Axiom is a free, open source computer algebra system
Switch branches/tags
Nothing to show
Clone or download
daly books/bookvolbib add references
Goal: Proving Axiom Sane

\index{Scott, Dana S.}
  author = "Scott, Dana S.",
  title = {{A Type-Theoretical Alternative to ISWIM, CUCH, OWHY}},
  journal = "Theoretical Computer Science",
  volume = 121,
  number = "1-2",
  year = "1993",
  pages = "411-440",
  abstract =
    "The paper (first written in 1969 and circulated privately) concerns
    the definition, axiomatization, and applications of the hereditarily
    monotone and continuous functionals generated from the integers and
    the Booleans (plus “undefined” elements). The system is formulated as
    a typed system of combinators (or as a typed λ-calculus) with a
    recursion operator (the least fixed-point operator), and its proof
    rules are contrasted to a certain extent with those of the untyped
    λ-calculus. For publication (1993), a new preface has been added, and
    many bibliographical references and comments in footnotes have been
  paper = "Scot93.pdf",
  keywords = "printed"


\index{Winkler, Franz}
  author = "Winkler, Franz",
  title = {{Reducing the Complexity of the Knuth-Bendix Completion
            Algorithm: A  ``Unification'' of Different Approaches}},
  booktitle = "European Conference on Computer Algebra (EUROCAL 85)",
  pages = "378-389",
  publisher = "Springer",
  year = "1985",
  isbn = "978-3-540-15983-4",
  abstract =
    "The Knuth-Bendix completion procedure for rewrite rule systems is
    of wide applicability in symbolic and algebraic computation.
    Attempts to reduce the complexity of this completion algorithm are
    reported in the literature. Already in their seminal 1967 paper
    D.E. Knuth and P.B. Bendix have suggested to keep all the rules
    iterreduced during the execution of the algorithm. G. Huet has
    presented a version of the completion algorithm in which every
    rewrite rule is kept in reduced form with respect to all the other
    rules in the system. Borrowing an idea of Buchberger's for the
    completion of bases of polynomial ideals the author has proposed
    in 1983 a criterion for detecting ``unnecessary'' critical
    pairs. If a critical pair is recognized as unnecessary then one
    need not apply the costly process of computing normal forms to
    it. It has been unclear whether these approaches can be
    combined. We demonstrate that it is possible to keep all the
    rewrite rules interreduced and still use a criterion for
    eliminating unnecessary critical pairs.",
  paper = "Wink85.pdf",
  keywords = "printed"


\index{Kandri-Rody, Abdelilah}
\index{Kapur, Deepak}
\index{Winkler, Franz}
  author = "Kandri-Rody, Abdelilah and Kapur, Deepak and Winkler, Franz",
  title = {{Knuth-Bendix Procedure and Buchberger Algorithm - A Synthesis}},
  booktitle = "ISSAC'89",
  publisher = "ACM Press",
  pages = 55-67",
  year = "1989",
  isbn = "0-89791-325-6",
  abstract =
    "The Knuth-Bendix procedure for the completion of a rewrite rule
    system and the Buchberger algorithm for computing a Groebner basis
    of a polynomial ideal are very similar in two respects: they both
    start wtih an arbitrary specification of an algebraic structure
    (axioms for an equational theory and a basis for a polynomial
    ideal, respectively) which is transformed to a very special
    specification of this algebraic structure (a complete rewrite rule
    system and a Groebner basis of the polynomial ideal, respectively).
    This special specification allows to decide many problems concerning
    the given algebraic structure. Moreover, both algorithms achieve
    their goals by employing the same basic concepts: formation of
    critical pairs and completion.

    Although the two methods are obviously related, the exact nature
    of this relation remains to be clarified. Based on previous work
    we show how the Knuth-Bendix procedure and the Buchberger algorithm
    can be seen as special cases of a more general completion procedure.",
  paper = "Kand89.pdf",
  keywords = "printed"


\index{Winkler, Franz}
  author = "Winkler, Franz",
  title = {{Computer Algebra}},
  booktitle = "Encyclopedia of Physical Science and Technology",
  volume = "4",
  year = "1992",
  publisher = "Academic Press",
  paper = "Wink92.pdf",
  keywords = "printed"


\index{Winkler, Franz}
  author = "Winkler, Franz",
  title = {{Computer Algebra - Problems and Developments}},
  booktitle = "Computers in Industry 2 (SCAFI'92)",
  pages = "1-17",
  year = "1995",
  isbn = "978-0471955290",
  paper = "Wink95.pdf"


\index{Winkler, Franz}
  author = "Winkler, Franz",
  title = {{Advances and Problems in Algebraic Computation}},
  type = "technical report",
  number = "99-49",
  year = "1999",
  institution = "RISC, Linz",
  abstract =
    "In the last years there has been dramatic progress in all areas
     of algebraic computation. Many working mathematiians have access
     to and actually use algebraic software systems for their research
     No end of this development is in sight. We report on some active
     topics in algebraic computation, namely the theory of integration
     and symbolic solution of systems of differential equations, the
     solution of systems of algebraic equations, the factorization of
     polynomials, and the design and analysis of algebraic curves and
  paper = "Wink99.pdf",
  keywords = "printed"


\index{Winkler, Franz}
  author = "Winkler, Franz",
  title = {{Computer Algebra and Geometry - Some Interactions}},
  booktitle = "Proc. of Coll. on Constructive Algebra and System
               Theory (CAST)",
  year = "2006",
  publisher = "unknown",
  isbn = "90-6984-477-x",
  pages = "127-138",
  abstract =
    "Algebraic curves and surfaces have been studied intensively in
    algebraic geometry for centuries. Thus, there exists a huge amount
    of theoretical knowledge about these geometric objects. Recently,
    algebraic curves and surfaces play an important and ever
    increasing role in computer aided geometric design, computer
    vision, computer aided manufacturing, coding theory, and
    cryptography, just to name a few application areas. Consequently,
    theoretical results need to be adapted to practical needs. We need
    efficient algorithms for generating, representing, manipulating,
    analyzing, rendering algebraic curves and surfaces. Exact computer
    algebraic methods can be employed effectively for dealing with
    these geometric problems.",
  paper = "Wink06.pdf"


\index{Davies, Rowan}
\index{Pfenning, Frank}
  author = "Davies, Rowan and Pfenning, Frank",
  title = {{A Modal Analysis of Staged Computation}},
  journal = "J. ACM",
  volume = "48",
  number = "3",
  pages = "555-604",
  year = "2001",
  abstract =
    "We show that a type system based on the intuitionistic modal logic S4
    provides an expressive framework for specifying and analyzing
    computation stages in the context of typed $\lambda$-calculi and
    functional languages. We directly demonstrate the sense in which our
    $\lambda_e^{\rightarrow\box}$-calculus captures staging, and also give
    a conservative embedding of Nielson and Nielson's two-level functional
    language in our functional language Mini-ML${}^\box$, thus proving that
    binding-time correctness is equivalent to modal correctness on this
    fragment. In addition, Mini-ML${}^\box$ can also express immediate
    evaluation and sharing of code across multiple stages, thus supporting
    run-time code generation as well as partial evaluation.",
  paper = "Davi01.pdf",
  keywords = "printed"


\index{Pfenning, Frank}
\index{Davies, Rowan}
  author = "Pfenning, Frank and Davies, Rowan",
  title = {{A Judgemental Reconstruction of Modal Logic}},
  journal = "Mathematical Structures in Computer Science",
  volume = "11",
  pages = "511-540",
  year = "2001",
  abstract =
    "We reconsider the foundations of modal logic, following Martin-Lof's
    methodology of distinguishing judgments from propositions. We give
    constructive meaning explanations for necessity and possibility which
    yields a simple and uniform system of natural deduction for
    intuitionistic modal logic which does not exhibit anomalies found in
    other proposals. We also give a new presentation of lax logic and nd
    that the lax modality is already expressible using possibility and
    necessity. Through a computational interpretation of proofs in modal
    logic we further obtain a new formulation of Moggi's monadic
  paper = "Pfen01.pdf",
  keywords = "printed"


\index{Lamport, Leslie}
  author = "Lamport, Leslie",
  title = {{The Future of Computing: Logic or Biology}},
  link =
  year = "2003",
  paper = "Lamp03.pdf",
  keywords = "printed"


\index{Glymour, Clark}
\index{Serafin, Luke}
  author = "Glymour, Clark and Serafin, Luke",
  title = {{Mathematical Metaphysics}},
  link = "\url{}",
  year = "2015",
  paper = "Glym15.pdf",
  keywords = "printed"


\index{Reynolds, John C.}
  author = "Reynolds, John C.",
  title = {{The Discoveries of Continuations}},
  journal = "Lisp and Symbolic Computation",
  volume = "6",
  pages = "233-248",
  year = "1993",
  abstract =
    "We give a brief account of the discoveries of continuations and
    related concepts.",
  paper = "Reyn93.pdf",
  keywords = "printed"


\index{Oles, Frank Joseph}
  author = "Oles, Frank Joseph",
  title = {{A Category-Theoretic Approach to the Semantics of
            Programming Languages}},
  school = "Syracuse University",
  year = "1982",
  abstract =
    "Here we create a framework for handling the semantics of fully
    typed programming languages with imperative features, higher-order
    ALGOL-like procedures, block structure, and implicit
    conversions. Our approach involves introduction of a new family of
    programming languages, the {\sl coercive typed $\lambda$-calculi},
    denoted by $L$ in the body of the dissertation. By appropriately
    choosing the linguistic constants (i.e. generators) of $L$, we can
    view phrases of variants of ALGOL as syntactically sugared phrases
    of $L$.

    This dissertation breaks into three parts. In the first part,
    consisting of the first chapter, we supply basic definitions and
    motivate the idea that functor categories arise naturally in the
    explanation of block structure and stack discipline. The second
    part, consisting of the next three chapters, is dedicated to the
    general theory of the semantics of the coercive typed
    $\lambda$-calculus; the interplay between posets, algebras, and
    Cartesian closed categories is particularly intense here. The
    remaining four chapters make up the final part, in which we apply
    the general theory to give both direct and continuation semantics
    for desugared variants of ALGOL. To do so, it is necessary to show
    certain functor categories are Cartesian closed and to describe a
    category $\Sigma$ of store shapes. An interesting novelty in the
    presentation of continuation semantics is the view that commands
    form a procedural, rather than a primitive, phrase type.",
  paper = "Oles82.pdf"


\index{Abdali, S. Kamal}
\index{Winkler, Franz}
  author = "Abdali, S. Kamal and Winkler, Franz",
  title = {{A Lambda-Calculus Model for Generating Verification Conditions}},
  type = "technical report",
  institution = "Rensselaer Polytechnic Institute",
  number = "CS-8104",
  year = "1981",
  abstract =
    "A lambda-calculus-based method is developed for the automatic
    generation of verification conditions. A programming language is
    specified in which inductive assertions associated with a program
    are incorporated within the body of the program by means of assert
    and maintain-while statements. This programming language includes
    the following features: assignments, conditionals, loops,
    compounds, ALGOL-type block structure. A model is developed which
    consists of rules to translate each statement in this programming
    language into the lamba-calculus. The model is such that the
    lambda-expression translation of any program reduces to a list
    (tuple) of lambda-expression representations of all verification
    conditions of the program. A proof of this property is given.",
  paper = "Abda81.pdf",
  keywords = "printed"


\index{Abdali, S. Kamal}
  author = "Abdali, S. Kamal",
  title = {{A Simple Lambda-Calculus Model of Programming Languages}},
  isbn = "978-1332196029",
  publisher = "Forgotten Books",
  year = "1973",
  abstract =
    "We present a simple correspondence between a large subset of
    ALGOL 60 language and lambda-calculus. With the aid of this
    correspondence, a program can be translated into a single
    lambda-expression. In general, the representation of a program is
    specified by means of a system of simultaneous conversion
    relations among the representations of the program
    constituents. High-level programming language features are treated
    directly, not in terms of the representations of machine-level
    operations. The model includes input-output in such a way that
    when the representation of a (convergent) program is applied to
    the input item representations, the resulting combination reduces
    to a tuple of the representations of the output items. This model
    does not introduce any imperative notions into the calculus; the
    descriptive programming constructs, such as expressions, and the
    imperative ones, such as assignment and jumps, are translated
    alike into pure lambda expressions. The applicability of the model
    to the problems of proving program equivalence and correctness is
    illustrated by means of simple examples."
  paper = "Abda73.pdf"


\index{Abdali, S. Kamal}
  author = "Abdali, S. Kamal",
  title = {{A Lambda-Calculus Model of Programming Languages -
           I. Simple Constructs}},
  journal = "J. of Computer Languages",
  volume = "1",
  pages = "287-301",
  year = "1974",
  abstract =
    "A simple correspondence is presented between the elementary
    constructs of programming languages and the lambda-calculus. The
    correspondence is obtained by using intuitive, functional
    interpretation of programming language constructs, completely
    avoiding the notions of machine memory and address. In particular,
    the concepts of program variable and assignments are accounted for
    in terms of the concepts of mathematical variable and
    substitution, respectively. Lambda-expression representations are
    given of assignments, conditional and compound statements,
    input-output, and blocks. Algol 60 is used as the illustrative
  paper = "Abda74.pdf",
  keywords = "printed"


\index{Abdali, S. Kamal}
  author = "Abdali, S. Kamal",
  title = {{A Lambda-Calculus Model of Programming Languages -
           II. Jumps and Procedures}},
  journal = "J. of Computer Languages",
  volume = "1",
  pages = "303-320",
  year = "1974",
  abstract =
    "The correspondence between programming languages and the
    lambda-calculus presented in Part I of the paper is extended there
    to include iteration statements, jumps, and procedures. Programs
    containing loops are represented by lambda-expressions whose
    components are specified recursively by means of systems of
    simultaneous conversion relations. The representation of
    call-by-name and side-effects in a program is accomplished without
    any resort to the concepts of memory andaddress by making use of a
    number of variables in addition to those declared by the programs;
    with the aid of these additional variables, the parameter linkage
    operations can be simulated by pure substitution. The
    applicability of the model to the problems of proving program
    correctness and equivalence is demonstrated by means of examples.",
  paper = "Abda74a.pdf",
  keywords = "printed"


\index{Abdali, S. Kamal}
  author = "Abdali, S. Kamal",
  title = {{A Combinatory Logic Model of Programming Languages}},
  school = "University of Wisconsin",
  year = "1974",
  abstract =
    "A simple correspondence is presented between a large subset of the
    ALGOL 60 language and the combinatory logic.  With the aid of this
    correspondence, a program can be translated into a single combinatory
    object.  The combinatory object representing a program is specified,
    in general, by means of a system of reduction relations among the
    representations of the program constituents.  This object denotes, in
    terms of the combinatory logic, the function that the program is
    intended to compute.

    The model has been derived by using intuitive, functional
    interpretations of the constructs of programming languages, completely
    avoiding the notions of machine command and address.  In particular,
    the concepts of program variable, assignment, and procedure have been
    accounted for in terms of the concepts of mathematical variable,
    substitution, and function, respectively.

    High-level programming language features are represented in the
    combinatory logic directly, not in terms of the representations of
    machine-level operations.  Input-output is treated in such a
    manner that when the representation of a program is applied to the
    representations of the input items, the resulting combination reduces
    to a tuple of the representations of the output items.

    The applicability of the model to the problems of
    proving program equivalence and correctness is illustrated
    by means of examples.",
  paper = "Abda74b.pdf"


\index{Krall, Andreas}
  author = "Krall, Andreas",
  title = {{Implementation Techniques for Prolog}},
  link = "\url{}",
  year = "unknown",
  abstract =
    "This paper is a short survey about currently used implementation
    techniques for Prolog. It gives an introduction to unification and
    resolution in Prolog and presents the memory model and a basic
    execution model. These models are expanded to the Vienna Abstract
    Machine (VAM) with its two versions, the VAM${}_{2p}$ and the
    VAM${}_{1p}$, and the most famous abstract machine, the Warren
    Abstract Machine (WAM). The continuation passing style model of
    Prolog, binary Prolog, leands to the BinWAM. Abstract
    interpretation can be applied to gather information about a
    program. This information is used in the generation of very
    specialized machine code and in optimizations like clause indexing
    and instruction scheduling on each kind of abstract machine.",
  paper = "Kralxx.pdf",
  keywords = "printed"


\index{Abadi, Martin}
\index{Cardelli, Luca}
\index{Curien, Pierre-Louis}
\index{Levy, Jean-Jacques}
  author = "Abadi, Martin and Cardelli, Luca and Curien, Pierre-Louis
            and Levy, Jean-Jacques",
  title = {{Explicit Substitutions}},
  booktitle = "Symp. of Principles of Programming Languages",
  publisher = "ACM",
  year = "1990",
  link = "\url{}",
  abstract =
    "The $lambda\sigma$-calculus is a refinement of the
    $lambda$-calculus where substitutions are manipulated
    explicitly. The $\lambda\sigma$-calculus provides a setting for
    studying the theory of substitutions, with pleasant mathematical
    properties. It is also a useful bridge between the classical
    $\lambda$-calculus and concrete implementations.",
  paper = "Abad90.pdf",
  keywords = "printed"


\index{Beeson, Michael}
  author = "Beeson, Michael",
  title = {{Formalizing Constructive Mathematics: Why and How?}},
  journal = "Lecture Notes in Mathematics",
  volume = "873",
  pages = "146-190",
  year = "1980",
  paper = "Bees80a.pdf"


\index{Goodman, Nicolas D.}
  author = "Goodman, Nicolas D.",
  title = {{Reflections on Bishop's Philosophy of Mathematics}},
  journal = "Lecture Notes in Mathematics",
  volume = "873",
  pages = "135-145",
  year = "1980",
  paper = "Good80.pdf",
  keywords = "printed"


\index{Rosenkranz, Markus}
  author = "Rosenkranz, Markus",
  title = {{Gr\"obner Bases in Symbolic Analysis}},
  publisher = "Walter de Gruyter, Berlin Germany",
  year = "2007",
  isbn = "978-3-11.019323-7",
  keywords = "axiomref, TPDref"


\index{Daly, Timothy}
  author = "Daly, Timothy",
  title = {{Literate Programming}},
  link = "\url{}",
  year = "2006",
  keywords = "axiomref, TPDref"


\index{Prengel, Alex}
  author = "Prengel, Alex",
  title = {{MIT axiom-math_v8.14}},
  link = "\url{}",
  year = "2016",
  keywords = "axiomref, TPDref"


\index{Jenks, Richard D.}
  author = "Jenks, Richard D.",
  title = {{The 2011 Richard D. Jenks Memorial Prize}},
  journal = "ACM Communications in Computer Algebra",
  volume = "45",
  number = "4",
  year = "2011",
  abstract =
    "The 2011 Richard D. Jenks Memorial Prize for Excellence in
    Software Engineering Applied to Computer Algebra was presented by
    members of the Prize Selection Committee and its chair, Erich
    Kaltofen, at ISSAC and SNC in San Jose, CA, on June 9, 2011 to the
    Maple Project at the University of Waterloo.",
  paper = "Jenk11.pdf",
  keywords = "axiomref"


\index{Jenks, Richard D.}
  author = "Jenks, Richard D.",
  title = {{The 2013 Richard D. Jenks Memorial Prize}},
  journal = "ACM Communications in Computer Algebra",
  volume = "47",
  number = "4",
  year = "2013",
  abstract =
    "The 2013 Richard D. Jenks Memorial Prize for Excellence in
    Software Engineering Applied to Computer Algebra was announced by
    members of the Prize Selection Committee, Mark Giesbrecht
    representing its chair Erich Kaltofen, at ISSAC in Boston, MA, on
    June 28, 2013 to have been awared to Professor William Arthur
    Stein of the Sage Project at the University of Washington.",
  paper = "Jenk13.pdf",
  keywords = "axiomref"


\index{Jenks, Richard D.}
  author = "Jenks, Richard D.",
  title = {{The 2015 Richard D. Jenks Memorial Prize}},
  journal = "ACM Communications in Computer Algebra",
  volume = "49",
  number = "4",
  year = "2015",
  abstract =
    "The 2015 Richard D. Jenks Memorial Prize was awarded on October
    30, 2015 at the Fields Institute in Toronto during the Major
    Thematic Program on Computer Algebra to Professor Victor Shoup for
    NTL: A Library for doing Number Theory.",
  paper = "Jenk15.pdf",
  keywords = "axiomref"


\index{Jenks, Richard D.}
  author = "Jenks, Richard D.",
  title = {{The 2017 Richard D. Jenks Memorial Prize}},
  journal = "ACM Communications in Computer Algebra",
  volume = "51",
  number = "4",
  year = "2017",
  abstract =
    "The 2017 Richard D. Jenks Memorial Prize for Excellence in
    Software Engineering Applied to Computer Algebra has been awarded
    to Stephen Wolfram for Wolfram-Alpha and Mathematica.",
  paper = "Jenk17.pdf",
  keywords = "axiomref"


\index{Lazard, Daniel}
  author = "Lazard, Daniel",
  title = {{Solving Systems of Algebraic Equations}},
  journal = "ACM SIGSAM Bulletin",
  volume = "35",
  number = "3",
  year = "2001",
  abstract =
    "Let $f_1,\ldots,f_k$ be $k$ k multivariate polynomials which have
    a finite number of common zeros in the algebraic closure of the ground
    field, counting the common zeros at infinity. An algorithm is
    given and proved which reduces the computations of these zeros to
    the resolution of a single univariate equation whose degree is the
    number of common zeros. This algorithm gives the whole algebraic
    and geometric structure of the set of zeros (multiplicities,
    conjugate zeros,...). When all the polynomials have the same degree,
    the complexity of this algorithm is polynomial relative to the generic
    number of solutions.",
  paper = "Laza01.pdf"


\index{Abramov, S.A.}
  author = "Abramov, S.A.",
  title = {{A Note on the Number of Division Steps in the Euclidean
  journal = "ACM SIGSAM",
  volume = "34",
  number = "4",
  year = "2000",
  paper = "Abra00.pdf",
  keywords = "printed"


\index{Essex, Christopher}
\index{Davison, Matt}
\index{Schulzky, Christian}
  author = "Essex, Christopher and Davison, Matt and Schulzky, Christian",
  title = {{Numerical Monsters}},
  journal = "ACM SIGSAM",
  volume = "34",
  number = "4",
  year = "2000",
  abstract =
    "When the results of certain computer calculations are shown to be not
    simply incorrect but dramatically incorrect, we have a powerful reason
    to be cautious about all computer-based calculations. In this paper we
    present a 'Rogue's Gallery' of simple calculations whose correct
    solutions are obvious to humans but whose numerical solutions are
    incorrect in pathological ways. We call these calculations, which can
    be guaranteed to wreak numerical mayhem across both software
    packages and hardware platforms, 'Numerical Monsters'. Our monsters
    can be used to provide deep insights into how computer calculations
    fail, and we use t h e m to engender appreciation for the subject of
    numerical analysis in our students. Although these monsters are based
    on well-understood numerical pathologies, even experienced numerical
    analysts will find surprises in their behaviour and ,can use the
    lessons they bring to become even better masters of their tools.",
  paper = "Esse00.pdf"


\index{Buchberger, Bruno}
  author = "Buchberger, Bruno",
  title = {{Computer Algebra: The End of Mathematics?}},
  journal = "ACM SIGSAM",
  volume = "36",
  number = "1",
  year = "2002",
  abstract =
    "Mathematical software systems, such as Mathematica, Maple, Derive,
    and so on, are substantially based on enormous advances in the area of
    mathematics known as Computer Algebra or Symbolic Mathematics. In
    fact, everything taught in high school and in the first semesters of a
    university mathematical education, is available in these systems 'at
    the touch of the button'. Will mathematics become unnecessary because
    of this? In the three sections of this essay, I answer this question
    for non-mathematicians, for mathematicians and for (future) students
    of mathematics.",
  paper = "Buch02.pdf"


\index{Barnett, Michael P.}
  author = "Barnett, Michael P.",
  title = {{Reasoning in Symbolic Computation}},
  journal = "ACM Communications in Symbolic Algebra",
  volume = "42",
  number = "1",
  year = "2008",
  abstract =
    "I discuss notations for some styles of mathematical reasoning that
    include analogy. These notations extend the conventions of the
    mathematica package mathscape that I reported recently in the
    Journal of Symbolic Computation. The paper introduces the reasoning
    objects that I call 'precursors' and 'consequences lists'.",
  paper = "Barn08.pdf",
  keywords = "printed"


\index{Lichtblau, Daniel}
  author = "Lichtblau, Daniel",
  title = {{Symbolic Definite (and Indefinite) Integration: Methods and
           Open Issues}},
  journal = "ACM Comm. in Computer Algebra",
  volume = "45",
  number = "1",
  year = "2011",
  link = "\url{}",
  abstract = "
    The computation of definite integrals presents one with a variety of
    choices. There are various methods such as Newton-Leibniz or Slater's
    convolution method. There are questions such as whether to split or
    merge sums, how to search for singularities on the path of
    integration, when to issue conditional results, how to assess
    (possibly conditional) convergence, and more. These various
    considerations moreover interact with one another in a multitude of
    ways. Herein we discuss these various issues and illustrate
    with examples.",
  paper = "Lich11.pdf"


\index{Zengler, Christoph}
\index{Kubler, Andreas}
\index{Kuchlin, Wolfgang}
  author = "Zengler, Christoph and Kubler, Andreas and Kuchlin, Wolfgang",
  title = {{New Approaches to Boolean Quantifier Elimination}},
  journal = "ACM Comm. in Computer Algebra",
  volume = "45",
  number = "2",
  year = "2011",
  abstract =
    "We present four different approaches for existential Boolean
    quantifier elimination, based on model enumeration, resolution,
    knowledge compilation with projection, and substitution. We point out
    possible applications in the area of verification and we present
    preliminary benchmark results of the different approaches.",
  paper = "Zeng11.pdf",
  keywords = "printed"


\index{Stoutemyer, David R.}
  author = "Stoutemyer, David R.",
  title = {{Ways to Implement Computer Algebra Compactly}},
  journal = "ACM Comm. in Computer Algebra",
  volume = "45",
  number = "4",
  year = "2011",
  abstract =
    "Computer algebra had to be implemented compactly to fit on early
    personal computers and hand-held calculators. Compact implementation
    is still important for portable hand-held devices. Also, compact
    implementation increases comprehensibility while decreasing develop-
    ment and maintenance time and cost, regardless of the platform. This
    article describes several ways to achieve compact implementations,
    \item Exploit evaluation followed by interpolation to avoid
    implementing a parser, such as in PicoMath
    \item Use contiguous storage as an expression stack to avoid garbage
    collection and pointer-space overhead, such as in Calculus Demon
    and TI-Math-Engine
    \item Use various techniques for saving code space for linked-storage
    representation of expressions and functions, such as muMath and Derive
  paper = "Stou11.pdf",
  keywords = "axiomref"


\index{Roche, Daniel S.}
  author = "Roche, Daniel S.",
  title = {{Chunky and Equal-Spaced Polynomial Multiplication}},
  journal = "J. Symbolic Computation",
  volume = "46",
  pages = "791-806",
  year = "2011",
  abstract =
    "Finding the product of two polynomials is an essential and basic
    problem in computer algebra. While most previous results have
    focused on the worst-case complexity, we instead employ the
    technique of adaptive analysis to give an improvement in many
    ‘‘easy’’ cases. We present two adaptive measures and methods
    for polynomial multiplication, and also show how to effectively
    combine them to gain both advantages. One useful feature of these
    algorithms is that they essentially provide a gradient between
    existing ‘‘sparse’’ and ‘‘dense’’ methods. We prove that these
    approaches provide significant improvements in many cases but
    in the worst case are still comparable to the fastest existing
  paper = "Roch11.pdf"


\index{Bailey, David H.}
\index{Borwein, Jonathan M.}
\index{Kaiser, Alexander D.}
  author = "Bailey, David H. and Borwein, Jonathan M. and
            Kaiser, Alexander D.",
  title = {{Automated Simplification of Large Symbolic Expressions}},
  journal = "J. Symbolic Computation",
  volume = "60",
  pages = "120-136",
  year = "2014",
  abstract =
    "We present a set of algorithms for automated simplification of
    symbolic constants of the form
    $\sum_i\alpha_i x_i$ with $\alpha_i$ rational and $x_i$
    complex. The included algorithms, called SimplifySum and
    implemented in Mathematica , remove redundant terms, attempt to make
    terms and the full expression real, and remove terms using repeated
    application of the multipair PSLQ integer relation detection
    algorithm. Also included are facilities for making substitutions
    according to user-specified identities. We illustrate this toolset by
    giving some real-world examples of its usage, including one, for
    instance, where the tool reduced a symbolic expression of
    approximately 100 000 characters in size enough to enable manual
    manipulation to one with just four simple terms.",
  paper = "Bail14.pdf"


\index{Bailey, David H.}
\index{Borwein, Jonathan M.}
  author = "Bailey, David H. and Borwein, Jonathan M.",
  title = {{High-precision Numerical Integration: Progress and Challenges}},
  journal = "J. Symbolic Computation",
  number = "46",
  pages = "741-754",
  year = "2011",
  abstract =
    "One of the most fruitful advances in the field of experimental
    mathematics has been the development of practical methods for very
    high-precision numerical integration, a quest initiated by Keith
    Geddes and other researchers in the 1980s and 1990s.These
    techniques, when coupled with equally powerful integer relation
    detection methods, have resulted in the analytic evaluation of many
    integrals that previously were beyond the realm of symbolic
    techniques.This paper presents a survey of the current state-of-the-art
    in this area (including results by the present authors and others),
    mentions some new results, and then sketches what challenges lie
  paper = "Bail11.pdf"


\index{Huang, Zongyan}
\index{England, Matthew}
\index{Wilson, David}
\index{Davenport, James H.}
\index{Paulson, Lawrence C.}
  author = "Huang, Zongyan and England, Matthew and Wilson, David and
            Davenport, James H. and Paulson, Lawrence C.",
  title = {{A Comparison of Three Heuristics to Choose the Variable
            Ordering for Cylindrical Algebraic Decomposition}},
  journal = "ACM Comm. in Computer Algebra",
  volume = "48",
  number = "3",
  year = "2014",
  abstraact =
    "Cylindrical algebraic decomposition (CAD) is a key tool for problems
    in real algebraic geometry and beyond. When using CAD there is often a
    choice over the variable ordering to use, with some problems
    infeasible in one ordering but simple in another. Here we discuss a
    recent experiment comparing three heuristics for making this choice on
    thousands of examples.",
  paper = "Huan14.pdf"


\index{Hamada, Tatsuyoshi}
  author = "Hamada, Tatsuyoshi",
  title = {{MathLibre: Personalizable Computer Environment for
            Mathematical Research}},
  volume = "48",
  number = "3",
  year = "2014",
  abstract =
    "MathLibre is a project to archive free mathematical software and free
    mathematical documents and offer them on Live Linux system. MathLibre
    Project is the direct descendant of KNOPPIX/Math Project. It provides
    a desktop for mathematicians that can be set up easily and quickly.",
  paper = "Hama14.pdf"


\index{Fateman, Richard}
  author = "Fateman, Richard",
  title = {{Algorithm Differentiation in Lisp: ADIL}},
  journal = "ACM Comm. in Computer Algebra",
  volume = "48",
  number = "3",
  year = "2014",
  abstract =
    "Algorithm differentiation (AD) is a technique used to transform a
    program $F$ computing a numerical function of one argument $F(x)$ into
    another program $G(p)$ that returns a pair,
    $\langle F (p), F^\prime(p)\rangle$ where by $F^\prime(p)$
    we mean the derivative of $F$ with respect to its argument $x$,
    evaluated at $x = p$. That is, we have a program AD that takes as input
    a program, and returns another: $G := AD(F)$. Over the years AD
    programs have been developed to allow $F$ to be expressed in some
    specialized variant of a popular programming language L (FORTRAN, C,
    Matlab, Python) and where $G$ is delivered in that language $L$ or some
    other. Alternatively, executing $F(p)$ is some environment will deliver
    $\langle F^\prime(p), F(p)\rangle$ directly. AD tools have also
    been incorporated in computer algebra systems (CAS) such as
    Maple. A CAS is hardly necessary for the task of writing the AD
    program, since the main requirement is a set of tools for
    manipulation of an internal (typically tree) form of a program.
    In Lisp, a normal program is already in this form and so the AD
    program in Lisp (ADIL), the target $F$ and the product $G$ can
    all be expressed compactly in Lisp. In spite of the brevity and
    extensibility of the ADIL program, we can provide
    features which are unsupported in other AD programs. In particular,
    recursive functions are easily accommodated. Our perspective here is
    to point out that for scientists who write programs in Lisp or any
    language that can be converted to Lisp, AD is easily at hand.",
  paper = "Fate14a.pdf"


\index{England, M.}
\index{Cheb-Terrab, E.}
\index{Bradford, R.}
\index{Davenport, J.H.}
\index{Wilson, D.}
  author = "England, M. and Cheb-Terrab, E. and Bradford, R. and
            Davenport, J.H. and Wilson, D.",
  title = {{Branch Cuts in MAPLE 17}},
  journal = "ACM Comm. in Computer Algebra",
  volume = "48",
  number = "1",
  year = "2014",
  abstract =
    "Accurate and comprehensible knowledge about the position of branch
    cuts is essential for correctly working with multi-valued functions,
    such as the square root and logarithm. We discuss the new tools in
    Maple 17 for calculating and visualising the branch cuts of such
    functions, and others built up from them. The cuts are described in an
    intuitive and accurate form, offering substantial improvement on the
    descriptions previously available.",
  paper = "Engl14c.pdf"


\index{Stoutemyer, David R.}
  author = "Stoutemyer, David R.",
  title = {{A Computer Algebra User Interface Manifesto}},
  journal = "ACM Communications in Computer Algebra",
  volume = "47",
  number = "4",
  year = "2013",
  abstract =
    "Many computer algebra systems have more than 1000 built-in functions,
    making expertise difficult. Using mock dialog boxes, this article
    describes a proposed interactive general-purpose wizard for organizing
    optional transformations and allowing easy fine grain control over the
    form of the result – even by amateurs. This wizard integrates ideas
    \item flexible subexpression selection;
    \item complete control over the ordering of variables and
    commutative operands, with well-chosen defaults;
    \item interleaving the choice of successively less main
    variables with applicable function choices to provide detailed
    control without incurring a combinatorial number of applicable
    alternatives at any one level;
    \item quick applicability tests to reduce the listing of
    inapplicable transformations;
    \item using an organizing principle to order the alternatives
    in a helpful manner;
    \item  labeling quickly-computed alternatives in dialog boxes
    with a preview of their results, using ellipsis elisions if
    necessary or helpful;
    \item allowing the user to retreat from a sequence of choices
    to explore other branches of the tree of alternatives – or to
    return quickly to branches already visited;
    \item allowing the user to accumulate more than one of the
    alternative forms;
    \item integrating direct manipulation into the wizard; and
    \item supporting not only the usual input-result pair mode, but
    also the useful alternative derivational and in situ replacement
    modes in a unified window.
  paper = "Stou13.pdf",
  kryeotfd = "printed"


\index{Sorenson, Jonathan}
  author = "Sorenson, Jonathan",
  title = {{Two Fast GCD Algorithms}},
  journal = "Journal of Algorithms",
  volume = "16",
  pages = "110-144",
  year = "1994",
  abstract =
    "We present two new algorithms for computing geatest common
    divisors: the right- and left-shift $k$-ary GCD algorithms. These
    algorithms are generalizations of the binary and left-shift binary
    algorithms. Interestingly, both new algorithms have sequential
    versions that are practical and efficient and parallel versions
    that rival the best previous parallel GCD algorithms. We show that
    sequential versions of both algorithms take $O(n^2/log~n)$ bit
    operations in the worst case to compute the GCD of two n-bit
    integers. This compares favorably to the Euclidean and moth
    precision versions of these GCD algorithms, and we found that both
    $k$-ary algorithms are faster than the Euclidean and binary
    algorithms on inputs of 100 to 1000 decimal digits in length. We
    show that parallel versions of both algorithms match the
    complexity of the best previous parallel GCD algorithm due to Chor
    and Goldreich. Specifically, if $log~n\le k \le 2^n$ and $k$ is a
    power of two, then both algorithms run in $O(n/log~n+log^2~n)$
    time using a number of processors bounded by a polynomial in $n$
    and $k$ on a common CRCW PRAM. We also discuss extended versions
    of both algorithms.",
  paper = "Sore94.pdf",
  keywords = "printed"


\index{Weber, Kenneth}
\index{Trevisan, Vilmar}
\index{Martins, Luiz Felipe}
  author = "Weber, Kenneth and Trevisan, Vilmar and Martins, Luiz Felipe",
  title = {{A Modular Integer GCD Algorithm}},
  journal = "J. of Algorithms",
  volume = "54",
  year = "2005",
  pages = "152-167",
  abstract =
    "This paper describes the first algorithm to compute the greatest
    common divisor (GCD) of two $n$-bit integers using a modular
    representation for intermediate values $U$, $V$ and also for the
    result. It is based on a reduction step, similar to one used in
    the accelerated algorithm when $U$ and $V$ are close to the same
    size, that replaces $U$ by $(U-bV)/p$, where $p$ is one of the
    prime moduli and $b$ is the unique integer in the interval
    $(-p/2,p/2)$ such that $b\equiv UV^{-1} (mod~p)$. When the
    algorithm is executed on a bit common CRCW PRAM with
    $O(n~log~n~log~log~log~n)$ processors, it takes $(O(n)$ time in
    the worst case. A heuristic model of the average case yields
    $O(n/log~n)$ time on the same number of processors.",
  paper = "Webe05a.pdf",
  keywords = "printed"


\index{Schonhage, A.}
  author = "Schonhage, A.",
  title = {{Probabilistic Computation of Integer Polynomial GCDs}},
  journal = "J. of Algorithms",
  volume = "9",
  pages = "365-371",
  year = "1988",
  abstract =
    "We describe a probabilistic algorithm for the computation of the
    gcd of two univariate integer polynomials of degrees $\le n$ with
    their $l^1$-norms being bounded by $2^h$ and estimate its expected
    running time by a worst-case bound of
  paper = "Scho88.pdf",
  keywords = "printed"


\index{LeGuin, Ursula K.}
  author = "LeGuin, Ursula K.",
  title = {{The Left Hand of Darkness}},
  publisher = "Penguin Random House",
  year = "1976",
  isbn = "978-1-101-66539-8"


\index{Shulman, Michael}
  author = "Shulman, Michael",
  title = {{Linear Logic for Constructive Mathematics}},
  link = "\url{}",
  year = "2018",
  abstract =
    "We show that numerous distinctive concepts of constructive
    mathematics arise automatically from an interpretation of 'linear
    higher-order logic' into intuitionistic higher-order logic via a Chu
    construction.  This includes apartness relations, complemented
    subsets, anti-subgroups and anti-ideals, strict and non-strict order
    pairs, cut-valued metrics, and apartness spaces. We also explain the
    constructive bifurcation of classical concept s using the choice
    between multiplicative and additive linear connectives. Linear logic
    thus systematically 'constructivizes' classical definitions and deals
    automatically with the resulting bookkeeping, and could potentially
    be used directly as a basis for constructive mathematics in place
    of intuitionistic logic.",
  paper = "Shul18.pdf",
  keywords = "printed"


\index{Khan, Muhammad Taimoor}
  author = "Khan, Muhammad Taimoor",
  title = {{On the Formal Verification of Maple Programs}},
  year = "2013",
  type = "technical report",
  number = "13-07",
  institution = "RISC Linz",
  abstract =
    "In this paper, we present an example-based demonstration of our
    recent results on the formal verification of programs written in the
    computer algebra language (Mini)Maple (a slightly modified subset of
    Maple).  The main goal of this work is to develop a verification
    framework for behavioral analysis of MiniMaple programs.  For
    verification, we translate an annotated MiniMaple program into the
    language Why3ML of the intermediate verification tool Why3 developed
    at LRI, France.  We generate verification conditions by the
    corresponding component of Why3 and later prove the correctness of
    these conditions by various supported by the Why3 back-end automatic
    and interactive theorem provers.  We have used the verification
    framework to verify some parts of the main test example of our
    verification framework, the Maple package DifferenceDifferential
    developed at our institute to compute bivariate difference-differential
    polynomials using relative Gr ̈obner bases.",
  paper = "Khan13.pdf",
  keywords = "printed"


\index{Daly, Timothy}
  author = "Daly, Timothy",
  title = {{Axiom Website:}},
  link = "\url{}",
  keywords = "axiomref",
  year = "2003"


\index{Dijkstra, Edsgar W.}
  author = "Dijkstra, Edsgar W.",
  title = {{Guarded Commands, Nondeterminancy and Formal Derivation
            of Programs}},
  journal = "Comm. of the ACM",
  volume = "18",
  number = "8",
  year = "1975",
  pages = "453-457",
  abstract =
    "So-called 'guarded commands' are introduced as a building block
    for alternative and repetitive constructs that allow
    nondeterministic program components for which at least the
    activity evoked, but possibly even the final state, is not
    necessarily uniquly determined by the initial state. For the
    formal derivation of programs expressed in terms of these
    constructs, a calculus will be shown.",
  paper = "Dijk75.pdf",
  keywords = "printed"


\index{van Emden, M.H.}
\index{Kowalski, R.A.}
  author = "van Emden, M.H. and Kowalski, R.A.",
  title = {{The Semantics of Predicate Logic as a Programming Language}},
  journal = "J. Association for Computing Machinery",
  volume = "23",
  number = "4",
  year = "1976",
  pages = "733-742",
  abstract =
    "Sentences in first-order predicate logic can be usefully interpreted
    as programs In this paper the operational and fixpomt semantics of
    predicate logic programs are defined, and the connections with the
    proof theory and model theory of logic are investigated It is
    concluded that operational semantics is a part of proof theory and
    that fixpolnt semantics is a special case of model-theoret:c
  paper = "Emde76.pdf",
  keywords = "printed"


\index{Tarski, Alfred}
  author = "Tarski, Alfred",
  title = {{Truth and Proof}},
  journal = "Scientific American",
  volume = "1969",
  pages = "63-77",
  year = "1969",
  paper = "Tars69.pdf",
  keywords = "printed"


\index{Apt, Krysztof R.}
\index{Bezem, Marc}
  author = "Apt, Krysztof R. and Bezem, Marc",
  title = {{Formulas as Programs}},
  booktitle = "The Logic Programming Paradigm",
  year = "1999",
  publisher = "Springer Berlin Heidelberg",
  pages = "75-107",
  isbn = "978-3-642-60085-2",
  abstract =
    "We provide here a computational interpretation of first-order
    logic based on a constructive interpretation of satisfiability
    w.r.t. a fixed but arbitrary interpretation. In this approach the
    formulas themselves are programs. This contrasts with the
    so-called formulas as types approach in which the proofs of the
    formulas are typed terms that can be taken as programs. This view
    of computing is inspired by logic programming and constraint logic
    programming but differs from them in a number of crucial aspects.

    Formulas as programs is argued to yield a realistic approach to
    programming that has been realized in the implemented programming
    language Alma-0 Apt, Brunekreef, Partington and Schaerf (1998)
    that combines the advantages of imperative and logic
    programming. The work here reported can also be used to reason
    about the correctness of non-recursive Alma-0 programs that do not
    include destructive assignment.",
  paper = "Aptx99.pdf",
  keywords = "printed"


\index{Mazur, Barry}
  author = "Mazur, Barry",
  title = {{When is One Thing Equal to Some Other Thing?}},
  link = "\url{}",
  year = "2007",
  paper = "Mazu07.pdf"


\index{Kant, Immanuel}
  author = "Kant, Immanuel",
  title = {{The Critique of Pure Reason}},
  publisher = "Cambridge University Press",
  year = "1998",
  isbn = "0-521-35402-1",
  link = "\url{}",
  paper = "Kant98.pdf"


\index{Smith, A.}
  author = "Smith, A.",
  title = {{The Knuth-Bendix Completion Algorithm and its Specification in Z}},
  type = "technical report",
  institution = "Ministry of Defence, RSRE Malvern WORCS",
  year = "1990",
  number = "RSRE Memorandum 4323",
  abstract =
    "Proving that something is a consequence of a set of axioms can be
    very difficult. The Knuth-Bendix completion algorithm attempts to
    automate this process when the axioms are equations. The algorithm
    is bound up in the area of term rewriting, and so this memorandum
    contains an introduction to the tehory of term rewriting, followed
    by an overview of the algorithm. Finally a formal specification of
    the algorithm is given using the language Z.",
  paper = "Smit90.pdf",
  keywords = "printed"


\index{Dick, A.J.J.}
  author = "Dick, A.J.J.",
  title = {{Automated Equational Reasoning and the Knuth-Bendix Algorithm:
            An Informal Introduction}},
  year = "1988",
  type = "technical report",
  institution = "Rutherford Appelton Laboratory",
  number = "RAL-88-043",
  paper = "Dick88.pdf",
  keywords = "printed"


\index{Dick, Jeremy}
  author = "Dick, Jeremy",
  title = {{An Introduction to Knuth-Bendix Completion}},
  journal = "The Computer Journal",
  volume = "34",
  number = "1",
  year = "1991",
  abstract =
    "An informal introduction is given to the underlying concepts of
    term rewriting. Topics covered are Knuth-Bendix completion,
    completion modulo equations, unfailing completion and theorem
    proving by completion.",
  paper = "Dick91.pdf",
  keywords = "printed"


\index{Knuth, Donald E.}
\index{Bendix, Peter B.}
  author = "Knuth, Donald E. and Bendix, Peter B.",
  title = {{Simple Word Problems in Unversal Algebras}},
  booktitle = "Computational Problems in Abstract Algebra",
  editor = "Leech, John",
  publisher = "Pergamon Press",
  isbn = "08-012975-7",
  year = "1970",
  pages = "263-298",
  paper = "Knut70.pdf",
  keywords = "printed"


\index{Leech, John}
  author = "Leech, John",
  title = {{Computational Problems in Abstract Algebra}},
  publisher = "Pergamon Press",
  year = "1970",
  isbn = "08-012975-7",
  paper = "Leec70.pdf"


\index{Dershowitz, Nachum}
  author = "Dershowitz, Nachum",
  title = {{Termination of Rewriting}},
  journal = "J. Symbolic Computation",
  volume = "3",
  year = "1987",
  pages = "69-116",
  abstract =
    "This survey describes methods for proving that systems of rewrite
    rules are terminating programs.  We illustrate the use in termination
    proofs of various kinds of orderings on terms, including polynomial
    interpretations and path orderings. The effect of restrictions, such
    as linearity, on the form of rules is also considered.  In general,
    however, termination is an undeeidable property of rewrite systems.",
  paper = "Ders87.pdf",
  keywords = "printed"


\index{Atkey, Robert}
  author = "Atkey, Robert",
  title = {{Parameterised Notions of Computation}},
  link = "\url{}",
  year = "2018",
  abstract =
    "Moggi’s Computational Monads and Power et al ’s equivalent notion of
    Freyd category have captured a large range of computational effects
    present in programming languages. Examples include non-termination,
    non-determinism, exceptions, continuations, side-effects and
    input/output. We present generalisations of both computational monads
    and Freyd categories, which we call parameterised monads and
    parameterised Freyd categories, that also capture computational
    effects with parameters. Examples of such are composable
    continuations, side-effects where the type of the state varies and
    input/output where the range of inputs and outputs varies. By also
    considering structured parameterisation, we extend the range of
    effects to cover separated side-effects and multiple independent
    streams of I/O. We also present two typed $\lambda$-calculi that
    soundly and completely model our categorical definitions — with and
    without symmetric monoidal parameterisation — and act as prototypical
    languages with parameterised effects.",
  paper = "Atke18.pdf",
  keywords = "printed"


\index{Bozman, Gerald}
\index{Buco, William}
\index{Daly, Timothy}
\index{Tetzlaff, William}
  author = "Bozman, Gerald and Buco, William and Daly, Timothy and
            Tetzlaff, William",
  title = {{Analysis of Free Storage Algorithms - Revisited}},
  journal = "IBM Systems Journal",
  volume = "23",
  number = "1",
  year = "1984",
  abstract =
    "Most research in free-storage management has centered around
    strategies that search a linked list and strategies that partition
    storage into predetermined sizes. Such algorithms are analyzed in
    terms of CPU efficiency and storage efficiency. The subject of this
    study is the free-storage management in the Virtual Machine/System
    Product (VM/SP) system control program. As a part of this study,
    simulations were done of established, and proposed, dynamic storage
    algorithms for the VM/SP operating system. Empirical evidence is given
    that simplifying statistical assumptions about the distribution of
    interarrival times and holding times has high predictive
    ability. Algorithms such as first-fit, modified first-fit, and
    best-fit are found to be CPU-inefficient. Buddy systems are found to
    be very fast but suffer from a high degree of internal
    fragmentation. A form of extended subpooling is shown to be as fast as
    buddy systems with improved storage efficiency. This algorithm was
    implemented for VM/SP, and then measured. Results for this algorithm
    are given for several production VM/SP systems.",
  paper = "Bozm84.pdf"

Latest commit 5054cfd Aug 12, 2018


"What matters the most is what you do for free" -- John Gorka

"...even more important, for the progress of mathematics in the computer
 age, is the beaver, who will build the needed infrastructure of computer
 mathematics, that would eventually enable us to solve many outstanding
 open problems, and many new ones. Consequently, the developers of computer
 algebra systems, and creators of algorithms, are even more important than
 both birds and frogs." --Doron Zeilberger

Listening to computer scientists argue, it seems that the standards of
proof is I've had two beers and there is this anecdote about a tribe
in New Guinea from one of Scott Birkins books that seems to be applicable.

The debate is hindered by low standards of proof.

-- Greg Wilson "What We Actually Know About Software Development"
When you teach, you do something useful.
When you do research, most days you don't.
-- June Huh IAS Princeton 

You've unpacked the Axiom source code to some directory. In this
document we'll call that directory /home/me/axiom. Note that the path
cannot contain spaces. 

================= MAKING AXIOM ========================================

Axiom builds a system-specific version based on a string we'll call
the SYSNAME. Currently recognized SYSNAME strings can be found on the
Axiom website at:

Replace SYSNAME below with the likely name of your system.

We also assume that you downloaded AXIOM to someplace. Suppose
that place is /home/me/axiom, then:

cd /home/me/axiom                         << where you unpacked the sources
export AXIOM=/home/me/axiom/mnt/SYSNAME   << which axiom to build
export PATH=$AXIOM/bin:$PATH
make                                      << build the system

A recent cause of likely build failures is SELinux.
See the faq file for possible solutions.

When the make completes you'll have an executable called $AXIOM/bin/axiom

================= INSTALLING AXIOM ====================================

You can install Axiom on your system by typing (as root):

make install

This will put Axiom into /usr/local/axiom 
and the axiom command in /usr/local/bin/axiom

You can change these defaults to anything thus:

make INSTALL=/home/me/myaxiom COMMAND=/home/me/bin/myaxiom install


Documentation can be found at various places in the system or on the
Axiom website: <>

There is a book (available on
Jenks, Richard D. and Sutor, Robert S. "Axiom, The Scientific Computation
System" Springer-Verlag, NY, 1992, ISBN 0-387-97855-0

The book is automatically built as part of the make and lives in:


In general every directory will contain a Makefile.dvi file.
These files document facts about how Axiom is built.
The directory mnt/linux/doc will contain .dvi files as they are written.

Axiom is free and open source software. It is copyrighted code that
is released under the Modified BSD license. Much debate about this
topic has already been archived on the axiom-legal and axiom-developer
mailing lists. The mail archives are available at the Axiom website:

For the purposes of copyright, Axiom is to be considered 
"Joint Work". Specifically:

 "A joint work is a work prepared by two or more individuals, with the
 intention that their separate contributions be merged into a single work.
 A joint author can also be an organization or a corporation under the
 definition of "work for hire." A person who has merely contributed ideas
 without actually documenting those ideas generally cannot be considered
 an author.

 Authors own the work jointly and equally, unless the authors make an
 agreement otherwise. Each joint author has the right to exercise any or all 
 of the exclusive rights inherent in the joint work. Each author may:
   * Grant thirds parties permission to use the work on a nonexclusive
     basis without the consent of the other joint authors
   * Transfer his or her entire ownership interest to another person without
     the other joint author's consent
   * Update the work for his or her own purpose

 Additionally, each joint author must account to the other joint authors
 for any profits received from licensing their joint work."


Questions and comments should be sent to:

Tim Daly



Scratchpad was a large, general purpose computer algebra system that
was originally developed by IBM under the direction of Richard Jenks.
The project started in 1971 and evolved slowly.  Barry Trager was key
to the technical direction of the project. Scratchpad developed over a
20 year stretch and was basically considered as a research platform
for developing new ideas in computational mathematics. In the 1990s,
as IBM's fortunes slid, the Scratchpad project was renamed to Axiom,
sold to the Numerical Algorithms Group (NAG) in England and became a
commercial system.  As part of the Scratchpad project at IBM in
Yorktown I worked on all aspects of the system and eventually helped
transfer the product to NAG. For a variety of reasons it never became
a financial success and NAG withdrew it from the market in October,

NAG agreed to release Axiom as free software. The basic motivation was
that Axiom represents something different from other programs in a lot
of ways. Primarily because of its foundation in mathematics the Axiom
system will potentially be useful 30 years from now.  In its current
state it represents about 30 years and 300 man-years of research
work. To strive to keep such a large collection of knowledge alive
seems a worthwhile goal.

However, keeping Axiom alive means more than just taking the source
code and dumping it onto a public server. There are a lot of things
about the system that need to change if it is going to survive and
thrive for the next 30 years.

The system is complex and difficult to build. There are few people who
know how it is structured and why it is structured that way. Somehow
it needs to be documented deeply so others can contribute.

The mathematics is difficult. Unlike other free software you can't
just reach for the old saying ``read the source code''.  The source
code is plain, clear and about as close to the mathematical theory as
is practical. Unfortunately the mathematical theory is enshrined in
some research library where few people will have access. Somehow this
must change. The research work, the mathematics, the published papers,
and the source code have all got to be kept together for the next
generation to read, understand and modify.

The mathematics is narrow and very focused. This was due to the fact
that, while Axiom is a great research platform, we only had a limited
number of visitors at IBM Research. So there is very little in the way
of, say, infinite group theory in Axiom. We can add it. Doing so will
show up shortcomings in the system.  For example, how do you represent
an infinite object? There are many possible representations and they
depend on your goals. The system will have to change, expand, and,
hopefully, become cleaner as more thought is applied. Scratchpad
changed continuously while it was being used for research and we
expect Axiom to do the same.

The language (spad) is designed to let you write algorithms that are
very close to the mathematics. However, the algorithms as presented in
the current system have never been shown or proven (an important
distinction) to be correct.  It is vital that we undertake the huge
effort of verifying and validating the code. How else can we trust the
results and of what use is a system this complex without trust?
Somehow we have to extend the system to integrate program proof
techniques.  That is, we have to make computational mathematics hold
to the same standard as the rest of mathematics.

All of which seems to integrate into a requirement for better
documentation. The key change which developers of Axiom will find with
this version is that the documentation is primary and the code is
secondary. Taking direction from Knuth and Dijkstra the system is now
in a literate programming style. The hope is that the next generation
of developers and users will be able to understand, maintain and
extend the system gracefully. And that eventually papers submitted to
journals (an Axiom Journal?) will be easily imported into the system
with their running code made available automatically.

There is no guarantee that this attempt to change the culture of
computational mathematicians is going to succeed. But it is our firm
belief that current systems have reached a complexity plateau and we
need to find new techniques to push the envelope.

In general, we need to consider changes to the system with a 30 year
horizon rather than the current write-ship-debug mentality of software
development. This is, after all, mathematics, the queen of the
sciences. It deserves all of the time, talent and attention we can
bring to bear on the subject.

Tim Daly -- September 3, 2002


All of these people have, in some way or other, been associated with
Scratchpad and Axiom. If you contribute, add your name.  The names are
in alphabetical order as we make no attempt to quantify the relative
merit of the contributions.

In books/bookvol5.pamphlet is a variable called credits
which contains this list. Typing 
at the axiom command prompt will prettyprint the list.

"An alphabetical listing of contributors to AXIOM:"
"Michael Albaugh        Cyril Alberga          Roy Adler"
"Christian Aistleitner  Richard Anderson       George Andrews"
"S.J. Atkins            Jeremy Avigad          Henry Baker"
"Martin Baker           Stephen Balzac         Yurij Baransky"
"David R. Barton        Thomas Baruchel        Gerald Baumgartner"
"Gilbert Baumslag       Michael Becker         Nelson H. F. Beebe"
"Jay Belanger           David Bindel           Fred Blair"
"Vladimir Bondarenko    Mark Botch             Raoul Bourquin"
"Alexandre Bouyer       Karen Braman           Wolfgang Brehm"
"Peter A. Broadbery     Martin Brock           Manuel Bronstein"
"Christopher Brown      Stephen Buchwald       Florian Bundschuh"
"Luanne Burns           William Burge          Ralph Byers"
"Quentin Carpent        Pierre Casteran        Robert Cavines"
"Bruce Char             Ondrej Certik          Tzu-Yi Chen"
"Bobby Cheng            Cheekai Chin           David V. Chudnovsky"
"Gregory V. Chudnovsky  Mark Clements          James Cloos"
"Jia Zhao Cong          Josh Cohen             Christophe Conil"
"Don Coppersmith        George Corliss         Robert Corless"
"Gary Cornell           Meino Cramer           Jeremy Du Croz"
"David Cyganski         Nathaniel Daly         Timothy Daly Sr."
"Timothy Daly Jr.       James H. Davenport     David Day"
"James Demmel           Didier Deshommes       Michael Dewar"
"Inderjit Dhillon       Jack Dongarra          Jean Della Dora"
"Gabriel Dos Reis       Claire DiCrescendo     Sam Dooley"
"Zlatko Drmac           Lionel Ducos           Iain Duff"
"Lee Duhem              Martin Dunstan         Brian Dupee"
"Dominique Duval        Robert Edwards         Heow Eide-Goodman"
"Lars Erickson          Mark Fahey             Richard Fateman"
"Bertfried Fauser       Stuart Feldman         John Fletcher"
"Brian Ford             Albrecht Fortenbacher  George Frances"
"Constantine Frangos    Timothy Freeman        Korrinn Fu"
"Marc Gaetano           Rudiger Gebauer        Van de Geijn"
"Kathy Gerber           Patricia Gianni        Gustavo Goertkin"
"Samantha Goldrich      Holger Gollan          Teresa Gomez-Diaz"
"Laureano Gonzalez-Vega Stephen Gortler        Johannes Grabmeier"
"Matt Grayson           Klaus Ebbe Grue        James Griesmer"
"Vladimir Grinberg      Oswald Gschnitzer      Ming Gu"
"Jocelyn Guidry         Gaetan Hache           Steve Hague"
"Satoshi Hamaguchi      Sven Hammarling        Mike Hansen"
"Richard Hanson         Richard Harke          Bill Hart"
"Vilya Harvey           Martin Hassner         Arthur S. Hathaway"
"Dan Hatton             Waldek Hebisch         Karl Hegbloom"
"Ralf Hemmecke          Henderson              Antoine Hersen"
"Nicholas J. Higham     Hoon Hong              Roger House"
"Gernot Hueber          Pietro Iglio           Alejandro Jakubi"
"Richard Jenks          Bo Kagstrom            William Kahan"
"Kyriakos Kalorkoti     Kai Kaminski           Grant Keady"
"Wilfrid Kendall        Tony Kennedy           David Kincaid"
"Keshav Kini            Ted Kosan              Paul Kosinski"
"Igor Kozachenko        Fred Krogh             Klaus Kusche"
"Bernhard Kutzler       Tim Lahey              Larry Lambe"
"Kaj Laurson            Charles Lawson         George L. Legendre"
"Franz Lehner           Frederic Lehobey       Michel Levaud"
"Howard Levy            J. Lewis               Ren-Cang Li"
"Rudiger Loos           Craig Lucas            Michael Lucks"
"Richard Luczak         Camm Maguire           Francois Maltey"
"Osni Marques           Alasdair McAndrew      Bob McElrath"
"Michael McGettrick     Edi Meier              Ian Meikle"
"David Mentre           Victor S. Miller       Gerard Milmeister"
"Mohammed Mobarak       H. Michael Moeller     Michael Monagan"
"Marc Moreno-Maza       Scott Morrison         Joel Moses"
"Mark Murray            William Naylor         Patrice Naudin"
"C. Andrew Neff         John Nelder            Godfrey Nolan"
"Arthur Norman          Jinzhong Niu           Michael O'Connor"
"Summat Oemrawsingh     Kostas Oikonomou       Humberto Ortiz-Zuazaga"
"Julian A. Padget       Bill Page              David Parnas"
"Susan Pelzel           Michel Petitot         Didier Pinchon"
"Ayal Pinkus            Frederick H. Pitts     Frank Pfenning"
"Jose Alfredo Portes    E. Quintana-Orti       Gregorio Quintana-Orti"
"Beresford Parlett      A. Petitet             Andre Platzer"
"Peter Poromaas         Claude Quitte          Arthur C. Ralfs"
"Norman Ramsey          Anatoly Raportirenko   Guilherme Reis"
"Huan Ren               Albert D. Rich         Michael Richardson"
"Jason Riedy            Renaud Rioboo          Jean Rivlin"
"Nicolas Robidoux       Simon Robinson         Raymond Rogers"
"Michael Rothstein      Martin Rubey           Jeff Rutter"
"Philip Santas          Alfred Scheerhorn      William Schelter"
"Gerhard Schneider      Martin Schoenert       Marshall Schor"
"Frithjof Schulze       Fritz Schwarz          Steven Segletes"
"V. Sima                Nick Simicich          William Sit"
"Elena Smirnova         Jacob Nyffeler Smith   Matthieu Sozeau"
"Ken Stanley            Jonathan Steinbach     Fabio Stumbo"
"Christine Sundaresan   Klaus Sutner           Robert Sutor"
"Moss E. Sweedler       Eugene Surowitz        Max Tegmark"
"T. Doug Telford        James Thatcher         Laurent Thery"
"Balbir Thomas          Mike Thomas            Dylan Thurston"
"Francoise Tisseur      Steve Toleque          Raymond Toy"
"Barry Trager           Themos T. Tsikas       Gregory Vanuxem"
"Kresimir Veselic       Christof Voemel        Bernhard Wall"
"Stephen Watt           Andreas Weber          Jaap Weel"
"Juergen Weiss          M. Weller              Mark Wegman"
"James Wen              Thorsten Werther       Michael Wester"
"R. Clint Whaley        James T. Wheeler       John M. Wiley"
"Berhard Will           Clifton J. Williamson  Stephen Wilson"
"Shmuel Winograd        Robert Wisbauer        Sandra Wityak"
"Waldemar Wiwianka      Knut Wolf              Yanyang Xiao"
"Liu Xiaojun            Clifford Yapp          David Yun"
"Qian Yun               Vadim Zhytnikov        Richard Zippel"
"Evelyn Zoernack        Bruno Zuercher         Dan Zwillinger"

Pervasive Literate Programming

I think David Diamond said it best (Datamation, June 1976, pg 134):

The fellow who designed it is working far away.
The spec's not been updated for many a livelong day.
The guy who implemented it is promoted up the line.
And some of the enhancements didn't match to the design.
They haven't kept the flowcharts, the manual's a mess.
And most of what you need to know, you'll simply have to guess.

and with respect to Axiom:

The research that it's based on is no longer to be had.
And the theory that it's based on has changed by just a tad.
If we keep it all together then at least there is a hope.
That the people who maintain it will have a chance to cope.

To quote Fred Brooks, "The Mythical Man-month"

  "A basic principle of data processing teaches the folly of trying to
   maintain independent files in synchronization... Yet our practice in
   programming documentation violates our own teaching. We typically
   attempt to maintain a machine-readable form of a program and an
   independent set of human-readable documentation, consisting of prose
   and flowcharts ... The solution, I think, is to merge the files, to
   incorporate the documentation in the source program."

   "A common fallacy is to assume authors of incomprehensilbe code
    will somehow be able to express themselves lucidly and clearly
    in comments." -- Kevlin Henney

   "A programmer who cannot explain their ideas clearly in natural
    language is incapable of writing readable code." -- Tim Daly

As you can already see from this document the whole of the Axiom
effort is structured around literate programs. Every directory has a
Makefile.pamphlet file which explains details of that directory. The
whole source tree hangs from the Makefile tree. (Some of the
Makefile.pamphlet files contain only text if executable code is not
needed).  Every source file is embedded in a pamphlet file.

Which begs the question: ``What is a pamphlet file?''.  Basically it
is a tex document with some additional markup tags that surround
source code. Pamphlet files are intended to document one particular
subject. Pamphlet files can be later combined into ``booklet'' files
as one would embed chapters into books.

Which begs the question: ``Why bother with pamphlet files?''.  Clearly
you didn't read the philosophy rant above. In more detail there have
been two traditional methods of documenting source code. The first is
to sprinkle random comments into the code.  This suffers from the
problem that the comments assume you already understand the purpose of
the code and why an enlightened comment like ``This oughta work'' is
perfectly clear and compelling. The second method is to write a
document as a separate file. They get written half-heartedly because
the lack of source code allows you to skip over explaining ugly
implementation details (where all of the real confusion lies). This
form of documentation never gets updated and gradually becomes
uninteresting history.

Pamphlet files overcome neither of these limitations if you don't make
the effort to do it right. Ask yourself the question ``What would
Knuth do?'' or ``Will this be clear 30 years from now?''.

Which begs the question: ``Why go to all this trouble?''.  Because
you're having a conversation with people who are far removed from you
in time, space, and understanding.  Because someone else will have to
maintain your code.  Because you are part of a community of
mathematicians who hold you to high standards. Because if you can't
explain it clearly maybe YOU don't understand it or it isn't as clear
as you think it is.

Lets imagine that we'd like to receive a pamphlet file from a
colleague. It contains a new theory and spiffy new algorithm.  We'd
like to be able to put the pamphlet file into the system and have
everything magically happen to integrate the new pamphlet into the
system. What would that imply? Well, lets assume that the pamphlet
file has certain required sections. We'd like to get more than the
technical paper and the code. We'd also like to see the help
information, test cases, example code, cross-references to other
pamphlets that would get automatically included, have the proof
available and automatically checkable, etc. In the best of all
possible worlds we have a front-end system that knows nothing except
how to deconstruct and integrate properly formed pamphlet files. If
this were true we could be sure that all of the mathematics is
documented and external to the system. There are no ``rabbits'' (as
Dijkstra called surprises or special knowledge) that we pull out of
our hat. Conceptually, given an underlying Lisp system, it is clear we
can built such a system.

The General Directory Structure

The Top Level directory structure contains 7 directories which are
explained in detail below. Three of the directories (license, zips,
and lsp) are not part of the essential core of Axiom.

The other four directories (src, int, obj, and mnt) comprise the
system build environment. Each directory has a specific purpose. Lets
look at the essential directories first.


The src directory consists of human-written, system-independent
code. You can copy this directory (and the top-level makefiles) and
have a complete build system. Nothing in this directory is ever
changed by the Makefiles and it can be mounted in a read-only fashion
during the build process. An example file would be the lisp source

The int directory consists of machine-generated, system-independent
code. Consider this directory as a cache. Nothing in this directory is
necessary for a clean system build but once the build completes the
information in this directory can significantly shorten rebuilds.
Since this information is system-independent we can use the cache no
matter what architecture we target. An example file would be the dvi
files generated from the tex sources.

The obj directory consists of machine-generated, system-dependent
code. This directory is "scratch" space for the compiler and other
tools. Nothing in this directory is useful once the system is
built. An example file would be the .o files from the C compiler.

The mnt directory consists of machine-generated, system-dependent code
that will comprise the "shipped system". You can copy this directory
and have a complete, running Axiom system. If the end user will see it
or need it in any way it belongs here. Executables are generally built
in obj and moved here. Example files would be the final executable
images, the input files, etc.

The four directories above make it possible to do a system build for
one system (say, a linux system) which will fill in the int
subdirectory. Then you can NFS mount the src and int directories
read-only on a Solaris machine and do a solaris system build.  The
original Axiom could build across many different architectures,
compilers, and operating systems.

The license directory

The license directory collects all of the licenses for material that
is included in this distribution. Source files contain a line that
refers to one of these license files. Some people are of the belief
that including the full license text in every source file magically
strengthens the license but this is not so.  Imagine including the
full text of the copyright at the beginning of every section of a book.

The LICENSE.AXIOM file is a Modified BSD-style license that covers all
of the files released as part of this distribution except as noted in
particular files. Copyright information that might have shown up in
source files that were released from NAG are also collected here and
noted at the top of the files they cover.

The zips directory

The zips directory contains particular distributions of
network-available software that is known to work with this release of
Axiom. Newer versions may work and, if so, should be added to this
directory. The makefiles that handle these files will unpack them into
the correct locations in the src directory tree. These files exist to
increase the stability of the distribution so we can guarantee that
the code works. We encourage testing the latest distributions so that
we can remain with the leading edge and give feedback to the
individual package developers if problems arise.

The lsp directory

Axiom lives on top of Common Lisp, specifically Gnu Common Lisp (GCL)

Steps to build Axiom

The Initial Distribution files

The initial distribution contains several top level files. These are:

1) Makefile.pamphlet
     This is the noweb source for the Makefile file. All changes to
     the Makefile should occur here and the

3) Makefile This is the actual Makefile that will create Axiom. 

In general the distribution will contain the pamphlet files for each
source file and the source file itself. Modifications should be made
and explained in the pamphlet files. The document command should be
run to rebuild the source file and the dvi file.

Steps in the build process

The sequence of steps necessary to build a clean Axiom is simply:

  export AXIOM=(path-including-this-directory)/mnt/SYSNAME
  export PATH=$AXIOM/bin:$PATH

If this fails check the FAQ for possible problems and their fixes.