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Axiom (computer algebra system)

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Axiom
Developer(s)Independent group of people
Stable release
Continuous using Docker
Repository
Operating systemCross-platform
TypeComputer algebra system
LicenseModified BSD License
Websiteaxiom-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.[1] 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.[2] 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 In 2001, it was withdrawn from the market and re-released under the Modified BSD License.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[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).

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 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.

  • HyperDoc displaying the available operations for a domain
    HyperDoc displaying the available operations for a domain
  • Axiom displaying a surface
    Axiom displaying a surface
  • Axiom Firefox Browser Interface
    Axiom Firefox Browser Interface
  • Axiom simplifying a heat equation
    Axiom simplifying a heat equation
  • Axiom matrix manipulation
    Axiom matrix manipulation
  • Axiom computing a Risch integral
    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

References

  1. ^ Axiom Homepage
  2. ^ Richard Dimick Jenks - Biographical Information
  3. ^ open-axiom.org
  4. ^ fricas.sourceforge.net
  5. ^ History of FriCAS
  6. ^ Why Literate Programming? at axiom-developer.org
  7. ^ Axiom-developer.org
  8. ^ a b "Axiom Computer Algebra System Information Sources". YouTube. November 30, 2008.

Further reading

  • James H. Griesmer; Richard D. Jenks (1971). "SCRATCHPAD/1: An interactive facility for symbolic mathematics": 42–58. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • 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". 1. Brunel University: 457–476. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • James H. Griesmer; Richard D. Jenks (1972). "SCRATCHPAD: A capsule view". 7 (10). ACM: 93–102. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • Richard D. Jenks (1974). "The SCRATCHPAD Language". 9 (4). ACM: 101–111. ISSN 0362-1340. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • Arthur C. Norman (1975). "Computing with Formal Power Series". TOMS. 1 (4). ACM: 346–356. doi:10.1145/355656.355660. ISSN 0098-3500.
  • Richard D. Jenks (1976). "A pattern compiler": 60–65. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • E. Lueken (1977). Ueberlegungen zur Implementierung eines Formelmanipulationssystems (Masters thesis) (in German). Germany: Technischen Universitat Carolo-Wilhelmina zu Braunschweig.
  • George E. Andrews (1984). "Ramanujan and SCRATCHPAD". Schenectady: General Electric: 383–408. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • James H. Davenport; P. Gianni; Richard D. Jenks; V. Miller; Scott Morrison; M. Rothstein; C. Sundaresan; Robert S. Sutor; Barry Trager (1984). ""Scratchpad"". Mathematical Sciences Department, IBM Thomas J. Watson Research Center. {{cite journal}}: Cite journal requires |journal= (help)
  • 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". Springer: 123–147. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • Robert S. Sutor (1985). "The Scratchpad II Computer Algebra Language and System". Springer: 32–33. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • Rüdiger Gebauer; H. Michael Möller (1986). "Buchberger's algorithm and staggered linear bases". ACM: 218–221. ISBN 0-89791-199-7. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • 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". ACM: 228–232. ISBN 0-89791-199-7. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • J. Purtilo (1986). "Applications of a software interconnection system in mathematical problem solving environments". ACM: 16–23. ISBN 0-89791-199-7. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • 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) (in French). IMAG, Grenoble Institute of Technology. 640-M.
  • Robert S. Sutor; Richard D. Jenks (1987). Richard L. Wexelblat (ed.). "The Type Inference and Coercion Facilities in the Scratchpad II Interpreter". ACM: 56–63. doi:10.1145/29650.29656. ISBN 0-89791-235-7. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • George E. Andrews (1988). R. Janssen (ed.). "Application of SCRATCHPAD to problems in special functions and combinatorics". Lecture Notes in Computer Science (296). Springer: 159–166. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • 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. doi:10.1016/s0747-7171(88)80048-8. ISSN 0747-7171.
  • Fritz Schwarz (1988). R. Janssen (ed.). "Programming with abstract data types: the symmetry package (SPDE) in Scratchpad". Lecture Notes in Computer Science. Springer: 167–176. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • 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. doi:10.1016/s0747-7171(88)80047-6.
  • Hans-J. Boehm (1989). "Type inference in the presence of type abstraction". Sigplan. 24 (7): 192–206. doi:10.1145/74818.74835.
  • Manuel Bronstein (1989). "Simplification of real elementary functions". ACM: 207–211. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • Claire Dicrescenzo; Dominique Duval (1989). P. Gianni (ed.). "Algebraic extensions and algebraic closure in Scratchpad II". Springer: 440–446. {{cite journal}}: Cite journal requires |journal= (help); Unknown parameter |booktitle= ignored (help)
  • 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
Wikimedia Commons has media related to Axiom.

Software forks:

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