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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 Griesmer[1] 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]

The design is principally due to Richard D. Jenks (IBM Research), James H. Davenport (University of Bath), Barry M. Trager (IBM Research), David Y.Y. Yun (Southern Methodist University) and Victor S. Miller (IBM Research). Early consultants on the project were David Barton (University of California, Berkeley) and James W. Thatcher (IBM Research). Implementation included Robert Sutor (IBM Research), Scott C. Morrison (University of California, Berkeley), Christine J. Sundaresan (IBM Research), Patrizia Gianni (University of Pisa), Albrecht Fortenbacher (Universitaet Karlsruhe), Stephen M. Watt (IBM Research and University of Waterloo), Josh Cohen (Yale University) and Michael Rothstein (Kent State University).[3]

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 In 2001, it was withdrawn from the market and re-released under the Modified BSD License.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[4] and FriCAS,[5] following "serious disagreement about project goals".[6] The Axiom project continued to be developed by Tim Daly.

The current research direction is "Proving Axiom Sane", that is, logical, rational, judicious, and sound.

Axiom is a literate program. Why Literate Programming? at website 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:

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)
Volume 15: The Axiom SANE Compiler
Bibliography: Axiom Bibliography—Literature references
Bug List: Axiom Bug List-Bug List
Reference Card: Axiom Reference Card--Useful function summary


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 [2] and the commands:

docker pull daly/axiom

docker run -i -t daly/axiom axiom


In Axiom, each object has 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 × 4 {\displaystyle 4\times 4} 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.

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 specialized interaction mode for Emacs, as well as a plugin for the TeXmacs editor.

HyperDoc displaying the available operations for a domain

Axiom displaying a surface

Axiom Firefox Browser Interface

Axiom simplifying a heat equation

Axiom matrix manipulation

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

Free and open-source software portal

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


Axiom Homepage
Richard Dimick Jenks - Biographical Information
History of FriCAS

"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.
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". ACM SIGPLAN Notices. 7 (10): 93–102. doi:10.1145/942576.807019.
Richard D. Jenks (1974). "The SCRATCHPAD language". ACM SIGSAM Bulletin. 8 (2): 20–30. doi:10.1145/1086830.1086834.
Arthur C. Norman (1975). "Computing with Formal Power Series". ACM Transactions on Mathematical Software. 1 (4): 346–356. doi:10.1145/355656.355660. ISSN 0098-3500.
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: 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. pp. 218–221. ISBN 978-0-89791-199-3.
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. pp. 228–232. ISBN 978-0-89791-199-3.
J. Purtilo (1986). Applications of a software interconnection system in mathematical problem solving environments. ACM. pp. 16–23. ISBN 978-0-89791-199-3.
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). "The type inference and coercion facilities in the scratchpad II interpreter". Papers of the Symposium on Interpreters and interpretive techniques - SIGPLAN '87. pp. 56–63. doi:10.1145/29650.29656. ISBN 978-0-89791-235-8.
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. 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.
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". ACM SIGPLAN Notices. 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
Wikimedia Commons has media related to Axiom.

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)

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia



Hellenica World - Scientific Library

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