Multiple Congruence Relations, First-Order Theories on Terms, and the Frames of the Applied Pi-Calculus
Conference paper
Abstract
We investigate the problem of deciding first-order theories of finite trees with several distinguished congruence relations, each of them given by some equational axioms. We give an automata-based solution for the case where the different equational axiom systems are linear and variable-disjoint (this includes the case where all axioms are ground), and where the logic does not permit to express tree relations x = f(y,z). We show that the problem is undecidable when these restrictions are relaxed. As motivation and application, we show how to translate the model-checking problem of \(\mathcal{A}\pi \mathcal{L}\), a spatial equational logic for the applied pi-calculus, to the validity of first-order formulas in term algebras with multiple congruence relations.
Keywords
Function Symbol Equational Theory Congruence Relation Critical Pair Ground Term
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References
- 1.Abadi, M., Cortier, V.: Deciding Knowledge in Security Protocols under Equational Theories. Theoretical Computer Science 367(1-2), 2–32 (2006)MathSciNetCrossRefMATHGoogle Scholar
- 2.Abadi, M., Fournet, C.: Mobile values, new names, and secure communication. In: POPL 2001 (2001)Google Scholar
- 3.Baader, F., Snyder, W.: Unification theory. In: Handbook of Automated Reasoning, vol. I, ch. 8, pp. 445–532. Elsevier and MIT Press (2001)Google Scholar
- 4.Blumensath, A., Grädel, E.: Automatic structures. In: Logic in Computer Science, Santa Barbara, CA, pp. 51–62 (June 2000)Google Scholar
- 5.Caron, A.-C., Coquide, J.-L., Dauchet, M.: Encompassment properties and automata with constraints. In: Kirchner, C. (ed.) RTA 1993. LNCS, vol. 690, pp. 328–342. Springer, Heidelberg (1993)CrossRefGoogle Scholar
- 6.Comon, H.: Solving symbolic ordering constraints. IJCS 1(4), 387–412 (1990)MathSciNetMATHGoogle Scholar
- 7.Comon, H., Dauchet, M., Gilleron, R., Löding, C., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (2007), http://www.grappa.univ-lille3.fr/tata (release October 12, 2007)
- 8.Comon, H., Delor, C.: Equational formulae with membership constraints. Information and Computation 112(2), 167–216 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 9.Comon, H., Haberstrau, M., Jouannaud, J.-P.: Syntacticness, cycle-syntacticness and shallow theories. Information and Computation 111(1), 154–191 (1994)MathSciNetCrossRefMATHGoogle Scholar
- 10.Comon, H., Lescanne, P.: Equational problems and disunification. Journal of Symbolic Computation 7, 371–425 (1989)MathSciNetCrossRefMATHGoogle Scholar
- 11.Dauchet, M., Tison, S., Heuillard, T., Lescanne, P.: Decidability of the confluence of ground term rewriting systems. In: LICS, pp. 353–359 (1987)Google Scholar
- 12.Dauchet, M., Tison, S., Heuillard, T., Lescanne, P.: Decidability of the confluence of finite ground term rewrite systems and of other related term rewrite systems. Information and Computation 88(2), 187–201 (1990)MathSciNetCrossRefMATHGoogle Scholar
- 13.Godoy, G., Huntingford, E., Tiwari, A.: Termination of rewriting with right-flat rules. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 200–213. Springer, Heidelberg (2007)CrossRefGoogle Scholar
- 14.Gurevich, Y., Veanes, M.: Logic with equality: Partisan corroboration and shifted pairing. Information and Computation 152(2), 205–235 (1999)MathSciNetCrossRefMATHGoogle Scholar
- 15.Huet, G.: Confluent reductions: Abstract properties and applications to term rewriting systems. J. ACM 27(4), 797–821 (1980)MathSciNetMATHGoogle Scholar
- 16.Hüttel, H., Pedersen, M.D.: A logical characterisation of static equivalence. Electronic Notes in Theoretical Computer Science 173, 139–157 (2007)CrossRefMATHGoogle Scholar
- 17.Jouannaud, J.-P., Okada, M.: Satisfiability of systems of ordinal notation with the subterm property is decidable. In: Leach Albert, J., Monien, B., Rodríguez-Artalejo, M. (eds.) ICALP 1991. LNCS, vol. 510, pp. 455–468. Springer, Heidelberg (1991)CrossRefGoogle Scholar
- 18.Kuncak, V., Rinard, M.C.: Structural subtyping of non-recursive types is decidable. In: Logic in Computer Science, Ottawa, Canada, pp. 96–107 (June 2003)Google Scholar
- 19.Lozes, É., Villard, J.: A spatial equational logic for the applied π-calculus. In: van Breugel, F., Chechik, M. (eds.) CONCUR 2008. LNCS, vol. 5201, pp. 387–401. Springer, Heidelberg (2008)CrossRefGoogle Scholar
- 20.Lozes, É., Villard, J.: A spatial equational logic for the applied π-calculus. Distributed Computing 23(1), 61–83 (2010)CrossRefMATHGoogle Scholar
- 21.Maher, M.J.: Complete axiomatizations of the algebras of finite, rational and infinite trees. In: LICS, Edinburgh, Scotland, pp. 348–357 (July 1988)Google Scholar
- 22.Malc’ev, A.I.: Axiomatizable classes of locally free algebras of various type. In: The Metamathematics of Algebraic Systems: Collected Papers 1936–1967, ch. 23 (1971)Google Scholar
- 23.Marcinkowski, J.: Undecidability of the ∃ * ∀ * part of the theory of ground term algebra modulo an AC symbol. In: Narendran, P., Rusinowitch, M. (eds.) RTA 1999. LNCS, vol. 1631, pp. 92–102. Springer, Heidelberg (1999)CrossRefGoogle Scholar
- 24.Su, Z., Aiken, A., Niehren, J., Priesnitz, T., Treinen, R.: The first-order theory of subtyping constraints. In: POPL, pp. 203–216. ACM, New York (2002)Google Scholar
- 25.Treinen, R.: A new method for undecidability proofs of first order theories. Journal of Symbolic Computation 14(5), 437–457 (1992)MathSciNetCrossRefMATHGoogle Scholar
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