Abstract
We consider the problem of building satisfiability procedures for unions of disjoint theories. We briefly review the combination schemas proposed by Nelson-Oppen, Shostak, and others. Three inference systems are directly derived from the properties satisfied by the theories being combined and known results from the literature are obtained in a uniform and abstract way. This rational reconstruction is the starting point for further investigations. We introduce the concept of extended canonizer and derive a modularity result for a new class of theories (larger than Shostak and smaller than Nelson-Oppen theories) which is closed under disjoint union. This is in contrast with the lack of modularity of Shostak theories. We also explain how to implement extended canonizers by using the basic building blocks used in Shostak schema or by means of rewriting techniques.
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References
Armando, A., Ranise, S., Rusinowitch, M.: A Rewriting Approach to Satisfiability Procedures. Info. and Comp. 183(2), 140–164 (2003)
Bachmair, L., Tiwari, A., Vigneron, L.: Abstract Congruence Closure. Journal of Automated Reasoning 31(2), 129–168 (2003)
Barrett, C.W., Dill, D.L., Stump, A.: A generalization of shostak’s method for combining decision procedures. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, pp. 132–147. Springer, Heidelberg (2002)
Conchon, S., Krstić, S.: Strategies for combining decision procedures. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 537–553. Springer, Heidelberg (2003)
Cyrluk, D., Lincoln, P., Shankar, N.: On Shostak’s decision procedure for combinations of theories. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 463–477. Springer, Heidelberg (1996)
Déharbe, D., Ranise, S.: Light-Weight Theorem Proving for Debugging and Verifying Units of Code. In: Press, I.C.S. (ed.) Proc. of the Int. Conf. on Software Engineering and Formal Methods, SEFM 2003 (2003)
Dershowitz, N., Jouannaud, J.-P.: Handbook of Theoretical Computer Science. In: Rewrite Systems, vol. B, ch. 6, pp. 244–320 (1990)
Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: A Theorem Prover for Program Checking. Technical Report HPL-2003-148, HP Laboratories (2003)
Downey, P.J., Sethi, R., Tarjan, R.E.: Variations on the Common Subexpression Problem. J. of the ACM 27(4), 758–771 (1980)
Enderton, H.B.: A Mathematical Introduction to Logic. Ac. Press, Inc., New York (1972)
Filliâtre, J.-C., Owre, S., Rueß, H., Shankar, N.: ICS: Integrated canonizer and solver. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 246–249. Springer, Heidelberg (2001)
Ganzinger, H.: Relating semantic and proof-theoretic concepts for polynomial time decidability of uniform word problems. In: Proc. 16th IEEE Symp. on Logic in Computer Science, pp. 81–92. IEEE Comp. Soc. Press, Los Alamitos (2001)
Ganzinger, H.: Shostak light. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 332–346. Springer, Heidelberg (2002)
Kapur, D.: A rewrite rule based framework for combining decision procedures. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, pp. 87–102. Springer, Heidelberg (2002)
Kapur, D.: Shostak’s congruence closure as completion. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232. Springer, Heidelberg (1997)
Kapur, D., Nie, X.: Reasoning about Numbers in Tecton. In: Proc. 8th Inl. Symp. Methodologies for Intelligent Systems, pp. 57–70 (1994)
Krstić, S., Conchon, S.: Canonization for disjoint unions of theories. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 197–211. Springer, Heidelberg (2003)
Manna, Z., Zarba, C.G.: Combining decision procedures. In: Aichernig, B.K., Maibaum, T. (eds.) Formal Methods at the Crossroads. From Panacea to Foundational Support. LNCS, vol. 2757, pp. 381–422. Springer, Heidelberg (2003)
Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. on Programming Languages and Systems 1(2), 245–257 (1979)
Oppen, D.C.: Complexity, convexity and combinations of theories. Theoretical Computer Science 12, 291–302 (1980)
Ranise, S., Ringeissen, C., Tran, D.-K.: Nelson-Oppen, Shostak and the Extended Canonizer: A Family Picture with a Newborn (Full Version), Available at http://www.loria.fr/~ranise/pubs/long-ictac04.ps.gz
Rueß, H., Shankar, N.: Deconstructing Shostak. In: Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, June 2001, pp. 19–28. IEEE Computer Society, Los Alamitos (2001)
Shankar, N., Rueß, H.: Combining shostak theories. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 1–18. Springer, Heidelberg (2002)
Shostak, R.E.: Deciding combinations of theories. J. of the ACM 31, 1–12 (1984)
Tinelli, C., Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures. Theoretical Computer Science 290(1), 291–353 (2003)
van Hentenryck, P., Graf, T.: Standard Forms for Rational Linear Arithmetics in Constraint Logic Programming. Annals of Mathematics and Artificial Intelligence 5, 303–319 (1992)
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Ranise, S., Ringeissen, C., Tran, DK. (2005). Nelson-Oppen, Shostak and the Extended Canonizer: A Family Picture with a Newborn. In: Liu, Z., Araki, K. (eds) Theoretical Aspects of Computing - ICTAC 2004. ICTAC 2004. Lecture Notes in Computer Science, vol 3407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31862-0_27
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DOI: https://doi.org/10.1007/978-3-540-31862-0_27
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