Axiom (computer algebra system)

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

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.


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 In 2001, it was withdrawn from the market and re-released under the Modified BSD License.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.


Axiom is a literate program.[6] The source code is becoming available in a set of volumes which are available on the website. These volumes contain the actual source code of the system.

The currently available documents are:


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


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


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.


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.

See also[edit]


Further reading[edit]

  • James H. Griesmer; Richard D. Jenks (1971). "SCRATCHPAD/1: An interactive facility for symbolic mathematics": 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". 1. Brunel University: 457–476. 
  • James H. Griesmer; Richard D. Jenks (1972). "SCRATCHPAD: A capsule view". 7 (10). ACM: 93–102. 
  • Richard D. Jenks (1974). "The SCRATCHPAD Language". 9 (4). ACM: 101–111. ISSN 0362-1340. 
  • Arthur C. Norman (1975). "Computing with Formal Power Series". TOMS. ACM. 1 (4): 346–356. ISSN 0098-3500. doi:10.1145/355656.355660. 
  • Richard D. Jenks (1976). "A pattern compiler": 60–65. 
  • 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. 
  • 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. 
  • 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. 
  • Robert S. Sutor (1985). "The Scratchpad II Computer Algebra Language and System". Springer: 32–33. 
  • Rüdiger Gebauer; H. Michael Möller (1986). "Buchberger's algorithm and staggered linear bases". ACM: 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". ACM: 228–232. ISBN 0-89791-199-7. 
  • J. Purtilo (1986). "Applications of a software interconnection system in mathematical problem solving environments". ACM: 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) (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. ISBN 0-89791-235-7. doi:10.1145/29650.29656. 
  • 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. 
  • James H. Davenport; Yvon Siret; Evelyne Tournier (1993) [1988]. Computer Algebra: Systems and Algorithms for Algebraic Computation. Academic Press. ISBN 978-0122042300. 
  • Rüdiger Gebauer; H. Michael Möller (1988). "On an installation of Buchberger's algorithm". Journal of Symbolic Computation. 6 (2-3): 275–286. ISSN 0747-7171. doi:10.1016/s0747-7171(88)80048-8. 
  • 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. 
  • David Shannon; Moss Sweedler (1988). "Using Gröbner 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. 
  • Claire Dicrescenzo; Dominique Duval (1989). P. Gianni, ed. "Algebraic extensions and algebraic closure in Scratchpad II". Springer: 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

External links[edit]

Software forks: