Axiom (computer algebra system)

Axiom

Developer(s)	Independent group of people
Stable release	Continuous using Docker
Operating system	Cross-platform
Type	Computer algebra system
License	Modified BSD License
Website	axiom-developer.org

Axiom is a free, general-purpose computer algebra system. It consists of an interpreter environment, a compiler and a library, which defines a strongly typed, mathematically (mostly) correct type hierarchy.

History

Two computer algebra systems named Scratchpad were developed by IBM. The first one was started in 1965 by James Greismer at the request of Ralph Gomory, and written in Fortran. The development of this software was stopped before any public release. The second Scratchpad, originally named Scratchpad II, was developed from 1977 on, at Thomas J. Watson Research Center, under the direction of Richard Dimick Jenks. Other key early developers were Barry Trager, Stephen Watt, James Davenport, Robert Sutor, and Scott Morrison.

Scratchpad II was renamed Axiom when IBM decided, circa 1990, to make it a commercial product. A few years later, it was sold to NAG. In 2001, it was withdrawn from the market and re-released under the Modified BSD License. Since then, the project's lead developer has been Tim Daly.

In 2007, Axiom was forked twice, originating two different open-source projects: OpenAxiom and FriCAS, following "serious disagreement about project goals". The Axiom project continued to be developed by Tim Daly.

Documentation

Axiom is a literate program. The source code is becoming available in a set of volumes which are available on the axiom-developer.org website. These volumes contain the actual source code of the system.

The currently available documents are:

• Combined Table of Contents
• Volume 0: Axiom Jenks and Sutor—The main textbook
• Volume 1: Axiom Tutorial—A simple introduction
• Volume 2: Axiom Users Guide—Detailed examples of domain use (incomplete)
• Volume 3: Axiom Programmers Guide—Guided examples of program writing (incomplete)
• Volume 4: Axiom Developers Guide—Short essays on developer-specific topics (incomplete)
• Volume 5: Axiom Interpreter—Source code for Axiom interpreter (incomplete)
• Volume 6: Axiom Command—Source code for system commands and scripts (incomplete)
• Volume 7: Axiom Hyperdoc—Source code and explanation of X11 Hyperdoc help browser
  • Volume 7.1 Axiom Hyperdoc Pages—Source code for Hyperdoc pages
• Volume 8: Axiom Graphics—Source code for X11 Graphics subsystem
  • Volume 8.1 Axiom Gallery—A Gallery of Axiom images
• Volume 9: Axiom Compiler—Source code for Spad compiler (incomplete)
• Volume 10: Axiom Algebra Implementation—Essays on implementation issues (incomplete)
  • Volume 10.1: Axiom Algebra Theory—Essays containing background theory
  • Volume 10.2: Axiom Algebra Categories—Source code for Axiom categories
  • Volume 10.3: Axiom Algebra Domains—Source code for Axiom domains
  • Volume 10.4: Axiom Algebra Packages—Source code for Axiom packages
  • Volume 10.5: Axiom Algebra Numerics—Source code for Axiom numerics
• Volume 11: Axiom Browser—Source pages for Axiom Firefox browser front end
• Volume 12: Axiom Crystal—Source code for Axiom Crystal front end (incomplete)
• Volume 13: Proving Axiom Correct—Prove Axiom Algebra (incomplete)
• Bibliography: Axiom Bibliography—Literature references

Videos

The Axiom project has a major focus on providing documentation. Recently the project announced the first in a series of instructional videos, which are also available on the axiom-developer.org website. The first video provides details on the Axiom information sources.

Philosophy

The Axiom project focuses on the “30 Year Horizon”. The primary philosophy is that Axiom needs to develop several fundamental features in order to be useful to the next generation of computational mathematicians. Knuth's literate programming technique is used throughout the source code. Axiom plans to use proof technology to prove the correctness of the algorithms (such as Coq and ACL2).

Axiom uses Docker Containers as part of a continuous release process. The latest image is available on any platform using boot2docker [1] and the commands:

docker pull daly/axiom

docker run -i -t daly/axiom axiom

Design

In Axiom, all objects have a type. Examples of types are mathematical structures (such as rings, fields, polynomials) as well as data structures from computer science (e.g., lists, trees, hash tables).

A function can take a type as argument, and its return value can also be a type. For example, Fraction is a function, that takes an IntegralDomain as argument, and returns the field of fractions of its argument. As another example, the ring of $4\times 4$ matrices with rational entries would be constructed as SquareMatrix(4, Fraction Integer). Of course, when working in this domain, 1 is interpreted as the identity matrix and A^-1 would give the inverse of the matrix A, if it exists.

Several operations can have the same name, and the types of both the arguments and the result are used to determine which operation is applied (cf. function overloading).

Axiom comes with an extension language called SPAD. All the mathematical knowledge of Axiom is written in this language. The interpreter accepts roughly the same language. SPAD was further developed under the name A# and later Aldor. Axiom no longer supports this language extension.

Features

Within the interpreter environment, Axiom uses type inference and a heuristic algorithm to make explicit type annotations mostly unnecessary.

It features 'HyperDoc', an interactive browser-like help system, and can display two and three dimensional graphics, also providing interactive features like rotation and lighting. It also has a specialised interaction mode for Emacs, as well as a plugin for the TeXmacs editor.

• AxiomHyperDoc.png

  HyperDoc displaying the available operations for a domain

• AxiomGraphics.png

  Axiom displaying a surface

• Axiomfirefox.jpg

  Axiom Firefox Browser Interface

• Heatequation.jpg

  Axiom simplifying a heat equation

• Matrixinmatrix.jpg

  Axiom matrix manipulation

• RischIntegration.jpg

  Axiom computing a Risch integral

Axiom has an implementation of the Risch algorithm for elementary integration, which was done by Manuel Bronstein and Barry Trager.

See also

• A# programming language
• Aldor programming language
• List of computer algebra systems

Further reading

• Timothy Daly "Axiom -- Thirty Years of Lisp"
• Timothy Daly "Axiom" Invited Talk, Free Software Conference, Lyon, France, May, 2002
• Timothy Daly "Axiom" Invited Talk, Libre Software Meeting, Metz, France, July 9–12, 2003

External links

• Axiom Homepage
• Online sandbox to try Axiom
• Source code repositories: Github, SourceForge, GNU Savannah
• Jenks, R.D. and Sutor, R. "Axiom, The Scientific Computation System"
• Daly, T. "Axiom Volume 1: Tutorial"

Software forks:

• OpenAxiom (SourceForge)
• FriCAS (SourceForge)