Advertisement

Symbolic Computation: Computer Algebra and Logic

Chapter
Part of the Applied Logic Series book series (APLS, volume 3)

Abstract

In this paper we present our personal view of what should be the next step in the development of symbolic computation systems. The main point is that future systems should integrate the power of algebra and logic. We identify four gaps between the future ideal and the systems available at present: the logic, the syntax, the mathematics, and the prover gap, respectively. We discuss higher order logic without extensionality and with set theory as a subtheory as a logic frame for future systems and we propose to start from existing computer algebra systems and proceed by adding new facilities for closing the syntax, mathematics, and the prover gaps. Mathematica seems to be a particularly suitable candidate for such an approach. As the main technique for structuring mathematical knowledge, mathematical methods (including algorithms), and also mathematical proofs, we underline the practical importance of functors and show how they can be naturally embedded into Mathematica.

Keywords

Computer Algebra Symbolic Computation Computer Algebra System High Order Logic Induction Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P.B. Andrews: An Introduction toMathematical Logic and Type Theory: To Truth Through Proof.Academic Press, London 1986.zbMATHGoogle Scholar
  2. B. Buchberger: Groebner Bases: An Algorithmic Method in Polynomial Ideal Theory. Chapter 6 in:Multidimensional Systems Theory, (N.K. Bose ed.). D. Reidel Publishing Company, Dordrecht, 1985.Google Scholar
  3. B. Buchberger: Induction Proofs in Equational Logic: A Case Study Using Mathematica.Internal Technical Report, The RISC Institute, A4232 Schloss Hagenberg, Austria, 1995.Google Scholar
  4. G.E. Collins: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition. Proceedings of the Second GI Conference on Automata Theory and Formal Languages.Lecture Notes in Computer Science, 33, pp. 515–532, Springer, Heidelberg, 1975.Google Scholar
  5. R. Constable:The Nuprl System.Lecture Notes of the Summer School on Logic of Computation, Marktoberndorf 1995. Edited by Institut für Informatik, Technische Universität München, 1995.Google Scholar
  6. G. Huet: The CIC System. Lecture Notes of the Summer School onLogic of Computation, Maxktoberndorf 1995. Edited by Institut für Informatik, Technische Universität München, 1995.Google Scholar
  7. D. Kapur, P. Narendran, H. Zhang: Automating Inductionless Induction using Test SetsJournal of Symbolic Computation, Vol. 11, No. 1&2, February 1991, pp. 83–111.CrossRefzbMATHMathSciNetGoogle Scholar
  8. N. Soiffer:Mathematical Typesetting in Mathematica.Proceedings of the ISSAC 1995 Conference, pp. 140–149.Google Scholar
  9. S. Wolfram: Mathematica:A System for Doing Mathematics by Computers.Addison- Wesley Publishing Company, Redwood, 1988.zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  1. 1.Research Institute for Symbolic ComputationLinzAustria

Personalised recommendations