Abstract
Given a theory \(\mathbb{T}\), a set of equations E, and a single equation e, the uniform word problem (UWP) is to determine if \(E\Rightarrow e\) in the theory \(\mathbb{T}\). We recall the classic Nelson-Oppen combination result for solving the UWP over combinations of theories and then present a constructive version of this result for equational theories. We present three applications of this constructive variant. First, we use it on the pure theory of equality (\(\mathbb{T}_{EQ}\)) and arrive at an algorithm for computing congruence closure of a set of ground term equations. Second, we use it on the theory of associativity and commutativity (\(\mathbb{T}_{AC}\)) and obtain a procedure for computing congruence closure modulo AC. Finally, we use it on the combination theory \(\mathbb{T}_{EQ}\cup\mathbb{T}_{AC}\cup\mathbb{T}_{PR}\), where \(\mathbb{T}_{PR}\) is the theory of polynomial rings, to present a decision procedure for solving the UWP for this combination.
Keywords
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.
Research supported in part by NSF grants CNS-0720721 and CNS-0834810 and NASA grant NNX08AB95A.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Armando, A., Ranise, S., Rusinowitch, M.: Uniform derivation of decision procedures by superposition. In: Fribourg, L. (ed.) CSL 2001 and EACSL 2001. LNCS, vol. 2142, pp. 513–527. Springer, Heidelberg (2001)
Audemard, G., Bertoli, P., Cimatti, A., Kornilowicz, A., Sebastiani, R.: A SAT based approach for solving formulas over boolean and linear mathematical propositions. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 195–210. Springer, Heidelberg (2002)
Bachmair, L., Ganzinger, H.: Buchberger’s algorithm: A constraint-based completion procedure. In: Jouannaud, J.-P. (ed.) CCL 1994. LNCS, vol. 845, pp. 285–301. Springer, Heidelberg (1994)
Bachmair, L., Ramakrishnan, I.V., Tiwari, A., Vigneron, L.: Congruence closure modulo Associativity-Commutativity. In: Kirchner, H. (ed.) FroCos 2000. LNCS (LNAI), vol. 1794, pp. 245–259. Springer, Heidelberg (2000)
Bachmair, L., Tiwari, A.: D-bases for polynomial ideals over commutative noetherian rings. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 113–127. Springer, Heidelberg (1997)
Bachmair, L., Tiwari, A.: Abstract congruence closure and specializations. In: McAllester, D. (ed.) CADE 2000. LNCS (LNAI), vol. 1831, pp. 64–78. Springer, Heidelberg (2000)
Barrett, C., Dill, D., Stump, A.: A generalization of Shostak’s method for combining decision procedures. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, p. 132. Springer, Heidelberg (2002)
Basu, S., Pollack, R., Roy, M.-F.: Algorithms in real algebraic geometry. Algorithms and Computation in Mathematics, vol. 10. Springer, Heidelberg (2003)
Bozzano, M., Bruttomesso, R., Cimatti, A., Junttila, T.A., Ranise, S., van Rossum, P., Sebastiani, R.: Efficient theory combination via boolean search. Information and Computation 204(10), 1493–1525 (2006)
Buchberger, B.: An algorithm for finding a basis for the residue class ring of a zero-dimensional ideal. PhD thesis, University of Innsbruck, Austria (1965)
Buchberger, B.: A critical-pair completion algorithm for finitely generated ideals in rings. In: Börger, E., Rödding, D., Hasenjaeger, G. (eds.) Rekursive Kombinatorik 1983. LNCS, vol. 171, pp. 137–161. Springer, Heidelberg (1984)
Collins, G.E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Cousot, P., Cousot, R.: Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: 4th ACM Symp. on Principles of Programming Languages, POPL 1977, pp. 238–252 (1977)
de Moura, L., Rueß, H., Sorea, M.: Lazy theorem proving for bounded model checking over infinite domains. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 438–455. Springer, Heidelberg (2002)
Dutertre, B., de Moura, L.: A fast linear-arithmetic solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)
Flanagan, C., Joshi, R., Ou, X., Saxe, J.: Theorem proving using lazy proof explication. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 355–367. Springer, Heidelberg (2003)
Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL(T): Fast decision procedures. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 175–188. Springer, Heidelberg (2004)
Gulwani, S., Tiwari, A.: Assertion checking over combined abstraction of linear arithmetic and uninterpreted functions. In: Sestoft, P. (ed.) ESOP 2006. LNCS, vol. 3924, pp. 279–293. Springer, Heidelberg (2006)
Gulwani, S., Tiwari, A.: Combining abstract interpreters. In: PLDI (June 2006)
Gulwani, S., Tiwari, A., Necula, G.C.: Join algorithms for the theory of uninterpreted symbols. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 311–323. Springer, Heidelberg (2004)
Harrison, J.: Theorem proving with the real numbers. Springer, Heidelberg (1998)
Hong, H.: Quantifier elimination in elementary algebra and geometry by partial cylindrical algebraic decomposition version 13 (1995), http://www.gwdg.de/~cais/systeme/saclib,www.eecis.udel.edu/~saclib/
Karr, M.: Affine relationships among variables of a program. Acta Informatica 6, 133–151 (1976)
Lynch, C., Morawska, B.: Automatic decidability. In: IEEE Symposium on Logic in Computer Science, LICS 2002, pp. 7–16. IEEE Society, Los Alamitos (2002)
Mayr, E.W., Meyer, A.R.: The complexity of the word problems for commutative semigroups and polynomial ideals. Advances in Mathematics 46, 305–329 (1982)
Microsoft Research. Z3: An efficient SMT solver, http://research.microsoft.com/projects/z3/
Nelson, G., Oppen, D.: Simplification by cooperating decision procedures. ACM Transactions on Programming Languages and Systems 1(2), 245–257 (1979)
Nieuwenhuis, R., Oliveras, A.: Decision procedures for SAT, SAT modulo theories and beyond. The Barcelogictools. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 23–46. Springer, Heidelberg (2005)
Oppen, D.: Complexity, convexity and combinations of theories. Theoretical Computer Science 12, 291–302 (1980)
Pratt, V.R.: Two easy theories whose combination is hard. Technical report, MIT (1977)
SRI International. Yices: An SMT solver, http://yices.csl.sri.com/
Stump, A., Barrett, C.W., Dill, D.L.: CVC: A cooperating validity checker. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 500–504. Springer, Heidelberg (2002)
Tarski, A.: A Decision Method for Elementary Algebra and Geometry, 2nd edn. University of California Press, Berkeley (1948)
Tiwari, A.: Decision procedures in automated deduction. PhD thesis, State University of New York at Stony Brook (2000)
Tiwari, A.: An algebraic approach for the unsatisfiability of nonlinear constraints. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 248–262. Springer, Heidelberg (2005)
Wolfman, S., Weld, D.: The LPSAT system and its application to resource planning. In: Proc. 16th Intl. Joint Conf. on Artificial Intelligence (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tiwari, A. (2009). Combining Equational Reasoning. In: Ghilardi, S., Sebastiani, R. (eds) Frontiers of Combining Systems. FroCoS 2009. Lecture Notes in Computer Science(), vol 5749. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04222-5_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-04222-5_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04221-8
Online ISBN: 978-3-642-04222-5
eBook Packages: Computer ScienceComputer Science (R0)