Abstract
Nominal unification is an extension of first-order unification that takes into account the \(\alpha \)-equivalence relation generated by binding operators, following the nominal approach. We propose a sound and complete procedure for nominal unification with commutative operators, or nominal C-unification for short, which has been formalised in Coq. The procedure transforms nominal C-unification problems into simpler (finite families) of fixed point constraints, whose solutions can be generated by algebraic techniques on combinatorics of permutations.
Work supported by the Brazilian agencies FAPDF (DE 193.001.369/2016), CAPES (Proc. 88881.132034/2016-01, 2nd author) and CNPq (PQ 307009/2013, 1st author).
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Notes
- 1.
Infix notation is adopted for commutative symbols: \(s * t\) abbreviates \(*\langle s,t\rangle \).
References
Aoto, T., Kikuchi, K.: A rule-based procedure for equivariant nominal unification. In: Pre-proceeding of Higher-Order Rewriting (HOR), pp. 1–5 (2016)
Aoto, T., Kikuchi, K.: Nominal confluence tool. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 173–182. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_12
Ayala-Rincón, M., Carvalho-Segundo, W., Fernández, M., Nantes-Sobrinho, D.: A formalisation of nominal equivalence with associative-commutative function symbols. ENTCS 332, 21–38 (2017)
Ayala-Rincón, M., de Carvalho-Segundo, W., Fernández, M., Nantes-Sobrinho, D.: On solving nominal fixpoint equations. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS (LNAI), vol. 10483, pp. 209–226. Springer, Cham (2017)
Ayala-Rincón, M., Fernández, M., Nantes-Sobrinho, D.: Nominal narrowing. In: Proceedings of the 1st International Conference on Formal Structures for Computation and Deduction (FSCD). LIPIcs, vol. 52, pp. 11:1–11:17 (2016)
Ayala-Rincón, M., Fernández, M., Rocha-oliveira, A.C.: Completeness in PVS of a nominal unification algorithm. ENTCS 323, 57–74 (2016)
Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge UP, New York (1998)
Braibant, T., Pous, D.: Tactics for reasoning modulo AC in Coq. In: Jouannaud, J.-P., Shao, Z. (eds.) CPP 2011. LNCS, vol. 7086, pp. 167–182. Springer, Heidelberg (2011)
Calvès, C.F.: Complexity and implementation of nominal algorithms. Ph.D Thesis, King’s College London (2010)
Calvès, C.F., Fernández, M.: Implementing nominal unification. ENTCS 176(1), 25–37 (2007)
Calvès, C., Fernández, M.: The first-order nominal link. In: Alpuente, M. (ed.) LOPSTR 2010. LNCS, vol. 6564, pp. 234–248. Springer, Heidelberg (2011)
Cheney, J.: \(\alpha \)Prolog Users Guide & Language Reference Version 0.3 DRAFT (2003)
Cheney, J.: Equivariant unification. J. Autom. Reasoning 45(3), 267–300 (2010)
Clouston, R.A., Pitts, A.M.: Nominal equational logic. ENTCS 172, 223–257 (2007)
Contejean, E.: A certified AC matching algorithm. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 70–84. Springer, Heidelberg (2004)
Fernández, M., Gabbay, M.J.: Nominal rewriting. Inf. Comput. 205(6), 917–965 (2007)
Fernández, M., Gabbay, M.J.: Closed nominal rewriting and efficiently computable nominal algebra equality. In: Proceedings of the 5th International Workshop on Logical Frameworks and Meta-languages: Theory and Practice (LFMTP). EPTCS, vol. 34, pp. 37–51 (2010)
Fernández, M., Gabbay, M.J., Mackie, I.: Nominal rewriting systems. In: Proceedings of the 6th International Conference on Principles and Practice of Declarative Programming (PPDP), pp. 108–119. ACM Press (2004)
Gabbay, M.J., Mathijssen, A.: Nominal (Universal) algebra: equational logic with names and binding. J. Logic Comput. 19(6), 1455–1508 (2009)
Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Formal Aspects Comput. 13(3–5), 341–363 (2002)
Kapur, D., Narendran, P.: Matching unification and complexity. SIGSAM Bull. 21(4), 6–9 (1987)
Kumar, R., Norrish, M.: (Nominal) Unification by recursive descent with triangular substitutions. In: Kaufmann, M., Paulson, L.C. (eds.) ITP 2010. LNCS, vol. 6172, pp. 51–66. Springer, Heidelberg (2010)
Schmidt-Schauß, M., Kutsia, T., Levy, J., Villaret, M.: Nominal unification of higher order expressions with recursive let. In: Hermenegildo, M.V., Lopez-Garcia, P. (eds.) LOPSTR 2016. LNCS, vol. 10184, pp. 328–344. Springer, Cham (2017)
Levy, J., Villaret, M.: An efficient nominal unification algorithm. In: Proceedings of the 21st International Conference on Rewriting Techniques and Applications (RTA). LIPIcs, vol. 6, pp. 209–226 (2010)
Nipkow, T.: Equational reasoning in Isabelle. Sci. Comput. Program. 12(2), 123–149 (1989)
Pitts, A.M.: Nominal Sets: Names and Symmetry in Computer Science. Cambridge UP, Cambridge (2013)
Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd edn. Springer, New York (2001)
Siekmann, J.: Unification of commutative terms. In: Ng, E.W. (ed.) Symbolic and Algebraic Computation. LNCS, vol. 72, pp. 22–29. Springer, Heidelberg (1979). https://doi.org/10.1007/3-540-09519-5_53
Urban, C.: Nominal unification revisited. In: Proceedings of the 24th International Workshop on Unification (UNIF). EPTCS, vol. 42, pp. 1–11 (2010)
Urban, C., Pitts, A.M., Gabbay, M.J.: Nominal unification. Theor. Comput. Sci. 323(1–3), 473–497 (2004)
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Ayala-Rincón, M., de Carvalho-Segundo, W., Fernández, M., Nantes-Sobrinho, D. (2018). Nominal C-Unification. In: Fioravanti, F., Gallagher, J. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2017. Lecture Notes in Computer Science(), vol 10855. Springer, Cham. https://doi.org/10.1007/978-3-319-94460-9_14
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