Abstract
We consider the problem of efficiently building extended canonizers, which are capable of solving the uniform word problem for some first-order theories. These reasoning artifacts have been introduced in previous work to solve the lack of modularity of Shostak combination schema while retaining its efficiency. It is known that extended canonizers can be modularly combined to solve the uniform word problem in unions of theories. Unfortunately, little is known about efficiently implementing such canonizers for component theories, especially those of interest for verification like, e.g., those of uninterpreted function symbols or lists. In this paper, we investigate this problem by adapting and combining work on rewriting-based decision procedures for satisfiability in first-order theories and SER graphs, a graph-based method defined for abstract congruence closure. Our goal is to build graph-based extended canonizers for theories which are relevant for verification. Based on graphs our approach addresses implementation issues that were lacking in previous rewriting-based decision procedure approaches and which are important to argue the viability of extended canonizers.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: On a rewriting approach to satisfiability procedures: extension, combination of theories and an experimental appraisal. In: Gramlich, B. (ed.) Frontiers of Combining Systems. LNCS (LNAI), vol. 3717, Springer, Heidelberg (2005)
Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. J. of Information and Computation 183(2), 140–164 (2003)
Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, Cambridge (1998)
Bachmair, L., Tiwari, A., Vigneron, L.: Abstract congruence closure. J. of Automated Reasoning 31(2), 129–168 (2003)
Conchon, S., Kristic, S.: Canonization for disjoint unions of theories. Information and Computation 199(1-2), 87–106 (2005)
Jouannaud, J., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM J. on Computing 15(4), 1155–1194 (1986)
Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebras. In: Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, Oxford (1970)
Lynch, C., Strogova, P.: SOUR graphs for efficient completion. Journal of Discrete Mathematics and Theoretical Computer Science 2(1), 1–25 (1998)
Nelson, C.G., Oppen, D.C.: Fast Decision Procedures based on Congruence Closure. Journal of the ACM 27(2), 356–364 (1980)
Nelson, C.G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. on Programming Languages and Systems 1(2), 245–257 (1979)
Oppen, D.C.: Reasoning About Recursively Defined Data Structures. J. ACM 27(3), 403–411 (1980)
Rusinowitch, M.: Theorem-proving with resolution and superposition. J. Symb. Comput. 11(1-2), 21–49 (1991)
Ranise, S., Ringeissen, C., Tran, D.: Nelson-Oppen, Shostak and the Extended Canonizer: A Family Picture with a Newborn. In: Liu, Z., Araki, K. (eds.) ICTAC 2004. LNCS, vol. 3407, pp. 372–386. Springer, Heidelberg (2005)
Ranise, S., Tinelli, C.: Satisfiability Modulo Theories. IEEE Magazine on Intelligent Systems 21(6), 71–81 (2006)
Ruess, H., Shankar, N.: Deconstructing Shostak. In: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, Boston, Massachusetts, USA, pp. 19–28. IEEE Computer Society, Los Alamitos (2001)
Scharff, C., Bachmair, L.: On the Combination of Congruence Closure and Completion. In: Buchberger, B., Campbell, J.A. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 103–117. Springer, Heidelberg (2004)
Shostak, R.E.: Deciding Combinations of Theories. Journal of the ACM 31, 1–12 (1984)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ranise, S., Scharff, C. (2007). Building Extended Canonizers by Graph-Based Deduction. In: Jones, C.B., Liu, Z., Woodcock, J. (eds) Theoretical Aspects of Computing – ICTAC 2007. ICTAC 2007. Lecture Notes in Computer Science, vol 4711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75292-9_30
Download citation
DOI: https://doi.org/10.1007/978-3-540-75292-9_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75290-5
Online ISBN: 978-3-540-75292-9
eBook Packages: Computer ScienceComputer Science (R0)