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Etude du typage dans le sytème de calcul scientifique Aldor (Study of types in the Aldor scientific computation system) — by Bill Page — last modified Nov 18, 2005 07:29 PM

Apparu vers le milieu des années 70, le langage de calcul formel Axiom [13], initialement Scratchpad, faisait montre de caractéristiques peu communes dans l'univers des langages de calcul formel, comparativement aux langages contemporains ou ultérieurs tels Reduce, Macsyma, Mathematica, Maple, Magma, Mupad. Aldor est le successeur direct du langage Axiom. Comme tous les langages de calcul formel, il autorise la manipulation des objets de base du calcul mathématique : entiers, flottants, booléens, chaînes de caractères, etc, en programmation impérative ou fonctionnelle. ( Appearing towards the middle of the seventies, the Axiom language of formal calculation [13], initially Scratchpad, displayed not very common characteristics in the world of languages of formal calculation, compared with the contemporary or subsequent languages such as Reduce, Macsyma, Mathematica, Maple, Magma, Mupad. Aldor is the direct successor of language Axiom. As all the languages of formal calculation, it allows the handling of basic objects of mathematical counting: integer, floating point, boolean, character strings, etc, in imperative or functional programming. )

Applying AXIOM to Partial Differential Equations — by Bill Page — last modified Nov 18, 2005 07:29 PM

We present an AXIOM environment called JET for geometric computations with partial differential equations within the framework of the jet bundle formalism. This comprises especially the completion of a given differential equation to an involutive one according to the Cartan-Kuranishi Theorem and the setting up of the determining system for the generators of classical and non-classical Lie symmetries. Details of the implementation are described and examples of applications are given. An appendix contains tables of all exported functions.

On the way to certify Computer Algebra Systems — by Bill Page — last modified Sep 19, 2005 02:50 PM

ABSTRACT (max 200 words): "The Foc project aims at supporting, within a coherent software system, the entire process of mathematical computation, starting with proved theories, ending with certified implementations of algorithms. In this paper, we explain our design requirements for the implementation, using polynomials as a running example. Indeed, proving correctness of implementations depends heavily on the way this design allows due mathematical properties to be truly handled at the programming level. " DATE: 1997 AUTHORS:S. Boulmé, T. Hardin, D. Hirschko V. Ménissier-Morain, and R. Rioboo Laboratoire d'Informatique de Paris 6 (LIP6), Université Pierre et Marie Curie (Paris 6), 8, rue du Capitaine Scott, 75015 Paris, France. EMAIL: "Therese.Hardin@lip6.fr"

A New Algebra System — by Bill Page — last modified Aug 29, 2007 06:28 AM

The thesis of this note is that, in order to achieve uniform semantics between compiled and interpreted code, and to avoid exposing all sorts of internal hacks to the user, we require a model for the semantics of the new algebra system. Among the requirements of this model are that it be: 1. Simple 2. Powerful 3. Related to a user's perception of the system. It is not necessary that it be efficient. I propose, as I believe others have done, the Alist model for computer algebra systems, in which a domain is conceived of as a set of values (about which little more will be said), a set of attributes (I do not fully understand these, but believe that they will follow much the same lines as operations) and a set of operations. The set of operations lists all the operations that can be performed on elements of the domain, and the user is invited to view the system as searching this list for an operation of the appropriate signature, and then applying it.

Approaching Inheritance from a “Natural” Mathematical Perspective and from a Java Driven Viewpoint — by Bill Page — last modified Jun 21, 2005 05:44 PM

Approaching Inheritance from a “Natural” Mathematical Perspective and from a Java Driven Viewpoint: a Comparative Review. Marc Conrad/1, Tim French/1, Carsten Maple/1, and Sandra Pott/2 1/ University of Luton, LU1 3JU, UK marc.conrad@luton.ac.uk, tim.french@luton.ac.uk, carsten.maple@luton.ac.uk 2/ University of York, YO10 5DD, UK sp23@york.ac.uk Abstract. It is well-known that few object-oriented programming languages allow objects to change their nature at run-time. There have been a number of reasons presented for this, but it appears that their is a real need for matters to change. In this paper we discuss the need for object-oriented programming languages to reflect the dynamic nature of problems, particularly those arising in a mathematical context. It is from this context that we present a framework that realistically represents the dynamic and evolving characteristic of problems and algorithms.

Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire — by Bill Page — last modified May 24, 2007 04:08 PM

Abstract We develop a calculus for lazy functional programming based on recursion operators associated with data type deffinitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example functions in Bird and Wadler's "Introduction to Functional Programming" can be expressed using these operators.

Polynomial GCD Using Straight Line Program Representation — by Bill Page — last modified Sep 07, 2006 06:27 AM

Bill Naylor (PhD Thesis University of Bath, 2000): Summary: This thesis is concerned with calculating polynomial greatest common divisors using straight line program representation. In the Introduction chapter, we introduce the problem and describe some of the traditional representations for polynomials, we then talk about some of the general subjects central to the thesis, terminating with a synopsis of the category theory which is central to the AXIOM computer algebra system used during this research. The second chapter is devoted to describing category theory. We follow with a chapter detailing the important sections of computer code written in order to investigate the straight line program subject. The following chapter on evaluation strategies and algorithms which are dependant on these follows, the major algorithm which is dependant on evaluation and which is central to our thesis being that of equality checking. This is indeed central to many mathematical problems. Interpolation, that is the determination of coefficients of a polynomial is the subject of the next chapter. This is very important for many straight line program algorithms, as their non-canonical sttructure implies that it is relatively difficult to determine coefficients, these being the basic objects that many algorithms work on. We talk about three separate interpolation techniques and compar their advantages and disadvantages. The final two chapters describe some of the results we have obtained from this research an finally conclusions we have drawn as to the viability of the straight line program approach and possible extensions. Finally we terminate with a number of appendices discussing side subjects encountered during the thesis. Download 07-Sep-2006: http://www.cs.bath.ac.uk/~wn/thesis.ps.gz

Does Axiom Solve Systems of O.D.E.'s Like Mathematica? — by Bill Page — last modified Feb 10, 2005 08:18 PM

If I were demonstrating Axiom and were asked this question my reply would be "No, but I am not sure that this is a bad thing", and I would illustrate this with the following example.

The "Unknown" in Computer Algebra — by Bill Page — last modified Sep 19, 2006 07:14 AM

Computer algebra systems have to deal with the confusion between "programming variables" and "mathematical symbols". We claim that they should also deal with "unknowns", i.e. elements whose values are unknown, but whose type is known. For example $x^p \ne x$ if $x$ is a symbol but $x^p=x$ if $x\in \mathrm{GF}(p)$. We show how we have extended Axiom do deal with this concept. Download 18-Sep-2006: http://lists.nongnu.org/archive/html/axiom-developer/2004-06/dviwYbMdboRdU.dvi PDF format: http://wiki.axiom-developer.org/public/TheUnknownInComputerAlgebra.pdf

Utilisation de logiciels libres pour la réalization do TP MT26 — by Bill Page — last modified Nov 18, 2005 07:29 PM

 

Book Review - Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena by Jon Barwise and Lawrence Moss — by Bill Page — last modified Jun 23, 2005 04:48 PM

From http://cogprints.org/336/ "To a greater or lesser degree, every scientific advance marks some departure from the common sense that preceded it." These words of Irving M. Copi (Copi, 1979, p.195) apparently summarize the nature of Vicious Circles in the most concise fashion. Following the steps of Aczel's ground-breaking monograph (Aczel, 1988) which marked a departure from the `common sense' that is attributed to classical set theory, this new book (abbreviated as VC in the sequel) of Jon Barwise and Larry Moss not only offers an introduction to the revolutionary and fascinating topic of non-wellfounded sets (a.k.a. hypersets) but also becomes the most authoritative source for any serious researcher (mathematician, philosopher, or computer scientist alike) who wants to understand and further pursue this timely topic." And in a footnote the reviewer write: "It is hard to tell why circularity has always been regarded with doubt in mathematical circles. One reason may be that circular arguments or structures are thought to be diffcult to grasp, most probably due to our educational make-up in linear thinking. On a more personal note: I tend to think that popular works such as Escher's drawings may have helped to create the illusion that circularity invites nonsense."

Logic and Dependent Types in the Aldor Computer Algebra System — by Bill Page — last modified Nov 18, 2005 07:29 PM

We show how the Aldor type system can represent propositions of first-order logic, by means of the `propositions as types' correspondence. The representation relies on type casts (using pretend) but can be viewed as a prototype implementation of a modified type system with type evaluation reported elsewhere [9]. The logic is used to provide an axiomatisation of a number of familiar Aldor categories as well as a type of vectors.

Possible Enhancements for Aldordoc — by Bill Page — last modified Jul 21, 2006 02:11 AM

Authors: Balint Joo, Tony Kennedy. Aldordoc is an application, that extracts documentation from Aldor programs -- in particular .asy files. It produces XML output which can then be converted to various display formats, oun of the current supported being HTML. The appearance of the HTML output however is quite basic, and it is to be embedded into Aldor documentation comments. This document investigates some of these issues. Download 21-Jul-2006: http://www.ph.ed.ac.uk/~bj/paraldor/WWW/docs/discussion/aldordoc.pdf

Next Generation Computer Algebra Systems — by Bill Page — last modified Nov 18, 2005 07:29 PM

AXIOM and the Scratchpad Concept: Applications to Research in Algebra Larry Lambe Presented to the 21st Nordic Congress of Mathematicians, June, 1992, Lulea, Sweden

Pseudo Differential Operators and Integrable Systems in AXIOM — by Bill Page — last modified Nov 18, 2005 07:29 PM

An implementation of the algebra of pseudo differential operators in the computer algebra system AXIOM is described. In several examples the application of the package to typical computations in the theory of integrable systems is demonstrated.

A First Report on the A# Compiler — by Bill Page — last modified Nov 18, 2005 07:29 PM

Stephen M. Watt Peter A. Broadbery Samuel S. Dooley Pietro Iglio Scott ~C. Morrison* Jonathan M. Steinbach Robert S. Sutor IBM Thomas J. Watson Research Center P.O. BIDX 218, Yorktown Heights, NY 10598 USA

Adding the axioms to Axiom - Towards a system of automated reasoning in Aldor — by Bill Page — last modified Nov 18, 2005 07:29 PM

A number of combinations of theorem proving and computer algebra systems have been proposed; in this paper we describe another namely a way to incorporate a logic in the computer algebra system Axiom. We examine the type system of Aldor - the Axiom Library Compiler - and show that with some modifications we can use the dependent types of the system to model a logic using the Curry-Howard isomorphism. We give a number of example applications of the logic we construct.

Object-Oriented Mathematical Programming and Symbolic/Numeric Interface — by Bill Page — last modified Nov 18, 2005 07:29 PM

The Axiom language is based on the notions of packages, domains and categories.

Real Algebraic Closure of an Ordered Field, Implementation in Axiom — by Bill Page — last modified Nov 18, 2005 07:29 PM

Real algebraic numbers appear in many Computer Algebra problems. For instance the determination of a cylindrical algebraic decomposition for an euclidian space requires computing with real algebraic numbers. This paper describes an implementation for computations with the real roots of a polynomial. This process is designed to be recursively used, so the resulting domain of computation is the set of all real algebraic numbers. An implementation for the real algebraic closure has been done in Axiom (previously called Scratchpad).

SCRATCHPAD 1 AN INTERACTIVE FACILITY FOR SYMBOLIC MATHEMATICS — by Bill Page — last modified Nov 18, 2005 07:29 PM

The SCRATCHPAD/1 system is designed to provide an interactive symbolic computational facility for the mathematician user. The system features a user language designed to capture the style and succinctness of mathematical notation, together with a facility for conveniently introducing new notations into the language. A comprehensive system library incorporates symbolic capabilities provided by such systems as SIN, MATHLAB, and REDUCE.

JET - An AXIOM Environment for Geometric Computations with Differential Equations — by Bill Page — last modified Nov 18, 2005 07:29 PM

Geometric methods play an important role in the analysis of nonlinear differential equations. For example, symmetry methods provide the more or less only systematic approach to the construction of solutions. However, most geometric computations tend to be very tedious. Thus the use of computer algebra systems considerably helps in the application of these methods. JET is an environment within the computer algebra system AXIOM to perform such computations. The current implementation emphasizes the two key concepts involution and symmetry. It provides some packages for the completion of a given system of differential equations to an equivalent involutive one based on the Cartan- Kuranishi theorem and for setting up the determining equations for classical and non-classical point symmetries.

Object Oriented Method for Axiom. — by Bill Page — last modified Nov 18, 2005 07:29 PM

Axiom is a very powerful computer algebra system which combines two languages paradigms (functional and OOP). Mathematical world is complex and mathematician use abstraction to design it. This paper presents some aspects of the object oriented development in Axiom. The axiom programming is based on several new tools for object oriented development, it uses two levels of class and some operations such that coerce, retract or convert which permit the type evolution. These notions introduce the concept of multi-view.

Modeling Inheritance as Coercion in a Symbolic Computation System — by Bill Page — last modified Mar 03, 2006 08:01 AM

In this paper the analysis of the data structures used in a symbolic computation system, called Kenzo, is undertaken. We deal with the specification of the inheritance relationship since Kenzo is an object-oriented system, written in CLOS, the Common Lisp Object System. We focus on a particular case, namely the relationship between simplicial sets and chain complexes, showing how the order-sorted algebraic specifications formalisms can be adapted, through the "inheritance as coercion" metaphor, in order to model this Kenzo fragment.

Algorithms for Type Inference with Coercions — by Bill Page — last modified Mar 03, 2006 08:05 AM

This paper presents algorithms that perform a type inference for a type system occurring in thecontext of computer algebra. The type system permits various classes of coercions between types and the algorithms are complete for the precisely defined system, which can be seen as a formal description of an important subset of the type system supported by the computer algebra program AXIOM. Previously only algorithms for much more restricted cases of coercions have been described or the frameworks used have been so general that the corresponding type inference problems were known to be undecidable.

The Type Inference and Coercion Facilities in the Scratchpad II Interpreter