Order Sorted Computer Algebra and Co ercions submitted by Nicolas James Doye for the degree of PhD of the University of Bath COPYRIGHT Attention is drawn to the fact that copyright of this thesis rests with its author This copy of the thesis has b een supplied on the condition that anyone who consults it is understo o d to recognise that its copyright rests with its author and that no quotation from the thesis and no information derived from it may b e published without the prior written consent of the author This thesis maybemadeavailable for consultation within the University Library and may b e photo copied or lent to other libraries for the purp oses of consultation Signature of Author Nicolas James Doye Abstract Computer algebra systems are large collections of routines for solving mathematical problems algorithmically eciently 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 ordersorted 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 dene 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 Finallywe 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 Maple Reduce Axiom Economyofeort Interest Contents Functoriality Newsp eak Magma 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 Universal Algebra Term Algebras Ordersorted 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 ers work I denitions Web ers 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 dened theories Op erator symb ols and names Moving certain op erators Retyping certain sorts Sorts and their order Altering Axioms 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 denitions 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.