Abstract
If a set of equations E ∪ Ax is such that E is confluent, terminating, and coherent modulo Ax, narrowing with E modulo Ax provides a complete E ∪ Ax-unification algorithm. However, except for the hopelessly inefficient case of full narrowing, nothing seems to be known about effective narrowing strategies in the general modulo case beyond the quite depressing observation that basic narrowing is incomplete modulo AC. In this work we propose an effective strategy based on the idea of the E ∪ Ax-variants of a term that we call folding variant narrowing. This strategy is complete, both for computing E ∪ Ax-unifiers and for computing a minimal complete set of variants for any input term. And it is optimally variant terminating in the sense of terminating for an input term t iff t has a finite, complete set of variants. The applications of folding variant narrowing go beyond providing a complete E ∪ Ax-unification algorithm: computing the E ∪ Ax-variants of a term may be just as important as computing E ∪ Ax-unifiers in recent applications of folding variant narrowing such as termination methods modulo axioms, and checking confluence and coherence of rules modulo axioms.
S. Escobar has been partially supported by the EU (FEDER) and the Spanish MEC/MICINN under grant TIN 2007-68093-C02-02. J. Meseguer and R. Sasse have been partially supported by NSF Grants CNS 07-16638, CNS 08-31064, and CNS 09-04749.
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References
Alpuente, M., Escobar, S., Iborra, J.: Modular termination of basic narrowing. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 1–16. Springer, Heidelberg (2008)
Alpuente, M., Falaschi, M., Vidal, G.: Partial Evaluation of Functional Logic Programs. ACM Transactions on Programming Languages and Systems 20(4), 768–844 (1998)
Anantharaman, S., Narendran, P., Rusinowitch, M.: Unification modulo cui plus distributivity axioms. J. Autom. Reasoning 33(1), 1–28 (2004)
Comon-Lundh, H., Delaune, S.: The finite variant property: How to get rid of some algebraic properties. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 294–307. Springer, Heidelberg (2005)
Durán, F., Lucas, S., Meseguer, J.: Termination modulo combinations of equational theories. In: Ghilardi, S. (ed.) FroCoS 2009. LNCS, vol. 5749, pp. 246–262. Springer, Heidelberg (2009)
Escobar, S., Meadows, C., Meseguer, J.: Maude-npa: Cryptographic protocol analysis modulo equational properties. In: FOSAD 2007. LNCS, vol. 5705, pp. 1–50. Springer, Heidelberg (2009)
Escobar, S., Meseguer, J.: Symbolic model checking of infinite-state systems using narrowing. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 153–168. Springer, Heidelberg (2007)
Escobar, S., Meseguer, J., Sasse, R.: Effectively checking the finite variant property. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 79–93. Springer, Heidelberg (2008)
Escobar, S., Meseguer, J., Sasse, R.: Variant narrowing and equational unification. Electr. Notes Theor. Comput. Sci. 238(3), 103–119 (2009)
Escobar, S., Meseguer, J., Thati, P.: Natural narrowing for general term rewriting systems. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 279–293. Springer, Heidelberg (2005)
Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–108. Springer, Heidelberg (2001)
Goguen, J.A., Meseguer, J.: Equality, types, modules, and (why not?) generics for logic programming. J. Log. Program. 1(2), 179–210 (1984)
Hanus, M.: The Integration of Functions into Logic Programming: From Theory to Practice. Journal of Logic Programming 19&20, 583–628 (1994)
Hölldobler, S.: Foundations of Equational Logic Programming. In: Hölldobler, S. (ed.) Foundations of Equational Logic Programming. LNCS (LNAI), vol. 353. Springer, Heidelberg (1989)
Hullot, J.-M.: Canonical forms and unification. In: Bibel, W., Kowalski, R.A. (eds.) CADE 1980. LNCS, vol. 87, pp. 318–334. Springer, Heidelberg (1980)
Jouannaud, J.-P., Kirchner, C., Kirchner, H.: Incremental construction of unification algorithms in equational theories. In: Díaz, J. (ed.) ICALP 1983. LNCS, vol. 154, pp. 361–373. Springer, Heidelberg (1983)
Jouannaud, J.-P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM J. Comput. 15(4), 1155–1194 (1986)
Meseguer, J.: Conditional rewriting logic as a united model of concurrency. Theor. Comput. Sci. 96(1), 73–155 (1992)
Meseguer, J., Thati, P.: Symbolic reachability analysis using narrowing and its application to verification of cryptographic protocols. Higher-Order and Symbolic Computation 20(1-2), 123–160 (2007)
Middeldorp, A., Hamoen, E.: Completeness results for basic narrowing. Journal of Applicable Algebra in Engineering, Communication, and Computing 5, 213–253 (1994)
Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. J. ACM 28(2), 233–264 (1981)
Ryan, P.Y.A., Schneider, S.A.: An attack on a recursive authentication protocol. a cautionary tale. Inf. Process. Lett. 65(1), 7–10 (1998)
TeReSe (ed.): Term Rewriting Systems. Cambridge University Press, Cambridge (2003)
Viola, E.: E-unifiability via narrowing. In: Restivo, A., Rocca, S.R.D., Roversi, L. (eds.) ICTCS 2001. LNCS, vol. 2202, pp. 426–438. Springer, Heidelberg (2001)
Viry, P.: Equational rules for rewriting logic. Theor. Comput. Sci. 285(2), 487–517 (2002)
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Escobar, S., Sasse, R., Meseguer, J. (2010). Folding Variant Narrowing and Optimal Variant Termination . In: Ölveczky, P.C. (eds) Rewriting Logic and Its Applications. WRLA 2010. Lecture Notes in Computer Science, vol 6381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16310-4_5
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DOI: https://doi.org/10.1007/978-3-642-16310-4_5
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