# Axiom (computer algebra system)

Developer(s) Independent group of people May 2012 Cross-platform Computer algebra system Modified BSD License 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

Axiom has been in development since 1965. It was started by James Greismer at the request of Ralph Gomory,[1] originally as Scratchpad. The main effort was led by a group at IBM under the direction of Richard Dimick Jenks.[2] Other key early developers were Barry Trager, Stephen Watt, James Davenport, Robert Sutor, and Scott Morrison.

In the 1990s, it was sold to NAG and given its current name. In 2001, it was withdrawn from the market and re-released under the Modified BSD License.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[3] and FriCAS,[4] following "serious disagreement about project goals".[5] The Axiom project continued to be developed by Tim Daly.

## Documentation

Axiom is a literate program.[6] 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:

## 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[7] website. The first video[8] provides details on the Axiom information sources.[8]

## 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).

## 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. The latter can still be used as an alternative extension language. It is, however, distributed under a different license.

## 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.

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

## References

• James H. Griesmer; Richard D. Jenks (1971). "SCRATCHPAD/1: An interactive facility for symbolic mathematics". Proceedings of the second ACM symposium on Symbolic and algebraic manipulation (SYMSAC '71). pp. 42–58.
• Richard D. Jenks (1971). META/PLUS - The Syntax Extension Facility for SCRATCHPAD (Research report). IBM Thomas J. Watson Research Center. RC 3259.
• James H. Griesmer; Richard D. Jenks (1972). "Experience with an online symbolic mathematics system". Proceedings of the ONLINE72 Conference 1. Brunel University. pp. 457–476.
• James H. Griesmer; Richard D. Jenks (1972). "SCRATCHPAD: A capsule view". SIGPLAN Notices 7 (10). ACM. pp. 93–102.
• Richard D. Jenks (1974). "The SCRATCHPAD Language". SIGPLAN Notices 9 (4). ACM. pp. 101–111. ISSN 0362-1340.
• Arthur C. Norman (1975). "Computing with Formal Power Series". TOMS (ACM) 1 (4): 346–356. ISSN 0098-3500.
• Richard D. Jenks (1976). "A pattern compiler". Proceedings of the third ACM symposium on Symbolic and algebraic manipulation (SYMSAC '76). pp. 60–65.
• E. Lueken (1977). Ueberlegungen zur Implementierung eines Formelmanipulationssystems (Masters thesis). Germany: Technischen Universitat Carolo-Wilhelmina zu Braunschweig. (German)
• George E. Andrews (1984). "Ramanujan and SCRATCHPAD". Proceedings of the 1984 MACSYMA Users' Conference. Schenectady: General Electric. pp. 383–408.
• James H. Davenport; P. Gianni; Richard D. Jenks; V. Miller; Scott Morrison; M. Rothstein; C. Sundaresan; Robert S. Sutor et al. (1984). "Scratchpad". Mathematical Sciences Department, IBM Thomas J. Watson Research Center.
• Richard D. Jenks (1984). "The New SCRATCHPAD Language and System for Computer Algebra". Proceedings of the 1984 MACSYMA Users' Conference (Schenectady, New York): 409–416.
• Richard D. Jenks (1984). "A primer: 11 keys to New Scratchpad". Proceedings of EUROSAM '84. Springer. pp. 123–147.
• Robert S. Sutor (1985). "The Scratchpad II Computer Algebra Language and System". Proceedings of EUROCAL '85. Springer. pp. 32–33.
• Rüdiger Gebauer; H. Michael Möller (1986). "Buchberger's algorithm and staggered linear bases". Proceedings of the fifth ACM symposium on Symbolic and algebraic computation (SYMSAC '86). ACM. pp. 218–221. ISBN 0-89791-199-7.
• Richard D. Jenks; Robert S. Sutor; Stephen M. Watt (1986). Scratchpad II: an abstract datatype system for mathematical computation (Research report). IBM Thomas J. Watson Research Center. RC 12327.
• Michael Lucks; Bruce W. Char (1986). "A fast implementation of polynomial factorization". Proceedings of SYMSAC '86. ACM. pp. 228–232. ISBN 0-89791-199-7.
• J. Purtilo (1986). "Applications of a software interconnection system in mathematical problem solving environments". Proceedings of SYMSAC '86. ACM. pp. 16–23. ISBN 0-89791-199-7.
• William H. Burge; Stephen M. Watt (1987). Infinite Structure in SCRATCHPAD II (Research report). IBM Thomas J. Watson Research Center. RC 12794.
• Pascale Sénéchaud; Françoise Siebert; Gilles Villard (1987). "Scratchpad II: Présentation d'un nouveau langage de calcul formel". TIM (Research report) (IMAG, Grenoble Institute of Technology). 640-M. (French)
• Robert S. Sutor; Richard D. Jenks (1987). "The Type Inference and Coercion Facilities in the Scratchpad II Interpreter". In Richard L. Wexelblat. Proceedings of the SIGPLAN '87 Symposium on Interpreters and Interpretive Techniques. ACM. pp. 56–63. doi:10.1145/29650.29656. ISBN 0-89791-235-7.
• George E. Andrews (1988). "Application of SCRATCHPAD to problems in special functions and combinatorics". In R. Janssen. Trends in Computer Algebra. Lecture Notes in Computer Science (296). Springer. pp. 159–166.
• James H. Davenport; Yvon Siret; Evelyne Tournier (1993) [1988]. Computer Algebra: Systems and Algorithms for Algebraic Computation. Academic Press. ISBN 978-0122042300.
• R. Gebauer; H. M. Moller (1988). "On an installation of Buchberger's algorithm". Journal of Symbolic Computation 6 (2-3): 275–286. ISSN 0747-7171.
• Fritz Schwarz (1988). "Programming with abstract data types: the symmetry package (SPDE) in Scratchpad". In R. Janssen. Trends in Computer Algebra. Lecture Notes in Computer Science. Springer. pp. 167–176.
• D. Shannon; M. Sweedler (1988). "Using Groebner bases to determine algebra membership, split surjective algebra homomorphisms determine birational equivalence". Journal of Symbolic Computation 6 (2-3): 267–273.
• Hans-J. Boehm (1989). "Type inference in the presence of type abstraction". SIGPLAN 24 (7): 192–206.
• Manuel Bronstein (1989). "Simplification of real elementary functions". Proceedings of the International Symposium on Symbolic and Algebraic Computation (SIGSAM '89). ACM. pp. 207–211.
• Claire Dicrescenzo; Dominique Duval (1989). "Algebraic extensions and algebraic closure in Scratchpad II". In P. Gianni. Symbolic and Algebraic Computation. Springer. pp. 440–446.
• 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