Abstract
Shostak's congruence closure algorithm is demystified, using the framework of ground completion on (possibly nonterminating, non-reduced) rewrite rules. In particular, the canonical rewriting relation induced by the algorithm on ground terms by a given set of ground equations is precisely constructed. The main idea is to extend the signature of the original input to include new constant symbols for nonconstant subterms appearing in the input. A byproduct of this approach is (i) an algorithm for associating a confluent rewriting system with possibly nonterminating ground rewrite rules, and (ii) a new quadratic algorithm for computing a canonical rewriting system from ground equations.
Partially supported by the National Science Foundation Grant nos. CCR-9308016, and CCR-9404930.
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References
D. Craigen, S. Kromodimoelijo, I. Meisels. W. Pase, and M. Saaltink, “Eves system description,” Proc. Automated Deduction — CADE 11, LNAI 607 (ed. Kapur), Springer Verlag (1992), 771–775.
D. Cyrluk, P. Lincoln, and N. Shankar, “On Shostak's decision procedures for combination of theories,” Proc. Automated Deduction — CADE 13, LNAI 1104 (eds. McRobbie and Slaney), Springer Verlag (1996), 463–477.
P.J. Downey, R. Sethi, and R.E. Tarjan, “Variations on the common subexpression problem,” JACM, 27(4) (1980), 758–771
Z. Fülöp, and S. Vagvölgyi, “Ground term rewriting rules for the word problem of ground term equations,” Bulletin of the EATCS, 45 (1991), 186–201.
G. Huet and D. Lankford, On the Uniform Halting Problem for Term Rewriting Systems. INRIA Report 283, March 1978.
D. Kapur and M. Subramaniam, “New uses of linear arithmetic in automated theorem proving for induction,” J. Automated Reasoning, 16(1–2) (1996), 39–78
D. Kapur and M. Subramaniam, “Mechanically verifying a family of multiplier circuits,” Proc. Computer Aided Verification (CAV), New Jersey, Springer LNCS 1102 (eds. R. Alur and T.A. Henzinger) (1996), 135–146
D. Kapur and H. Zhang, “An overview of Rewrite Rule Laboratory (RRL), ” Computers and Math. with Applications, 29(2) (1995), 91–114.
D. Kozen, Complexity of Finitely Presented Algebras. Technical Report TR 76-294, Dept. of Computer Science, Cornell Univ., Ithaca, NY, 1976.
D. Knuth and P. Bendix, “Simple word problems in universal algebras,” in Computational Problems in Abstract Algebra (ed. Leech), Pergamon Press (1970), 263–297.
G. Nelson, and D.C. Oppen, “Simplification by cooperating decision procedures,” ACM Tran. on Programming Languages and Systems 1 (2) (1979) 245–257.
G. Nelson, and D.C. Oppen, “Fast decision procedures based on congruence closure,” JACM, 27(2) (1980), 356–364.
D. Plaisted, and A. Sattler-Klein, “Proof lengths for equational completion,” Information and Computation, 125 (1996), 154–170.
R.E. Shostak, “An algorithm for reasoning about equality,” Communications of ACM, 21(7) (1978), 583–585.
R.E. Shostak, “Deciding combination of theories,” Journal of ACM 31 (1), (1984) 1–12.
W. Snyder, “A fast algorithm for generating reduced ground rewriting system from a set of ground equations,” J. Symbolic Computation, 1992.
H. Zhang, “Implementing contextual rewriting,” Proc. Third International Workshop on Conditional Term Rewriting Systems, Springer LNCS 656 (eds. J. L. Remy and M. Rusinowitch), (1992), 363–377.
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© 1997 Springer-Verlag Berlin Heidelberg
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Kapur, D. (1997). Shostak's congruence closure as completion. In: Comon, H. (eds) Rewriting Techniques and Applications. RTA 1997. Lecture Notes in Computer Science, vol 1232. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62950-5_59
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DOI: https://doi.org/10.1007/3-540-62950-5_59
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