf has a direct embedding at its i th position g
f
be the set of direct emb edding p ositions of f
The next assumption is undesirable due to its di culty to guarantee in any implemen
tation However it is provable in the more common covariant case and we shall prove
it in the pro of of the coherence theorem
We are assuming that structural co ercions b ehave con uently The assumption b elow
is stated to o strongly We shall state the assumption we really need
which is more complicated immediately after Assumption gives the idea of what
we require whereas assumption gives us what is necessary It is an assumption
ab out typ e constructors
Assumption Let the typeconstructor f induce a structural coercion If t
then if there t f s s t f u u are coercions and t f t
n n
n
exists
F s s t t f s s t
n n n
f
n
t F u u t t f u u
n n n
f
n
the fol lowing holds
F s s t t F u u t t
n n n n
f f
n n
Chapter Coherence The coherence theorem
or equivalently the fol lowing diagrams commute
f s s t
n
F s s t t
n n
f
n
F u u t t
n n
f
n
f u u t f t t
n
n
Now what we actually require Supp ose that a typ e t is not constructed bythen ary
typ e constructor f which is contravariantinitsith p osition at which it also has a direct
emb edding
Supp ose also that t is co ercible to twotyp es whichmay b e directly embedded in f at
the ith p osition Let us directly embed them to gain two new typ es constructed by
f If these new typ es constructed by f can b oth b e structurally co erced to the same
typ e t also constructed by f then the two comp ositions of co ercions t t are the
same
Succinctly A structural co ercion of a direct emb edding of any other co ercion is unique
Assumption Let f be a type constructor c ontravariant at its i th position at
which it also has a direct embedding and let t be a type not constructed by f
If t a and t b are coercions and f a a f t t and
i i a n
a n
b
are structural coercions then f b b f t t
n
b
n
i i
a
b
b f b b a f a a
n n
or equivalently the fol lowing diagram commutes
Chapter Coherence The coherence theorem
t
a
b
a b
i i
i i
f b b f a a
n n
f a a f b b
n n
b
a
f t t
n
Notice that assumption is provable in the case that f is covariant at i and we
shall prove this in the pro of of the coherence theorem In the contravariant case
there is no link b etween t and t and it is this whichmake it an assumption
i
The following assumption is more easily ensurable We merely require that for two
typ es constructed by the same typ e constructor if there exists a co ercion between
them then it equals the structural co ercion between them which we now assume to
exist This is also an assumption on typ e constructors
Assumption If f s s f t t then there exists a structural coer
n n
cion f s s to f t t and it is the unique coercion f s s to
n n n
f t t
n
We are now nally in a p osition to state and prove the co ercion theorem
Theorem Coherence theorem Assume that al l coercions between ground
types are only built by one of the fol lowing mechanisms
coercions between base types
coercions induced by structural coercions
direct embeddings de nition in a type constructor
composition of coercions
identity function on ground types as coercions
Chapter Coherence The coherence theorem
If assumptions and are satis ed
then the set of ground types as objects and the coercions between them as arrows form
a category which is a preorder
Proof By assumption and lemma the set of ground typ es as ob jects and
the co ercions b etween them form a category
All our co ercions are either an individual co ercion or a comp osition of nitely many
co ercions from the list co ercions b etween base typ es structural co ercions direct em
b eddings de nition in a typ e constructor and identity functions
Thus wemay decomp ose any co ercion into such a nite comp osition We de ne len
co ercions in any decomp osition of the length of to be the minimum number of
Clearly there are in nitely manyvaluesoflen but there is a minimum value
For any two ground typ es t and t we will prove by induction on the length of the
co ercions b etween them that any co ercions b etween them are unique
Let t t b e co ercions
If the length of and is then replacing the unordered pair with we
need to lo ok at our eight cases and de ned ab ove
Cases and If isabasetyp e co ercion then t and t are base typ es and co ercions
between base typ es are unique by assumption
Case and If is an identity function on a ground typ e then t t then by
assumption
Case If and are structural co ercions then and are equal by assumption
Case If is a direct emb edding and is a structural co ercion then there are two
cases
Case a is covariant Then by assumption
Case b is contravariant Thus t f t t and there exists i and a co ercion
n
t t This can not happ en since no comp osition of our four basic co ercions can create
i
such a co ercion
y assumption Case If and are direct emb eddings then b
Wenowmay assume that any co ercions of length less than or equal to k are unique
Supp ose that t t are co ercions and max len len k Also supp ose
that and where for i in f g we have s t are single
i i
atomic co ercions Also t s and t s are unique co ercions since their
Chapter Coherence The coherence theorem
lengths are less than or equal to k
If the length of or is then we de ne or resp ectively to be the identity
function on t
As stated b efore there are eight cases and Wemay transp ose
and if we wish since their order is unimp ortant
Case is a co ercion b etween base typ es is a co ercion b etween base typ es Then
t s s are all base typ es Base typ es may only b e the targets of base typ e co ercions
hence t is a base typ e Thus and are co ercions between base typ es and are thus
equal by assumption
Case is a co ercion b etween base typ es is an identity function Again t and t
must b oth b e base typ es and and are equal by assumption
Case is a structural co ercion is a structural co ercion Let t f t t
n
If t is equal to some f t t then and are structural co ercions f t t to
n n
t f t t and hence are equal by assumption
n
Else we may assume that t f t t Since all our co ercions are built from
n
comp ositions of the four basic typ es of co ercion wemay consider and to b e chains
of comp ositions Without loss of generality assume that the length of is k and
the length of is l where l k
k
t h h h h t
k k
and
l
s s s t t s
l l
In b oth the comp osition chains of co ercions from t to t there must exist a minimal
elementofthechain which is constructed by f and this by assumption can not b e s
or h Also b ecause and are structural co ercions these minimal elements are not
h or s
k l
In the chain we shall assume that this elementish In the chain we shall assume
a
that this elementiss
b
Since h is constructed by f and h is not and is one of the four basic co ercions
a a a
it must be a direct emb edding It can not be a structural co ercion since h is not
a
constructed by f It is not a co ercion b etween base typ es b ecause h is constructed by
a
It is not an identity co ercion since h is not constructed by f whereas h is f
a a
Chapter Coherence The coherence theorem
Similarly is also a direct emb edding
b
Now by assumption the direct emb eddings must be at the same p osition We
shall assume that this p osition is the ith
Let h f a a and s f b b Call the unique structural co ercions
a n n
b
h t and s t Notice that h a and s b
a a a i i
b b b
Call the maps t h and t s where
a
b
a
b
a
a
and
b
b
If a or b equal then wemay insert an initial identity co ercion and lengthen the chain
byone This do es not a ect our induction on length since we are always in go o d shap e
with identity co ercions Neither a nor b maybe by our assumption on t
Thus wehave the following diagram and we wish to prove that it commutes
t
a
b
h a s b
a i i
b
i i
f a a f b b
n n
s f b b h f a a
n a n
b
b
a
f t t
n
Case a f is covariant in the ith p osition
Consider the co ercions a t and b t Recall that t f t t
i i
n
i i
These must exist b ecause and lift them covariantly resp ectively
a
b
i
The de nition of direct emb eddings guarantees the existence of
f t t
n
Chapter Coherence The coherence theorem
Nowby the de nitions ab ove
i
a
f a a a
n
By assumption
i i
a
f a a
f t t
n
n
Thus
i
a
f t t
n
By induction on length we know that the co ercion t to t is unique Hence
i
a b
So
i
b
f t t
n
Assumption gives us
i i
b
f b b
f t t
n
n
Thus
i
b
b f b b
n
whichby de nition means
Graphically the following is a commutative diagram
t
a
b
h a s b
a i i
b
i i
t
i
f a a f b b
n n
i
h f a a s f b b
a n n
b
f t t
n
b
a
t f t t
n
Case b f is contravariant in its ith p osition
Chapter Coherence The coherence theorem
There is nothing to link t and t in this case and wemust resort to assumption
i
t
a
b
t
i
h a s b
a i i
b
i i
f a a f b b
n n
h f a a s f b b
a n n
b
b
a
t f t t
n
Case is a direct emb edding is a structural co ercion
Case a is covariant Then bycase aabove
If t Case b is contravariant Let s f s s and t f t
n n
s is the co ercion lifted contravariantly by where t
i i
F s t
f
i
i
Also
i
f t t
n
The uniqueness of implies that
i
f s s
n
By assumption wehave
i i
F s t
f
i f s s
i
f t t
n
n
Thus
i i
F s t
f
i f s s
i
f t t
n
n
Chapter Coherence The coherence theorem
and therefore
i
F s t
f
i
i
f t t
n
Hence
Case is a structural co ercion is an identity function Let t f t t
n
By viewing as the identity function
F t t t t id id
f
t
t
n n
n
Thus by case
is a direct emb edding is a direct emb edding Assumption implies Case
that the direct emb eddings must b e at the same p osition i Therefore s s So by
the inductivehyp othesis By assumption Hence
Case is a direct emb edding is an identity function Let t f t t
n
By viewing as the identity structural co ercion
id F t t t t id
f
t
t
n n
n
Thus by case ab ove
Case is an identity function is an identity function Thus and
However since s s by the inductivehyp othesis Hence
Thus wehaveproved for all co ercions of length less than or equal to k Hence
the result hold for all co ercions by induction
In case b of the case when the length is we claimed that there can not be built a
co ercion from a typ e constructor to one of its arguments
From our description so far of Axiom one might think that Axiom would p ermit the
construction of the following typ e
Fraction Fraction Integer
which is the quotient eld of the quotient eldofintegers However the quotient eld
of a quotient eld is itself and hence
Fraction Integer Fraction Fraction Integer
The quotient eldoftheintegers Z is more commonly known as the rationals Q
Chapter Coherence Extending the coherence theorem
For this reason Axiom contains sp ecial co de in the interpreter to stop such pathological
typ es b eing instantiated A Bo ot function isValidType checks to see if one is trying
to create a typ e liketheoneabove or
Polynomial Polynomial Integer
But this is hand crafted sp ecial co de covering afewcases and mentions the typ es by
name It is conceivable that a user could create a new typ e called MyFraction which
is identical to Fraction This would not b e picked up b e isValidType and thus
MyFraction MyFraction Integer
could b e instantiated Since the typ e is then isomorphic to one of its arguments it is
feasible that a co ercion between the two could be de ned contradicting our claim in
case b of the case when the length is
This co ercion can still not be built from our four basic typ es thus de ning such a
co ercion contradicts our assumptions for the coherence theorem
Extending the coherence theorem
Firstly Web er s pro of of his conjecture relies on induction on com t This
assumes that there are no typ es of in nite complexity in our system This is not the
case in Axiom since one could de ne the following typ es in one le
R r Ring Ring r
D Ring R D
D Ring R D
though calling D would b e disastrous So we add the following extra assump
tion
Assumption There do not exist any types of non nite complexity
The pro of of the coherence theorem do es not rely on typ e having nite complexity
wever it is still a sensible assumption to make and can b e easily guaranteed in a real Ho
implementation It is also imp erative that this assumption holds if the automated
co ercion algorithm is to terminate
This would try to create the constant from the ring D which can not b e evaluated
Chapter Coherence Extending the coherence theorem
Assumption states that wemayhave only one direct emb edding into an algebra
However in Polynomial Integer one would wish to p erform the direct emb eddings
of b oth of Integer and Symbol yet this violates assumption and thus wewould
not b e able to prove that our algebra system is coherent
The reason for Web er s assumption is to stop co ercions like the following o ccurring
If A is a group then b eing able to co erce A A A whilst p otentially useful is
ambiguous As he points out in Web b Web a Web A can be co erced into
A A via the isomorphisms A A I or A I A where I is the trivial subgroup
of A
In this example the inclusions are ambiguous since using either co ercion the target of
the inclusion is a group However in many cases the typ es are incomparable eg
Integer and Symbol and thus the assumption seems to be to o strong The question
is what do wemeanby incomparable
Certainly there is no co ercion function Integer Symbol or Symbol Integer But
for two distinct non trivial prop er normal subgroups G H of A such that G H I
then there is no co ercion function A G A H or A H A G yet there then exists
A G is the two distinct co ercion functions A A G A H and coherency is lost
quotient group A factored out by G and is the set faG j a Ag where aG a G i
a a G Thus the condition of there not existing a co ercion function b etween our
twotyp es is not su cient for our de nition of incomparability
However we notice that there exists no typ e which can b e co erced to b oth Symbol and
Integer but there do es exist homomorphisms A A G and A A H So if we
cho ose the statement Typ es A and B are incomparable if there do es not exist a typ e S
whichcan b e co erced to b oth A and B asa de nition of incomparability then we are
in go o d shap e
We state this in the language of Web er as follows To replace the assumption we
rst need to alter de nition our previous replacement of de nition to the
following
De nition Let f be an n ary type constructor If i
n
f ng ground types t there exists a coercion
i i
i
t f t t
i n
f t t
n
then we say that f has a direct emb edding at its i th p osition
Chapter Coherence Extending the coherence theorem
Moreover let s bea ground type and i f ng and de ne
i
P i s s t t f t t
i i n
f t t
n
where is a coercion and also de ne
D f i s jP i s g
f
Note then that D fij s P i s g Wenow alter assumption to the following
f
Assumption L et f beann ary typeconstructor Then the fol lowing conditions
hold
i s j s D i j
f
i
Direct embedding coercions are unique i e if t f t t and
i n
f t t
n
i
t f t t then
i n
f t t
n
i i
f t t f t t
n n
So wemaynow extend the coherence theorem theorem by altering the assump
tion list to our new relaxed set of assumptions It is not necessary to reprove the entire
theorem but merely the cases whichinvolved assumption
Theorem Extended coherence theorem Assume that al l coercions between
ground types are only built by one of the fol lowing mechanisms
coercions between base types
coercions induced by structural coercions
direct embeddings de nition in a type constructor
composition of coercions
identity function on ground types as coercions
If assumptions and are satis
ed then the set of ground types as objects and the coercions between them as arrows
form a category which is a preorder
Chapter Coherence Conclusion
Proof The entire pro of of theorem is valid except where assumption was
used
First we deal with the length of and b eing The only case which relied on
assumption was case
Case If and are direct emb eddings then by assumption
Now the induction case The only cases which relied on assumption were cases
and
Case In the pro of of theorem in this case we relied on assumption to
show that the two direct emb eddings h h and s s were at the same
a a
b b
p osition
h and t s by assumption the direct emb eddings must be Now since t
a
b
at the same p osition
Thus the rest of the pro of of this case holds
Case is a direct emb edding is a direct emb edding Since t s and t s
assumption implies that the direct emb eddings must be at the same p osition
i Therefore s s So by the inductive hyp othesis By assumption
Hence
Inserting the rest of the pro of of theorem completes the pro of
Th us wehave relaxed an imp ortant one of the conditions of the coherence theorem
and proved that the theorem still holds
Conclusion
In this chapter we stated all the mathematics needed to state Web er s coherence con
jecture and give his pro of
Wehave also stated extra mathematics to correct the statement of the conjecture and
then prove it hence promoting it to a theorem We have then relaxed one of the
conditions and shown that the theorem still holds
This theorem is the cornerstone for ensuring correct co ercions
By having a coherent typ e system then provided we are careful in how we imple
ment our typ e constructors the must satisfy the assumptions and howwe create our
co ercions comp ositions of the four basic typ es all our co ercions are then unique
Chapter
The automated co ercion
algorithm
Intro duction
In this chapter we shall intro duce some extra mathematics which will allowustostate
the automated co ercion algorithm Furthermore this sound foundation will allowusto
demonstrate that all co ercions generated by the algorithm are in fact homomorphisms
Indeed wemayeven guarantee that these are co ercions in the sense of de nition
All the work in this chapter is the author s own
Finitely generated algebras
In this section we shall de ne what it means to say that an algebra is nitely generated
and also what it means to say that a nitely generated algebra is decomp osable
De nition Let beanorder sorted S signature We say that S has a principal
sort if h Si de nes an algebra which has one sort which without loss of generality
we shal l always assume to be S which is the most interesting of the algebra
By most interesting wemeanthata more na v e algebraist would consider this sort
to de ne the entire algebra
For example if we de ne to b e the Group algebra with sorts
the group a boolean sort an integer number system
Chapter The automated co ercion algorithm Finitely generated algebras
then the most interesting sort is the one for the group whichwould b e carried by
a set of all the group elements
All Axiom algebras must have a principal sort referred to as in the source co de One
can think of S as the sort that the Axiom co de constructs In the terminology
of chapter it is the typ e constructed by the typ e constructor Axiom views base
typ es as typ e constructors with no arguments
De nition We say that a theory hh Si Si is nitely generated i S has a
principal sort S and the fol lowing set is nite
n q S
n q
This states that given a nitely generated algebra hA i whichis a mo del for S
then there are a nite numb er of op erators which can return an element of the carrier
of the principal sort
This means that only nitely many functions directly construct the carrier of the
principal sort
The next de nition allows us to decomp ose any element of such a algebra into at least
two pieces Moreover there exists such a decomp osition for each one of the nitely
many constructors Furthermore decomp osing such an element and recombining using
the corresp onding constructor is equivalent to the identity function
De nition A nitely generated theory hh Si Si is decomp osable i
n q S
i f ng
i i S q n
i
id S
S
For example in a an algebra of lists cons is a constructor The corresp onding functions
which decomp ose an elementarecar and cdr The following equation holds
cons car x cdr x x
Notice that this equation is only well formed when x is not the empty list For the
empty list which is constructed by the ary function nil there are no decomp osers
and the equation is as follows
Chapter The automated co ercion algorithm Constructibility
nil x
It is imp ortant to note that the part functions are usually partial functions For
i
example car is not de ned on the empty list
Notice that wehave not yet de ned a way of di erentiating b etween which constructors
construct which element We shall do this in the following section
Constructibility
We now come to the most imp ortant concept of this thesis The aim of this thesis is
to provide a metho d for creating co ercions constructively We now supply the means
do so
As usual weneedtomake some de nitions rst We will state what wemeanbya n
algorithmically re constructible algebra What we really mean is that an algebra is
algorithmically re constructible if some subset of the elements of the most interesting
sort carrier of that algebra can b e built up from a nite family of op erators
This nite family consists of constructors used to create increasingly large elements
of the principal sort
W e shall also have equations in the theory linking the constructors to part functions
whichwe shall use to split large elements into an n tuple of smaller elements The con
structor applied to its asso ciated part functions acting on an element will b e equivalent
to the identity function
We shall b e able to tell how an element is constructed by using a query function That
is to say we mustbeable to know which constructor s may construct any particular
element of the principal sort
We require these de nitions so that if an algebra is a mo del of some algorithmically
reconstructible theory and another algebra contains some of the constructors of the
rst algebra then some subset of the elements of the most interesting sort carrier
We shall show this to be a can be co erced from the rst typ e to the second
homomorphism where is a particular signature to which b oth algebras b elong
We demand our extensions to be protecting extensions so that p erforming op erations
on an algebra or lo oking at equations of elements of that algebra when viewed as a
mo del of the original theory or the extension of the theory yield the same result
De nition We cal l a theory hh Si Si constructible i is a protecting exten
Chapter The automated co ercion algorithm Constructibility
sion of a nitely generated theory We then cal l a constructing signature and any
model for the theory a constructible algebra
Similarly
De nition We cal l a theory hh Si Si reconstructible i is a protecting ex
tension of a decomposable theory We then cal l a reconstructing signature and any
model for the theory a reconstructible algebra
Moreover
De nition If hh Xi Xi is the decomposable theory extended by the recon
structible hh Si Si we de ne the constructors of hh Si Si to be
C
We de ne the constant constructors to be the constructors of arity denoted by
and the non constant constructors to betheconstructors of al l other arities We denote
C C
this set by n
C
We also de ne the part functions of hh Si Si associatedwith to bethe
i
Constant constructors have no associated part functions
Final ly we de ne the constructor equations to be the fol lowing set and demand it to be
a subset of S and hence X given by
C
f id j g
n S
We also need to ensure that certain other relationships between constructors which
hold in our source algebra hold in our target algebra
If an element of the source algebra may b e constructed in more than one way we require
that reconstructing that element in the target algebra using any of those metho ds yields
the same result
Notice that since our theories are protecting extensions we are in go o d shap e since
any equation that holds in the protecting extension which concerns only the sorts from
must hold in the original signature Thus these the original unextended signature
equations will hold in the target algebra
We call these equations that link di erent constructors together the secondary con
structor equations
Chapter The automated co ercion algorithm Constructibility
De nition We cal l a reconstructible theory hh Si Si algorithmically recon
structible i S contains a boolean sort B contains a set of symbols cal led the
query functions denoted and de ned by
Q C
f j g
S B
and S contains the fol lowing equations cal led the query equations
C
a a T
n n q S n q S
n q S
C
We also demand that for each in
n q S
a a a a F
n n q S n
n q S
Final ly we demand that al l secondary constructor equations are de ned in the theory
A secondary constructor equation is any equation which has a constructor as the nal
symbol on both right and left sides
Wesay nal symb ol to mean this is op erator applied last In our notation functions
written on the left a nal symbol is written to the left of all the other elements of a
formula
For example and e e e b oth have as a nal symbol
In the algorithm we shall demand that the decomp osable theory whichbothtyp es mo del
is the smallest theory which they b oth mo del This is to ensure that the algorithm not
only creates a homomorphism but that it is a co ercion de nition The pro of of
this is given in corollary
Example We shall take the varietyof Lists as our example and we shall assume
that List is a memb er of this variety Then the constructors for the mo del are nil and
cons where nil is a constant constructor whereas cons is a non constant constructor
cons asso ciated part functions are car and cdr Thus we have the constructor equa
tion
cons car x cdr x id x
The query functions are null and consp and our query equations are
Chapter The automated co ercion algorithm Constructibility
null nil T
and for any list l and any element of the underlying typ e i
consp cons i l T
Notice also that consp nil F and also for the same l and i null cons i l F
It is nowworth discussing some of the ner p oints of homomorphism
For to be a homomorphism we require that the constructor functions are preserved
by homomorphism We do not require that the part or query functions are homomor
phically preserved indeed wedonoteven require that they are in the signature of the
algebra of the target of
Thus we do not require that b oth the source and target of are mo dels of the same
algorithmically reconstructible theory but that the source is mo del of an algorithmically
reconstructible theory T and that all the constructors of T are functions of the target
of inherited from the same signature theory
This is an imp ortant point As an example we shall consider p olynomial rings A
p olynomial is a function suchas
x y xy x y
That is a sum of pro ducts of an element of the underlying ring which in the ab ove
example might b e the integers and variables raised to non negativepowers A mono
mial is a p olynomial which is a pro duct of a non zero element of the underlying ring
and variables raised to non negativepowers
A p olynomial ring is ordered by an extension of the order on the variables In particular
is greater than another m if the exp onent of in Z x y if x y then a monomial m
x is greater in m than in m Should the exp onent of x be equal in both then the
exp onentofy is compared in the same manner
The leading monomial of a p olynomial is the largest monomial of a p olynomial If
x y in the ab ove p olynomial then x y is the leading monomial Should y xthen
the leading monomial would b e xy
The reductum of a p olynomial is the p olynomial less its leading monomial
Chapter The automated co ercion algorithm The algorithm
Co ercions between two p olynomial rings need only use leadingMonomial as a part
function which is not preserved via homomorphism since it dep ends on the order
ing given to the variables instead of some unnatural xed for all p olynomial rings
with variable from a xed domain most imp ortant monomial function whichwould
cho ose the same monomial regardless of variable ordering
Also this allows us to form the natural monomorphism from Q x to Z x which in
Axiom is
Polynomial Fraction Integer to Fraction Polynomial Integer
which are in Axiom s view without clever hackery in the interpreter two unrelated
s This may be constructed without having to force to be a constructor of Ring
Fraction Polynomial Integer or indeed leadingMonomial etc to be available in
Fraction Polynomial Integer
The algorithm
We are now in a p osition to state the automated co ercion algorithm
The algorithm to create the co ercion will b e stated in English It is to o implementation
dep endent to state any ner
The actual algorithm to co erce will b e stated as Lisp pseudo co de In Lisp a b c
means apply the function a to the arguments b c cond is the like the switch
statement in C or Java
condsor switches are equivalent to if then else statements if a condition is true
then we evaluate and return the following statement and leave the cond blo ck else
go to the next condition and rep eat
For example the line
x
S
S
means
if x then
S
S
The nal statement t is the default statement since t is always true This line will
only b e reached when all the other conditions have not b een satis ed and shows that
wehave failed to build a total automated co ercion function
Chapter The automated co ercion algorithm The algorithm
This could happ en if one fails to list all the constructors and their queries for a typ e
The lines of the form
x
S
S
check for the constants of the typ e For example the constant p olynomial can
not be constructed using a non zero number of parts using any normal construction
metho dology thus co ercing from Q x to Z x we mightsay
if zero x Polynomial Fraction Integer
then zero Fraction Polynomial Integer
returning the appropriate in Z x
The lines of the form
x x x
q q
n n
n q S
n q S n q S
n q S
are the constructing lines For example in a p olynomial ring wemightwrite
if x is the sum of a monomial and a p olynomial
then coerce leadingMonomial x coerce reductum x
where the addition function is that taken from the target domain
In this case leadingMonomial x and reductum x are b oth p olynomials In general
the parts of the element need not b e of the same typ e as the original element
For example in List algebras co ercing from List A to List B where there exists
or we can build a co ercion from A to B then wehave back in Lisp terminology
consp x cons coerce car x coerce cdr x
car x is the rst element of the list and is of typ e A rather than List A the typ e
ybeconsed on to the frontof of x So coerce car x is of typ e B and thus ma
coerce cdr x whichisoftyp e List B
Algorithm The Automated Co ercion Algorithm Let hA i be a model
for the algorithmical ly reconstructible theory hh Si Si
Chapter The automated co ercion algorithm The algorithm
Let hB i be a model of hh Di Di where hh Si Si is a protecting extension of
hh Di Di and some or al l of the constructors of hh Si Si are in fact in hh Di Di
Also we demand that theredoes not exist an extension of hh Di Di which both hA i
and hB i model
Then the fol lowing is an algorithm to coerce from hA i to hB i
The for i are the potential ly automated coercions from A to B from the
i i i
abstract type of S
i
The entire morphism created thus is cal led and not only is it a homomorphism it is
a coercion de nition
x cond
x
S
S
Repeat for each in
S
x x x
q q
n n
n q S
n q S n q S
n q S
C
Repeat for each in
n q S
t error
The algorithm to create the coercion is
createCoerce hA i hB i
determine hh Di Di error if doesn t exist
determine hh Si Si error if doesn t exist
determine
for
determine
determine
C
determine
C
for
determine
determine
determine for all relevant i
i
construct and return as defined above
So the algorithm presented ab ove allows us to algorithmically reconstruct elements of
one typ e as elements of another
Chapter The automated co ercion algorithm Existence of the co ercion
Existence of the co ercion
Before we prove that the automated co ercion algorithm constructs a co ercion when one
exists wemust prove that it do es not create some unnatural function when no co ercion
exists
We shall use the terminology of algorithm in this section
In the simplest case some of the functions will not exist in the target typ e and the
algorithm will error at an early stage This will be b ecause there is no hh Di Di
which b oth hA i and hB i mo del
If a homomorphism exists but not a co ercion then this means that there exists an
extension of hh Di Di whichboth hA i and hB i mo del In this case the algorithm
will error at an early stage
A i are available in hB i but no homomorphism co ercion If all the constructors of h
exists then this could b e b ecause of the at least one of the following causes
One of the co ercions to be used do es not exist
At least one of typ es which is required for construction or recursively required for
construction and whichisnot the carrier of the principal sort may not be co ercible
to its counterpart in the target Provided the algorithm checks at construction time
that every co ercion used directly or indirectly by exists or is constructible then we
i
may rep ort an error at an early stage
Non homomorphic constructors
The fact that hh Si Si is a protecting extension of hh Di Di and that all sec
ondary constructor equations hold in hh Si Si guarantees that these equations hold
in hh Di Di and hence in hB i
This not only ensures that the order of the lines in is unimp ortant but also that if
a certain relationship holds in hA i it must hold in hB i
For example if someone de nes that all nite eld algebras are constructed by and
succ the successor function the automated co ercion algorithm will not attempt to
create the co ercion Z Z This is b ecause there exists a secondary constructor
equation in the theory of nite elds of size n
Chapter The automated co ercion algorithm Proving co erciveness
n
succ
So in this example hA i Z and hB i Z The equation succ holds in
the theory of nite elds of size Z is a obviously a mo del for this theory
This equation is not true in any theory whichtheab ove theory extends and of which
Z is a mo del Otherwise the equation would hold in Z and that patently is not true
Thus the automated co ercion algorithm will error at an early stage from not b eing able
to nd hh Di Di
Proving homomorphicity and co erciveness
We now make some notational de nitions so that we may prove the nal result of
That is we shall show that algorithm constructs a homomorphism the chapter
theorem which is a co ercion corollary
Recall notation Thus for any constructor symbol the asso ciated construc
n q S
the query function is and for i f ng tor function in hA i is
n q S
n q S
the part functions are
i
n q S
De nition For a term t in a term algebra T X if
t X then length t
s
t then length t
s
t t t then length t length t length t
n n
This de nition of length of an element will form the basis of our inductive pro of of the
automated co ercion algorithm However in general we shall not b e dealing with term
algebras but their homomorphic images
De nition For x an element of a algebra we de ne
length x min ft T X j t xg
where is the unique homomorphism given in the rst universality theorem
Chapter The automated co ercion algorithm Proving co erciveness
corresponds to Assumption If one of the part functions
i
S S
n q S
thus q S then we demand that
i
t length t t T X length
i
n q S
where is the unique homomorphism given in the rst universality theorem
A couple of technical de nitions to make the next assumption easier to understand
De nition Suppose hA i is a constructe d algebra constructed by the S sorted
signature If for any i f jS jg we have that S appears in the arity of any of
i
the constructors of we say that A is required for construction by A
i
So for example in List Integer we have that Integer is required for construction
since it is an argument of cons Notice that List Integer is not required for con
struction itself since it is the carrier of S
De nition Suppose hA i is a constructed algebra c onstructed by the S sorted
signature Let A berequiredforconstruction by A Now consider A as the carrier
i i
of the principal sort of hB i a constructed algebra
If B is required for construction by B A hence j then we say that B is
j i j
recursively required for construction by A
Now suppose that the carrier D of the principal sort of aconstructed algebra hD i is
recursively required for construction by A If D is required for construction by D
k
then we also say that D is recursively required for construction by A
k
So for example in List List Integer the only typ e which is required for construc
tion is List Integer since this is the only argumentof cons which is not the carrier
of the principal sort List List Integer
Thus the only typ es which are recursively required for construction are List Integer
and the typ es which are recursively required for construction by List Integer
Since Integer is the only typ e required for construction by List Integer it is the
only typ e recursively required for construction by List Integer
List List Integer Thus the typ es which are recursively required for construction by
are List Integer and Integer
The following assumption is required so that wemayprove that the automated co ercion
algorithm terminates
Chapter The automated co ercion algorithm Proving co erciveness
Assumption We demand that A is not recursively required for construction
Theorem Let hh Si Si be an algorithmical ly reconstructible theory which is a
C
protecting extension of the theory hh Xi Xi such that
Let hB i bea model for hh Xi Xi and hA i be a model for hh Si Si
is a homomorphism hA i hB i Then the function given in algorithm
Proof
Let b e the correct homomorphism which is attempting to emulate
Since contains nitely many cases and covers all cases of constructors for we
need only consider one line from Also by induction on com t we can assume that
i f jS jg
i i
Since covers all cases of constructor for we do not need to consider the error line
since it will never b e reached
For we may induct on length and we are in go o d shap e by assumption on
length and on the interaction b etween length and com
Here is one line from
if a a a
c p c p
q c n n
Now assuming a we knowthat
q
a a a
p p
c n
then since is a homomorphism
a a a a
c p p p c p
n n c n c
Now we knowthat by our induction argument
a a a
c p c p c p c p
c n n c n n
a
whence for this line and therefore every line and the entire function a
a So by induction and thus is a homomorphism
Now by adding one extra condition we may prove that the automated co ercion algo
rithm generates co ercions in the sense of de nition
Chapter The automated co ercion algorithm Conclusion
Corollary Let hh Si Si be an algorithmical ly reconstructible theory which is
C
a protecting extension of the theory hh Xi Xi such that
Let hB i be a model for hh Xi Xi and hA i be a model for hh Si Si such that
there does not exist any extension of hh Xi Xi which both hA i and hB i model
Then the function given in algorithm is a coercion hA i hB i
Proof No other theories extend hh Xi Xi whichboth hA i and hB i mo del so
the only thing which could stop b eing a co ercion would b e for there to exist a theory
which b oth hA i and hB i weretomodelwhichwas not extended by hh Xi Xi
So supp ose for a contradiction hA i and hB i were to mo del a theory which
hh Xi Xi do es not extend then wemaymanufacture a new theory containing all the
sorts op erator symb ols and equations of b oth theories
We may have some duplication of sorts and op erators so this manufacturing pro cess
would need to be p erformed intelligently Explicitly and hh Xi Xi may b oth be
extensions of some theory We might but not always only wish our manufactured
theory to contain the sorts op erator symb ols and equations of once not twice
This new theory would clearly extend hh Xi Xi and hA i and hB i would b oth
mo del this new theory Hence wehave a contradiction and such an can not exist
So hh Xi Xi m ust sp ecify the unique smallest variety to which both hA i and
hB i b elong We already know by theorem that is a homomorphism by
de nition it must b e a co ercion
This is why algorithm lo oks for hh Di Di since it must sp ecify the smallest
variety to whichboth hA i and hB i b elong
Conclusion
In conclusion we have shown that it is p ossible to create a homomorphism between
yp es twotyp es from an imp ortant subset of t
Moreover this homomorphism may b e constructed algorithmically and hence the gen
eral construction of homomorphisms is implementable on a computer
Furthermore if the typ e system adheres to all the assumptions made and the homo
morphism is from the theory which sp eci es the smallest variety to whichbothtyp es
b elong then it is the unique coercion between the twotyp es
The theory of rings inherits from the theory of monoids twice
Chapter
Implementation details
Intro duction
In this chapter we shall discuss the implementation details of the automated co ercion
algorithm Axiom version a was chosen as the computer algebra system for this
work The only other options would have been
Magma
writing an entirely new computer algebra system from scratch
extending another computer algebra system eg Reduce
These were the only options since a computer algebra system with a notion of typ es
and typ es of typ es falling either into a category theory like framework or a universal
algebra set up was required
Axiom and Magma are the only two systems which are currently based on these sound
principles and it was thought that either extending Reduce to be Axiom esque or
writing a new order sorted algebra system were beyond the scop e of the pro ject
Since the source co de for Axiom a was made available byNAG to the author this
was the soundest choice
Bo ot and Axiom
Versions to a of Axiom were based on Austin Kyoto Common Lisp AKCL
Previous versions of Scratchpad were built on top of other Lisps and since AKCL is
Chapter Implementation details The top level
now GNU Common Lisp GCL and covered by the GNU Public Licence GPL future
versions of Axiom including the current version are to be based on Co demist
Common Lisp CCL
This has had a useful side e ect CCL is a far smaller faster and more mo dern
implementation of Common Lisp This means that Axiom s fo otprintisnow far smaller
than it has b een in the past
Axiom has b een said to have been b orn big it initially only ran on large IBM
mainframes With the advent of CCL based Axiom it now runs on the humble PC
This is true whether the PC is running an op erating system likeLinux or whether it is
running Windows or NT
AKCL based The hardware This research was p erformed on Axiom a which is
used was usually an IBM RS running AIX and o ccasionally a Sparc Server
running Solaris
Axiom is built in the following manner Bo ot is a syntactically sugared interpreted or
compiled form of Lisp Bo ot retains all the functionality of AKCL with a lot more
thrown in A lot of these extra features are there for historical reasons for example
there are plenty of functions to allow co de written in VM Lisp to continue to run Some
are there to genuinely extend the usability of the language
The Axiom interpreter is written in Bo ot and hence any alteration to the Axiom world
must b e written in Bo ot
Compiled libraries of co de Packages Domains Categories are either written in Spad
or Aldor These are the strict languages which de ne the typ es algebras and theories
of Axiom They may b e compiled and loaded into the Axiom interpreter
The Axiom interpreter itself contains many features not available to the Package etc
writer In b oth Spad and Aldor strong typ es are the order of the day The interpreter
is meant to b e a bit more user friendly Hence typ e inference and on the y co ercions
are available to the user in the interpreter
So for this research the co de had to b e written in Bo ot
The top level
The basic design was as follows To p erform a co ercion from a Domain A to a Domain
B one needs to be able to ascertain which constants queries constructors and part
functions are available to b oth A and B These functions should all come from the same
Chapter Implementation details Lab elling op erators
algorithmically reconstructible theory Category
The automated co ercion function should be created and then compiled and stored
away ready for fast access in case of future use Such items are called cached lamb
das or clams for short and are already part of the Axiom system Thus this was
implemented
The automated co ercion function should b e integrated with the current co ercion mech
anism in Axiom This was done by inserting the relevant line at the correct p oint of
the Bo ot function coerceInteractive
Lab elling op erators
part We had to decide on a mechanism to mark which op erators were constructors
functions etc
One metho d would have been to create sub typ es of the Axiom Domain Mapping
which would have b een infeasible requiring much rewriting of the Spad and Aldor
compilers This is also true of the metho d given in a later section
A Spad Domain contains the following parts
the name of the typ e constructor
a variable name Category pair for each of the parameters of the typ e con
structor
a Category or Join intersection of Categories to whichthe Domain b elongs
a list of additional op erator symb ols functions for the typ e
a list of metho ds for some of or all the op erator symb ols of the typ e
Every op erator symbol de nition in a Domain or Category and hence available in a
Domain of that Category has a certain number of sp ecial comments called com
ments
These comments as opp osed to ordinary comments in most languages and Axiom s
other style of comments comments are parsed by the compiler to pro duce the
Occasionally a variable name Domain pair For example IntegerMod p PositiveInteger
This is not the same as all the sorts of the signature One may declare extra unused sorts Also
wemay declare a function where the source or the target maycontain a ground typ e For example
functions which return a b o olean liketyp e are usually declared to return Boolean rather than a pair
where the Category is the Category of b o olean liketyp es
Chapter Implementation details Getting information from domains
do cumentation for Axiom s sophisticated on line help system Hyp erDo c Hyp erDo c
was one of the rst hyp ertext systems
This parsing of the comments allowed us to enter sp ecial keywords for each sp ecial
function whichwere read and stored at compile time for each Domain We could then
read them at run time using the Axiom interpreter s built in comments reader
Getting information from domains
The comments were gathered for a Domain by recursion up the Category inher
itance lattice using the GETDATABASE function We asked the database for all the
do cumentation for each Category and then asking which Categories it extended
The do cumentation includes all the functions and their resp ective comments This
allowed the automated searc hing for keywords and hence the sp ecial functions of a
Domain
Note that we only needed the comments for functions op erator symb ols declared in a
Category Functions declared in Domains are of no interest to the automated co ercion
algorithm This is b ecause these do not corresp ond to any of the sp ecial functions
constructors part functions or queries of a signature or theory
The keywords were checked quite simply b eing memb ers of the following list
list constant constructor part query
Part number checking was only slightly more di cult with each number having to be
converted from a Lisp string to a Lisp integer A restriction was placed on queries
that they must be called the same as their asso ciated constant or constructor but
with a app ended This restriction was not necessary and could havebeenworked
around easily
The sp ecial function lists from b oth Domainswere compared and the required functions
were extracted from b oth
Checking information from domains
For each sp ecial function a check was p erformed to see whether that function was
really available in the Domain
Chapter Implementation details Flaws in the implementation
The envisaged problem was that Axiom s Categories can be conditional that is to
say some function de nitions only exist if certain relationships hold for the Category s
parameters
GETDATABASE ignores the parameters to a Category and returns all the functions which
may b e exp orted byaDomain of that Category Hence the extra checkwas necessary
to ensure that the automated co ercion function did not try to include these unavailable
functions
Thus the homomorphism created by the automated co ercion algorithm was always
equivalent to de nition of a conditional homomorphism given in section
Flaws in the implementation
The implementation was originally envisaged as a fully working piece of co de of pro
duction standard and hence shippable with a commercial release of Axiom For various
reasons outlined b elow only a working prototyp e was implemented
Delib erate restrictions placed by the author on the design were
Query function names were restricted to b e the name of their asso ciated constant
function with a app ended
The numb er of non ary constructors was limited to one
Recursion through coerceInteractive was achieved using na ve means
Neither the existence of hh Si Si nor hh Di Di were checked See algo
rithm for the meaning of these
Both the rst two items could have easily b een worked around but would have required
a more complex comment reader This was considered to b e a minor detail which
would have required to o much implementation time
The third item was more problematic Attempts were made to remedy this situation
e re However the depth of knowledge needed to implement this correctly would hav
quired to o much time to learn This was the main factor in the downgrading of the
implementation to mere prototyp e status
It was a certain amount of na vety on the author s part to assume that a pro duction
standard implementation of the automated co ercion Bo ot package was within the scop e
of this pro ject
Chapter Implementation details Flaws in the implementation
The metho d to work around the complexities of Axiom s evaluation lo op was to add a
Spad Package which exp orted a function whichthencalledthe Bo ot function The
co de for this Package called NCoerce follows
Coercion Package
abbrev package NCOERCE NCoerce
NCoerce Source Type Target Type Exports Implementation where
Exports with
nCoerce Source Target
Implementation add
nCoerce x Source Target
nCreateCoerce x Source Target Lisp
To call a function from a Package is relatively easy in Bo ot and uses the same syntax
as calling a function from a Domain
The function nCoerce ab ove then uses the Axiom Lisp syntax for calling a function
de ned in Bo ot This breaking out of the Bo ot co de to call a Spad function which
then in turn calls Bo ot co de is most undesirable
However should an exp ert in the Axiom interpreter be given the Bo ot co de for the
automated co ercion algorithm then it is merely a matter of changing one line This
one line currently calls a function which creates lines of the form
nCoerce x NCoerce Source Target
This somehow needs to b e changed to something which p erforms
coerce x Target
is the Axiom syntax meaning the previous function should return the following
typ e In this case coerce x to the typ e Target
The automated co ercion Bo ot package was only implemented to deal with Spad co de
and not Aldor co de This is b ecause Bo ot handles co de written in the two di erent
Chapter Implementation details Flaws in the implementation
languages in a di erent way There are di erent interfaces to the two internal repre
sentations of a Domain or a Category or Package
This again could have b een remedied but would have required to o much time com
pared with the prosp ective gain in functionality
Occasionally Axiom s mo demap selection can get confused Mo demap selection is
Axiom s metho d for selecting metho ds from an overloaded op erator name This causes
the automated co ercion package to fall over at a premature stage
The fourth item is a bit more di cult to solve The existence of hh Si Si is implicit
if a Domain has sp ecial functions It would also b e p ossible to check if b oth source and
target are b oth memb ers of a common Category using GETDATABASE It would even
to check that all the constructors of the source are from this Category be p ossible
Finally itwould b e p ossible to ensure that this is the smallest such Category
However it would b e most problematical to determine whether hh Si Si is a protect
ing extension of hh Di Di or not
This is a more general problem than one which just applies to the automated co ercion
algorithm This is due to the fact that Axiom b elieves everything that you say One
may pro duce a Category and claim that it extends another Yet if Axiom s Categories
were really varieties sp eci ed by theories there would need to b e some wayofchecking
whether the equations in the original Category still held in the new one Similarly
Domains need not mo del the theory sp ecifying the variety Category to which they
have b een declared to b elong
The current situation in automated theorem proving means that Axiom will b elieveany
false assertions like those mentioned in the previous paragraph This is not a design
aw of either Axiom or the automated co ercion algorithm neither Axiom s designers
nor I had any choice automated theorem proving has not yet advanced su ciently
for us to take advantage of it Similarly and moreover Axiom will b elieve that any
function whichonemay implement called coerce is a valid co ercion
algorithm in the This means that in this implementation of the automated co ercion
current release of Axiom it would b e p ossible to create new co ercions b etween p o
tentially unrelated typ es or non homomorphically between similar typ es Using the
example from section we could create a co ercion Z Z
The present state of automated theorem proving technology with sp ecial reference
to Axiom is detailed in MS This pap er details recent research and near future
directions for the sub ject
Chapter Implementation details Conclusion
Other aws which shall b e dealt with in chapter include
Some Axiom Categories do not agree with our notion of varieties sp eci ed by
signatures or theories Sections
Axiom Domains have to o much in common with their Categories There is a
lack of distinction between op erator symbols and op erators Similarly between
sorts and carriers Sections
Conclusion
We conclude that the automated co ercion algorithm is probably implementable in the
Axiom interpreter It is certainly p ossible to create a version whichworks However it
would take someone with deep er knowledge of the co de to force a complete integration
of the current implementation with the Axiom interpreter
What wedohave is a pro of of concept If Category authors correctly lab el their op er
ators then the co ercion function do es the right thing It is only the p erfect integration
into the Axiom s kernel that has not b een demonstrated
Chapter
Making Axiom algebraically
correct
Intro duction
In this chapter we shall discuss various details which need to be changed to make
a computer algebra system in this case Axiom more algebraically correct By
this we mean so that Axiom s Categories behave more like our concept of varieties
sp eci ed by order sorted theories and that the Domains act more like our notion of
order sorted algebras more correctly
Explicitly de ned theories
In Axiom the Categories were originally intended to b e akin to varieties sp eci ed by
signatures The Domainswere then meanttobelike algebras of those signatures This
seems in retrosp ect to have beenanover ambitious aim
The Categories do b ehave and lo ok to the casual observer liketheywere sp eci ed by
multi sorted algebraic typ e theories but there are some clear di erences
Op erator symbols and names
In Axiom if a category de nes an op erator symbol then for all domains in that
n q s
of will b e category that op erator name
n q s n q s
n q s
Chapter Making Axiom algebraically correct Op erator symb ols and names
It may app ear to be not that much of a disadvantage to have such a restriction on
op erator names but an imp ortant case in mind is that of the monoid A ring has two
binary op erators over which it forms a monoid The only way in Axiom that we can
ensure this is to havetwo di erent monoidal categories A distinct disadvantage
If we allowed op erator symb ols to di er from op erator names then another Category
could extend Monoid in two di erentways
Syntactically this would be most di cult We would need a way of declaring the
two op erator symbols in Ring traditionally and as b eing from di erent op erator
symb ols in Group which extends Monoid and Monoid
For example we could write something like the following
Ring Category with M Monoid G Group
Operators from Monoid
M
M
Operators from Group
G
G
inv inv G
This would identify and with the binary op erator and identity element resp ectively
from Monoid Whereas and inv would be identi ed with the binary op erator
identity element and inverse function from Group
Notice that both the op erator symbol eg and both the source arity eg
and target eg sort are needed to uniquely identify the op erator These corresp ond
to q and s resp ectively We do not need to know n since this is deducible from q
A function in Axiom in either a Domain or Category is currently represented by name
target source comments Function declarations in Categories are normally of
the form
comments s q
s and q are reversed in Axiom s internal representation of a function
Chapter Making Axiom algebraically correct Op erator symb ols and names
and once exp orted by a ground typ e ie genuine Axiom Domain with all parameters
to the Domain xed it b ecomes
s q
A A comments
n q s
where
n q s
For the co ercion algorithm we inserted sp ecial words in the comments eld to try and
mimic an op erator symb ol
Another approachmighthave b een to declare other typ es eg Constructor a b which
are sub typ es of Mapping a b
A far b etter approach is to extend the information contained within the function con
struct For initial de nitions in Categories that is new op erators whichhave come
from this Category not ones which this Category may extend then the current
declaration metho d su ces
However as in the ab ove Ring example for Categories whic h extend others then a
b etter metho d maybe
s q comments
where is the new op erator symb ol corresp onding to amember of where this
n q s
signature extends
We haveusedvariable names to avoid confusion One may wish to de ne a Category
called DoubleMonoid where G is a Monoid instead of a Group Obviously no inv
op erator would b e available to sucha Category
The same terminology could also be used to de ne sp eci c algebras Domains For
Zg is an additive example for all n in N nZ whose elements is the set fnz j z
group whereas S n the symmetric group on n elements is usually considered to b e
amultiplicative group
So for nZ wemight write
IntegersTimes n PositiveInteger Group
inv
Chapter Making Axiom algebraically correct Op erator symb ols and names
However for S n we might write
SymmetricGroup n PositiveInteger Group
inv inv
In this example we can see the following prop osed metho dology for declaring op erator
names which corresp ond to op erator symbols
s q comments
n q s
The signature do es not need to be mentioned in each function de nition since the
algebra is only declared to b e a mo del of one signature
This do es create more unnecessary confusion for Domain Category writers but cor
rectness should overrule ease of use It is also conceivable that this will have a negative
wever this is true of all e ect on the amount of time it takes to compile a Domain Ho
typ e checking compilers
More imp ortantly it should not have any e ect on run time sp eed of execution This
should de nitely b e the case if the internal representation is moved to something ap
proximating Axiom s current order
s q comments
n q s
This puts the target and source closer and less op erations away from b eing discovered
Clearly in this prop osed metho dology the carriers of the source and target are not
mentioned explicitly in the function representation
To sp eed up function typ e lo ok ups mo demap selection each Domain could provide a
hash of sorts and carriers Indeed then the internal representation could b e
s q
A A comments
n q s
and then mo demap selection would b e as fast as in the currentversion of Axiom but
the hash could be used going in the other direction to determine s and q for the
automated co ercion algorithm
Chapter Making Axiom algebraically correct Moving certain op erators
Moving certain op erators
Another factor which stops Axiom acting totally homomorphically is that certain op er
ators app ear in Categories to o far up the inheritance lattice This is b est illustrated
by an example
In Axiom one often would like to b e able to convert Liststo Sets The Domain whose
items are nite sets or actually sometimes classes over some particular typ e is called
Set in Axiom
Sets and Lists in Axiom are both certainly nite collections which can be built by
adding in another elementatatime In List this maybeachieved using either cons or
append though obviously cons is far more e cient whereas Setscan b e built using
the sometimes non e ective command insert
Axiom knows that Lists are sorted whereas Sets are not The problem which is
vious is that adding in a new element to a List will always increase immediately ob
the length of the List This is not true of Axiom s nite Sets They b oth however get
the same element counting op erator from the same Category This function comes
from the the Category Aggregate whichis the most general typ e of collection in
Axiom
List S for some xed S is a mo del of ListAggregate and Set S is a mo del of
FiniteSetAggregate Both of these theories are extensions of the theory Aggregate
Therefore if we had a homomorphism from Liststo Sets then the following equation
should hold yet it clearly do es not
since the left hand side is
and the right hand side is
f g
and unless the carrier sort of collection length is not NonNegativeInteger but a dif
ferent typ e one in which all elements are equal we will not be able to create any
homomorphisms from List to Set
This op eration inserts a p otentially new elementinto a set insert x s s fxg s x
Chapter Making Axiom algebraically correct Retyping certain sorts
In section we demanded that if the automated co ercion algorithm was to b e appli
cable in this case from List S to Set S then ListAggregate would need to be a
protecting extension of Aggregate
It is clear that in ListAggregate the following equation holds
cons a b b
If this were a protecting extension then this equation would hold in Aggregate How
ever as wehave already observed
insert a b b
when a b
Thus ListAggregate is not a protecting extension of Aggregate and the automated
co ercion algorithm can not b e applied
There are three obvious solutions
Disallow co ercions from List to Set Although we would still allow a non
homomorphic convert op erator
Move the element counting op eration further down the Category inheritance lat
tice until it do es not app ear in any Categories to which both List and Set
b elong Or if they do b oth b elong to this Category ensure that none of the part
functions constants queries or constructors are from this Category or anyofits
ancestors
Move the adding new element op eration further down the lattice in a similar
manner Thus cons and insert would not know anything ab out each other In
this case the automated co ercion algorithm would still not b e applicable
This is merely one example of manysuch op erators which could require moving
Retyping certain sorts
As in section this is b est illustrated by an example
In Axiom the Category Ring exp orts a function
Or an extension of Aggregate which b oth ListAggregate and FiniteSetAggregate extends
Chapter Making Axiom algebraically correct Sorts and their order
characteristic NonNegativeInteger
rememb ering that Axiom do es not di erentiate b etween sorts and carriers So imagine
the natural ring epimorphism Z Z Then the following equation should hold
characteristic characteristic
yet this is clearly is not the case since must send the Void sort to Void and thus the
left hand side must equal the characteristic of Z which is Whereas for the right
hand side wehave This is the natural map from N f g to itself This is of
course the identity map and hence the right hand side has the value
There is clearly something very wrong here There are two solutions the rst of which
is highly unsatisfactory
Stop characteristic b eing a function One could p ersuade it to b e an attribute
of the Ring
Alter the carrier of the return typ e of characteristic from NonNegativeInteger
to b eing a typ e with the same elements but a di erent idea of equality
This is one of many such cases in Axiom
Sorts and their order
We have b een discussing Axiom s Category inheritance system as if it were a true
attempt at mo delling order sorted algebra There are however two areas which are
distinctly missing from the Axiom mo del these are the sorts and their order
There is a distinct confusion in Axiom b etween sorts and carriers The author b elieves
that all Category de nitions the signatures or theories whichspecifyvarieties should
not use genuine typ es at any p oint in the signatures or any other p oint These should
only o ccur in the Domainsofthat Category
For example many Categories assume that the only b o olean like typ e is Boolean
Similarly the typ es NonNegativeInteger PositiveInteger and Integer are often
assumed and are not parameters of a typ e
For the typ es Boolean NonNegativeInteger and PositiveInteger this is not nor
mally a complete disadvantage There are not likely to b e other algebras in our system
which b ehave similarly enough to replace these typ es
Chapter Making Axiom algebraically correct Sorts and their order
However there are co ercions between NonNegativeInteger PositiveInteger and
Integer in the obvious directions This then imp oses some order on the sorts of the
typ e which is not explicitly mentioned
Some of the sorts of an algebra are de ned is always the principal sort and any
parameters of the Category are there to o Others however are merely mentioned in
function prototyp es signatures All should b e listed in a sort list and or sort lattice
see b elow
Neglected sorts normally include those which are carried by the four typ es mentioned
ab ove However other typ es are often neglected to o These include List which is often
present so that elements of the typ e may b e constructed The underlying representa
tion of most typ es in a Lisp based system are often lists More seriously particular
p olynomial representations are sometimes present
There is no real mechanism built into Axiom to order the sorts of an signature Category
The order is implicit in Categorical inheritances such as CoercibleTo and
RetractibleTo which give information on how the principal sort relates to some other
sorts
I b elieve that a more sensible waywould b e to declare a lattice like arrangementwith
the declaration of a Category Indeed this would then make the sort list clear to o
since this is never explicitly de ned either
For example we could add the syntax SortOrder to b e used as follows
ACategory a A b B Category
with SortOrder
a
b
exportedFunction List A
or b etter SortOrder could b e a Category which could b e de ned as assuming this
is compilable
According to Bro One problem is that it may not b e p ossible for the compiler to gure out
exactly what this construct will exp ort but it might still b e implementable
Chapter Making Axiom algebraically correct Altering Axiom s databases
SortOrder ls List Pair Type Category
for pr in ls repeat
coerce first pr second pr
This for lo op would not build the entire sort lattice for any particular category This
would b e b etter done by the compiler at compile time on a p er Category or p er Type
basis
Thus any Category which do es not extend SortOrder would be a non order sorted
signature
Altering Axiom s databases
Axiom has many built in databases for lo oking up comments functions attributes etc
from each Category or Domain These are usually text les with character number
keys for faster cross referencing
Database and hash table technology has certainly improved in vast amounts over re
cent years and is in fact quite an evolving area of computing Magma utilises many
di erent hand written in C databases for the large number of pre computed tables
of facts needed by the mo dern discrete mathematician
Each Magma database utilises the fastest lo ok up metho d available to the authors for
that particular task While Axiom s metho d is not excessively slow it is also not
particularly fast
More imp ortantly from our p oint of view Axiom could have some extra databases to
aid the automated co ercion algorithm Sp eci cally there could b e a database contain
ing for each Category a list of lists of sp ecial functions exp orted by that algebra
but not its ancestors This could just be also be a compiled fact ab out each algebra
but the lo ok up overhead would b e greater
The restriction on only having functions from that Category and not its ancestors
sp ecial functions allows for the dynamic nature of Axiom s Category system Without
this restriction altering a Category further up the lattice could cause the database
information to b ecome outdated for all of its descendants
Chapter Making Axiom algebraically correct Conclusion
Conclusion
Axiom behaves very similarly to a language based on order sorted theories and the
algebras that mo del them However there are some key areas in which Axiom di ers
to the mathematical notions
We summarise these areas in the following table
Mathematical Notion Axiom
Op erator names need not be the Op erator names are always the
same as op erator symbols same as op erator symb ols
Co ercions are homomorphisms and Co ercions need not act like ho
hence act homomorphically on all momorphisms at all If it were
op erators p ossible to implement a univer
sal equation checker then Axiom
would b e alright
A signature dep ends on its sort Category de nitions do not explic
list and hence a sort list is part itly list their sorts
of its de nition
No algebras are mentioned in a sig Category de nitions do dep end on
nature de nition only sorts sp eci c Domains
The order on the sorts is part of Category de nitions do not explic
the de nition of a signature itly order their sorts
Wehave presented metho ds for addressing all of these di erences
Chapter
Conclusions
Intro duction
In this chapter we shall summarise the work wehave done and what problems wehave
managed to overcome We shall also saywhatfuturework may b e p erformed to extend
the ideas presented here
Summary of work done
In this section we shall summarise the work done in this thesis and what problems
these ideas overcome We shall split this down into the following sections
Category theory and order sorted algebra bases for computer algebra systems
Section
Representation and syntax of order sorted algebra Section
Coherence of a typ e system Section
The automated co ercion algorithm Section
Category theory and order sorted algebras as the bases for
sound strongly typ ed computer algebra systems
W ehaveshown why b oth category theory and order sorted algebra b oth provide solid
mo dels for the typ es systems found in computer algebra
Chapter Conclusions Summary of work done
In sections and wehave demonstrated the correlation b etween a computer
algebra typ e system and category theory See also app endix B
In sections and we have shown how a computer algebra typ e system corre
lates with order sorted algebra In section we extended the notion of order sorted
signatures to cover conditional signatures which o ccur in Axiom
We have mentioned some of the corresp ondence between category theory and order
sorted algebra section We have usually used order sorted algebra as our mo del
This is b ecause order sorted algebra more readily corresp onds to the algebraic inheri
tance mechanism This is mainly due to the order on the sorts whichisnot available
higher order in category theory Also category theory has more di culty expressing
p olymorphism
Representation and syntax issues of an order sorted algebra
based typ e system
In chapter we demonstrated various metho dologies for extending Axiom s current
typ e system so that it may more closely mo del order sorted signatures and algebra
In section we showed how Axiom s syntax may be extended so that a signature
may extend another more than once We also showed that this syntax could b e used
to uniquely identify op erator names with op erator symbols This then allows op erator
symb ols to di er from op erator names
In section we discussed how b oth the sort list and the order on the sorts could b e
intro duced to Axiom s signatures
Sections and commented on how the Axiom signature tree may b e altered to
allow co ercions to act more homomorphically
Finally we discussed how Axiom signatures could b e compiled to contain extra informa
tion whichwould enhance the sp eed of the automated co ercion algorithm section
On coherence
In chapter we rst stated all the mathematics used by Web er to state and prove
Web er s coherence conjecture sections and
In section we then showed that this pro of is not correct Weshowed that altering
a de nition and adding a couple of new assumptions that let us prove the co ercion
Thus this solves an interesting class of multiple inheritance problem
Chapter Conclusions Future work and extensions
theorem
In section we then showed that it was p ossible to relax one of the assumptions
which Web er made and still have a coherent typ e system The extended coherence
theorem
The automated co ercion algorithm
In sections and we provided enough mathematics to show that the auto
mated co ercion algorithm section is an algorithm which returns a function
which is not only a homomorphism but a co ercion whichwe de ned in de nition
This homomorphism is unique in a coherent typ e system which we can guarantee
providing we can satisfy all the assumptions of the extended co ercion theorem
four basic typ es of co ercion given The homomorphism created may be built from the
in the statement of that theorem
In chapter we showed that a demonstration implementation of this algorithm is
p ossible in Axiom
Future work and extensions
Wehave presented in this thesis the basis for a mathematically sound computer algebra
system We have also shown that it would be p ossible to implement the automated
co ercion algorithm in such a system
At present there is no such system though Axiom and Magma come close
We have shown how Axiom s syntax could be extended to allow it to mo del an order
sorted algebra system the author would b e most interested in implementing either a
future release of Axiom or mayb e a freeware Axiom lik e system
To implement a totally new Axiom like system it would be nice to take advantage
of a language which already supp orts ob jects Thus such a system could be written
in C Str youto o Kin an implementation of EuLisp Level Pad or
Java Gra
However note that Java do es not allowmultiple inheritance amongst classes It do es
allow multiple inheritance from interfaces though Interfaces are always e ectively
abstract classes
Thus a Java implementation of an Axiom like computer algebra system would haveall
Chapter Conclusions Future work and extensions
Domains as classes and all Categories as interfaces
Axiom s categories are not always abstract that is they maycontain implementations
of functions This is a go o d thing as it enforces certain truths ab out a particular
function
Forcing any imp ortant facts by making a class which implements a particular interface
would then ruin anychance of reattaining multiple inheritance This e ectively means
that Javawould not make a sensible implementation language
A more serious omission of Java is that functions are not rst class ob jects This means
that the automated co ercion algorithm would b e unimplementable in an algebra system
with Java classes for ob jects
C is a tempting choice but the time for one p erson to implementtheentire system
in suchalowlevel language might make it infeasible The source co de for the Axiom
interpreter alone takes up over lines of Bo ot and Lisp co de
Youto o is an implementation of EuLisp Level whichdoes allow some forms of mul
a with tiple inheritance So while the temptation of using a fashionable language Jav
p otentially many more users is there to implement the new algebra system correctly
we see that Lisp is still the sensible choice
Access to OBJ and the Magma source co de would also prove interesting The author
would liketo see if the automated co ercion algorithm is easily implementable for any
system other than Axiom whichprovided all the original inspiration
Sticking with the current release of Axiom it would be nice to sp end a few hours
with one of the few exp erts who truly understand the Axiom interpreter This would
undoubtedly lead to a successful integration of the automated co ercion algorithm with
Axiom
However this would b e only available in the interpreter An implementation inside the
Aldor compiler would b e far more di cult and hence makeavery interesting research
pro ject
Future research by Martin MS and research students of Davenp ort should ensure
that Axiom s theory lattice is guaranteed to b e mathematically correct Eventually we
mayeven b e able to prove that one theory is a protecting extension of another though
this may b e some years from now
Until such research is completed the accuracy of the automated co ercion algorithm
dep ends on the abilityof Category and Domain authors to adhere to to the assumptions
The not equals function is de ned to b e not equals
Chapter Conclusions Future work and extensions
made in this thesis
App endix A
Extra category theory
A Intro duction
This app endix merely lists some extra category theory needed by app endix B
A Extra de nitions
De nition A The constant functor c sends al l objects of J to c and al l arrows
to
c
J
De nition A The diagonal functor C C sends each object c c the
constant functor al l arrows f c c of C to the natural transformation f c
c A limit for F J C is a universal arrow hr ui from to F
De nition A Acategory is complete if given any smal l category J and any func
tor F J C then F has a limit
App endix B
Set theory
B Intro duction
The work in this section and sections B B and B is not strictly necessary for the
rest of this thesis however the author b elieves that it is necessary if one wishes to have
a strict foundation to one s computer algebra system
I also b elieve this work to b e necessary if one wishes to form categories of categories in
Axiom See section B
B Basics
We need not go into a precise axiomatisation of set theory or indeed class theory in
this thesis but merely p oint the reader towards some suitable references Dev for
Zermelo Fraenkel set theory Ber for von Neumann Bernays and BHFL for a
deep exp osition on many formulations of set theory
Mac Lane ML avoids this by letting categories only form over small and large
sets also known as sets and classes in von Neumann Bernays set theory Brie y
to formulate set theory in this fashion one tak es na ve set theory and app ends the
existence of a xed set U called the Universe The universe has the following prop erties
x u U x U
u v U fu v g hu v i u v U
x U P x y U
y x
App endix B Set theory B Basics
the set of all nite ordinals is in U
if f a b is a surjective function with a U and b U thenb U
One then de nes a set to b e a smal l set i it is a member of U
Thus he refers to the category with all categories as ob jects and all functors as
arrows as the meta category of al l categories Meta categories are in his b o ok are
what would b e called categories if only their ob jects and arrows formed small or large
sets
Mac Lane s book b eing the de nitive volume on category theory also refers to other
formulations of set theory andeven mentions in passing Lawvere s pap er Law In
that pap er Lawvere formulates eight axioms for categories and proves that if one has
a complete category de nition A satisfying those axioms then it is equivalentto
the category of sets and mappings and moreover is unique
Lawvere continues further down this abstract path in his pap er Law which
attempts to axiomatise all of mathematics without recourse to any form of set theory
Herrlich and Strecker HS use von Neumann Bernays to overcome the di culties
of having categories and meta categories They consider as we have that a category
may b e de ned over any collection of ob jects and arrows
von Neumann Bernays set theory intro duces the notion of class in an attempt to over
come the problems asso ciated with trying to form the set of all sets Any collection of
sets is a class but if A is a proper class that is a class which is not representable as a
set then there exists no class which has A as a memb er
It is for this reason that the authors of HS intro duce the notion of a conglomerate
They lo osely de ne conglomerates to b e collections of classes of conglomerates and
say that they require that
Every class is a conglomerate
Conglomerates are closed under formation of ordered pairs unions intersections
complements disjoint unions and Cartesian pro ducts
However they also state that there is no conglomerate of all conglomerates and thus
they intro duce the term cartel for collections of conglomerates
It should b e noted that Lawvere s pap er is extremely abstract in content and that it do es not even
formulate a traditional form of the relationship Indeed A B is de ned to mean that there exists
a monomorphism A with co domain B and x A is de ned via B A x B x A x x
I b elievethattheymay only mean collections of classes or conglomerates representable by classes
and will take this meaning from here on
App endix B Set theory B Basics
Indeed we see that if we de ne the author s concept of inductive classes as follows
we will be in good shape
De nition B Inductive classes An entity X is cal led a class i X is
a set
An entity X is cal ledaclass where is any ordinal number i an ordinal
number al l members of X are class s
ordinal numbers X is a class X is a class
There is no reason for the following not to hold but the author can not prove it from
the de nitions alone Thus we state it here as an axiom
Axiom B A nite col lection of class s is a class
So if wenow de ne Cat as follows we will always b e able to nd a category in which
we can work rather than a meta category Unless of course one still wishes to form
a category of al l categories or a category with itself as an ob ject We are just saying
that we can always nd a category large enough in whichwe can work
Many theoretical computer scientists do not b elieve that there really exists a problem
at all Davenp ort states in Dav
many mathematical sets are not represented in computing languages
quite as we would like For example the set Z and m ore seriously no
computer can really represent R
yet we often pretend that they can For example in most computer algebra systems
allow the existence calculation and manipulation of bignums integers larger than
the largest representable within one machine word In a lazily evaluated system one
might even allow the creation of integers larger than the number of molecules in the
universe let alone the usable workspace of any current computer Indeed this is already
commonplace
So just b ecause one s computer cannot representevery elementof Z do es not mean we
cannot have atyp e of al l integers
Davenp ort also states in Dav
One might think of the universe U as b eing the set of all p ossible data
ob jects or bit patterns
App endix B Set theory B Inductive class approximation
Hence since everything with which we work is nite we would not be required to
do any more work at this p oint The author disagrees simply b ecause one s machine
contains only nitely may states do es not mean that we may assume that all of our
collections are small sets
B Inductive class approximation
In Axiom
S Set Any f g S insert S S
de nes S to be a nite collection with itself as its sole element however it most
umber such certainly is not a small set In fact there do es not exist an ordinal n
that S is a class
So what use is the idea of inductive classes in this situation The metho d of inductive
class approximation allows us to identify the S s on each side of a de ning equation
with di erent approximations to the value of S
To demonstrate the metho d of inductive class approximation let us x C to b e a class
Now alter C by letting
C C fC g
The op eration on the right hand side is the traditional function and should be
familiar to set theorists from one of the rst three axioms of general set theory Ber
It is also used to de ne successors in ordinal theory
As b efore there do es not exist an ordinal suchthat C is now a class What wemay
do is identify the C sontheright hand side with a copy of the original C andthus the
C on left is now a class For those myself included who do not like the idea of
the same symbol having di erentvalues in the same equation then here is a di erent
version
C C fC g
We now have an approximation to C Given any ordinal suchthatC is
an approximation to C then we de ne C the th approximation via
C C fC g
An ordinal is an ordinal numberifitisaset Men
App endix B Set theory B Inductive classes and category theory
Thus we can always nd an approximation to C containing a large enough but
di erent approximation to C
In traditional notation
Od C C C
If we could continue inductively approximating ad in nitum then we would reach a
limit ordinal Lim c However C may still not b e equal to C
Lim c
At rst glance this app ears to be neither a usable nor implementable idea however
for dealing with categories on a computer it is in fact a useful concept as we shall see
w b elo
B Inductive classes and category theory
The following mathematics presented here for the rst time de nes a way for us to
always have a category of all the categories in whichwe currently wish to work It
allows for Axiom s Category to b e a category and indeed by lazy representation to
b e a member of Category
De nition B For any ordinal number Cat is the category which has as ob
jects any category C for which both Ob j C and Arr C are representable as class s
The arrows of Cat are the functors between these categories
Prop osition B Cat is an object of Cat
Proof ordinal numb ers X is a class X is a class
Hence trivial
So for example is an ob ject of Cat and Set is an ob ject of Cat
B Inductive classes categories and Axiom
tity functor for the category of all Categories in Supp ose we wish to de ne the iden
Axiom We shall call this functor CopyCat
If we assume that Axiom s Category is normally a Cat andwe wished for a category
whichwas parameterised bya Category then the mathematical way of representing
CopyCat C Category Category C
App endix B Set theory B Inductive classes categories and Axiom
is that CopyCat takes C a Cat but the CopyCat category is a Cat
Thus wenowhave Category as b oth a Cat and a Cat dep ending on whether we
have any instantiations of CopyCat or not We may carry on inductively changing
the approximation of Category to the category of all Categories should we have
CopyCat CopyCat CopyCat etc ad in nitum and beyond but we shouldn t
need that on a computer
Obviously this is only a trivial example This mathematics is a neat trick and would
allow category theorists to p erform their research in Axiom
However Join is a functor from
Y
Category Category
n N
in Axiom so CopyCat de ned ab ove is equivalent to Join acting on one argument
Axiom do es not consider Join to b e a functor since it is only de ned in the compiler s
and acts solely as syntax in the construction of new Categories
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