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Completeness in PVS of a nominal unification algorithm. (English) Zbl 1394.68348
Benevides, Mario (ed.) et al., Proceedings of the 10th workshop on logical and semantic frameworks, with applications (LSFA 2015), Natal, Brazil, August 31 – September 1, 2015. Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 323, 57-74 (2016).
Summary: Nominal systems are an alternative approach for the treatment of variables in computational systems. In the nominal approach variable bindings are represented using techniques that are close to first-order logical techniques, instead of using a higher-order metalanguage. Functional nominal computation can be modelled through nominal rewriting, in which $$\alpha$$-equivalence, nominal matching and nominal unification play an important role. Nominal unification was initially studied by C. Urban et al. [Theor. Comput. Sci. 323, No. 1–3, 473–497 (2004; Zbl 1078.68140)] and then formalised by C. Urban [J. Autom. Reasoning 40, No. 4, 327–356 (2008; Zbl 1140.68061)] in the proof assistant Isabelle/HOL and by R. Kumar and M. Norrish [Lect. Notes Comput. Sci. 6172, 51–66 (2010; Zbl 1291.68356)] in HOL4. In this work, we present a new specification of nominal unification in the language of PVS and a formalisation of its completeness. This formalisation is based on a natural notion of nominal $$\alpha$$-equivalence, avoiding in this way the use of the intermediate auxiliary weak $$\alpha$$-relation considered in previous formalisations. Also, in our specification, instead of applying simplification rules to unification and freshness constraints, we recursively build solutions for the original problem through a straightforward functional specification, obtaining a formalisation that is closer to algorithmic implementations. This is possible by the independence of freshness contexts guaranteed by a series of technical lemmas.
For the entire collection see [Zbl 1342.68007].

##### MSC:
 68T15 Theorem proving (deduction, resolution, etc.) (MSC2010) 03B35 Mechanization of proofs and logical operations 68Q42 Grammars and rewriting systems
##### Software:
Isabelle/HOL; Nominal Isabelle; PVS; LCF
Full Text:
##### References:
 [1] Aydemir, B.; Bohannon, A.; Weihrich, S., Nominal reasoning techniques in coq (extended abstract), Electronic Notes in Theoretical Computer Science, 174, 5, 69-77, (2007) · Zbl 1278.68248 [2] Avelar, A. B.; Galdino, A. L.; de Moura, F. L.C.; Ayala-Rincón, M., First-order unification in the PVS proof assistant, Logic Journal of the IGPL, 22, 5, 758-789, (2014) [3] Ayala-Rincón, M.; Kamareddine, F., Unification via the λs e-style of explicit substitution, Logic Journal of the IGPL, 9, 4, 489-523, (2001) · Zbl 0987.03012 [4] Calvès, C., Unifying nominal unification, (24 th International Conference on Rewriting Techniques and Applications, RTA 2013, LIPIcs, vol. 21, (2013)), 143-157 · Zbl 1356.68119 [5] Calvès, C.; Fernández, M., The first-order nominal link, (Logic-Based Program Synthesis and Transformation - 20 th International Symposium, LOPSTR 2010, Revised Selected Papers, Lecture Notes in Computer Science, vol. 6564, (2010), Springer), 234-248 · Zbl 1326.68083 [6] Cheney, J., Relating nominal and higher-order pattern unification, (Proceedings of the 19 th International Workshop on Unification (UNIF 2005), (2005)), 104-119 [7] Constable, R.; Moczydlowski, W., Extracting the resolution algorithm from a completeness proof for the propositional calculus, Annals of Pure and Applied Logic, 161, 3, 337-348, (2009) · Zbl 1221.03015 [8] Dowek, G.; Hardin, T.; Kirchner, C., Higher-order unification via explicit substitutions, Information and Computation, 157, 1/2, 183-235, (2000) · Zbl 1005.03016 [9] Fernández, M.; Gabbay, M. J., Nominal rewriting, Information and Computation, 205, 6, 917-965, (June 2007) [10] Fernández, M.; Rubio, A., Nominal completion for rewrite systems with binders, (Automata, Languages, and Programming, 39th International Colloquium, ICALP 2012, Lecture Notes in Computer Science, vol. 7392, (2012)), 201-213, Part II · Zbl 1367.68130 [11] Galdino, A. L.; Ayala-Rincón, M., A formalization of the Knuth-bendix(-huet) critical pair theorem, Journal of Automated Reasoning, 45, 3, 301-325, (2010) · Zbl 1207.68338 [12] Huet, G. P., A unification algorithm for typed λ-calculus, Theoretical Computer Science, 1, 27-57, (1975) · Zbl 0332.02035 [13] Kumar, R.; Norrish, M., (nominal) unification by recursive descent with triangular substitutions, (Interactive Theorem Proving, First International Conference, ITP 2010, Lecture Notes in Computer Science, vol. 6172, (2010), Springer), 51-66 · Zbl 1291.68356 [14] Levy, J.; Villaret, M., An efficient nominal unification algorithm, (Proceedings of the 21 st International Conference on Rewriting Techniques and Applications, RTA 2010, LIPIcs, vol. 6, (2010)), 209-226 · Zbl 1236.68138 [15] Levy, J.; Villaret, M., Nominal unification from a higher-order perspective, ACM Transactions on Computational Logic, 13, 2, 10, (2012) · Zbl 1352.03017 [16] Paulson, L. C., Verifying the unification algorithm in LCF, Science of Computer Programming, 5, 2, 143-169, (1985) · Zbl 0567.68054 [17] Shankar, N.; Owre, S.; Rushby, J. M.; Stringer-Calvert, D. W.J., PVS prover guide, (2001), SRI International, Technical report [18] Urban, C.; Pitts, A. M.; Gabbay, M., Nominal unification, Theoretical Computer Science, 323, 1-3, 473-497, (2004) · Zbl 1078.68140 [19] Urban, C., Formalisation of nominal unification in isabelle/HOL, (2004), TU Munich, last visited April 2015 [20] Urban, C., Nominal techniques in isabelle/HOL, Journal of Automated Reasoning, 40, 4, 327-356, (2008) · Zbl 1140.68061 [21] Urban, C., Nominal unification revisited, Proceedings 24 th International Workshop on Unification, UNIF 2010, Electronic Proceedings in Theoretical Computer Science, 42, 1-11, (2010)
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