Abstract
The most effective theorem proving systems (such as PVS, Acl2, and HOL) provide a kind of two-level reasoning, where the knowledge of a given domain is treated by a special purpose reasoner and a generic reasoning module is used for the actual problem specification. To obtain an effective integration between these two levels of reasoning is far from being a trivial task. In this paper, we propose a combination of Window Reasoning and Constraint Contextual Rewriting to achieve an effective integration of such levels. The former supports hierarchical reasoning for arbitrarily complex expressions. The latter provides the necessary theorem proving support for domain specific reasoning. The two levels of reasoning cooperate by building and exploiting a context, i.e. a set of facts which can be assumed true while transforming a given subexpression. We also argue that the proposed combination schema can be useful for building sound simplifiers to be used in computer algebra systems.
This work is partly supported by the European Union “CALCULEMUS Project” (HPRN-CT-2000-00102). The author would like to thank A. Armando and M. Rusinowitch for many helpful discussions on issues related to this paper. The anonymous referees helped improving the paper.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
A. Armando and C. Ballarin. Maple’s Evaluation Process as Constraint Contextual Rewriting. In Proc. of the 2001 Int. Symp. on Symbolic and Algebraic Computation (ISSAC-01), pages 32–37, New York, July 22–25 2001. ACMPress.
A. Armando, L. Compagna, and S. Ranise. RDL—Rewrite and Decision procedure Laboratory. In Proceedings of the International Joint Conference on Automated Reasoning (IJCAR 2001), Siena, Italy, June 2001.
A. Armando and S. Ranise. Constraint Contextual Rewriting. In Proc. of the 2nd Intl. Workshop on First Order Theorem Proving (FTP’98), 1998.
A. Armando and S. Ranise. Termination of Constraint Contextual Rewriting. In Proc. of the 3rd Intl. W. on Frontiers of Comb. Sys.’s (FroCos’2000), LNCS 1794, 2000.
A. Armando, S. Ranise, and M. Rusinowitch. Uniform Derivation of Decision Procedures by Superposition. In Computer Science Logic (CSL01), Paris, France, 10–13 September, 2001.
P. Baumgartner. An Ordered Theory Resolution Calculus. In Logic Programming and Automated Reasoning, number 624 in LNAI, pages 119–130, 1992.
N. Bjørner. Integrating Decision Procedures for Temporal Verification. PhD thesis, Stanford University, 1999.
R.S. Boyer and J S. Moore. Integrating Decision Procedures into Heuristic Theorem Provers: A Case Study of Linear Arithmetic. Machine Intelligence, 11:83–124, 1988.
A. Church. A Formulation of the Simple Theory of Types. J. of Symbolic Logic, 5(1):56–68, 1940.
F. Corella. What Holds in a Context? J. of Automated Reasoning, 10:79–93, 1993.
A. Degtyarev and A. Voronkov. The Undecidability of Simultaneous Rigid EUnification. Theoretical Computer Science, 166(1–2):291–300, 1996.
H. B. Enderton. A Mathematical Introduction to Logic. Academic Pr., 1972.
J.-C. Filliâtre. Formal Proof of a Program: Find. Science of Computer Programming, To appear.
J. H. Gallier, S. Ratz, and W. Snyder. Theorem Proving Using Rigid E-Unification: Equational Matings. In Logic in Computer Science (LICS’87), Ithaca, New York, 1987.
Jim Grundy. A Method of Program Refinement. PhD thesis, University of Cambridge, Computer Laboratory, New Museums Site, Pembroke Street, Cambridge CB2 3QG, England, November 1993. Technical Report 318.
C. A. R. Hoare. Algorithm 65, Find. Comm. of the ACM, 4(7):321, July 1961.
C. A. R. hoare. Proof of a Program: FIND. Comm. of the ACM, 14(1):39–45, January 1971.
G. Huet, G. Kahn, and C. Paulin-Mohring. The Coq Proof Assistant: a tutorial. Technical Report 204, INRIA-Rocquencourt, 1997.
D. Kapur and X. Nie. Reasoning about Numbers in Tecton. Technical report, Department of Computer Science, State University of New York, Albany, NY 12222, March 1994.
R. Nieuwenhuis and A. Rubio. Paramodulation-based theorem proving. In A. Robinson and A. Voronkov, editors, Hand. of Automated Reasoning. 2001.
D. Prawitz. Natural Deduction. Acta Universitatis Stockholmiensis, Stockholm Studies in Philosophy 3. Almqvist & Wiksell, Stockholm, 1965.
P. J. Robinson and J. Staples. Formalising the Hierarchical Structure of Practical Mathematical Reasoning. Journal of Logic and Computation, 3(1):47–61, February 1993.
S. Schulz. System Abstract: E 0.61. In R. Goré, A. Leitsch, and T. Nipkow, editors, Proc. of the 1st IJCAR, Siena, number 2083 in LNAI, pages 370–375. Springer, 2001.
C. Weidenbach and A. Nonnengart. Small clause normal form. In A. Robinson and A. Voronkov, editors, Hand. of Automated Reasoning. 2001.
H. Zhang. Contextual Rewriting in Automated Reasoning. Fundamenta Informaticae, 24(1/2):107–123, 1995.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ranise, S. (2002). Combining Generic and Domain Specific Reasoning by Using Contexts. In: Calmet, J., Benhamou, B., Caprotti, O., Henocque, L., Sorge, V. (eds) Artificial Intelligence, Automated Reasoning, and Symbolic Computation. AISC Calculemus 2002 2002. Lecture Notes in Computer Science(), vol 2385. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45470-5_27
Download citation
DOI: https://doi.org/10.1007/3-540-45470-5_27
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43865-6
Online ISBN: 978-3-540-45470-0
eBook Packages: Springer Book Archive