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Nominal Unification and Matching of Higher Order Expressions with Recursive Let (2102.08146v4)

Published 16 Feb 2021 in cs.LO and cs.AI

Abstract: A sound and complete algorithm for nominal unification of higher-order expressions with a recursive let is described, and shown to run in nondeterministic polynomial time. We also explore specializations like nominal letrec-matching for expressions, for DAGs, and for garbage-free expressions and determine their complexity. We also provide a nominal unification algorithm for higher-order expressions with recursive let and atom-variables, where we show that it also runs in nondeterministic polynomial time. In addition we prove that there is a guessing strategy for nominal unification with letrec and atom-variable that is a trade-off between exponential growth and non-determinism. Nominal matching with variables representing partial letrec-environments is also shown to be in NP.

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References (49)
  1. Nominal Unification of Higher Order Expressions with Recursive Let. In: Hermenegildo MV, López-García P (eds.), Logic-Based Program Synthesis and Transformation - 26th International Symposium, LOPSTR 2016, Edinburgh, UK, September 6-8, 2016, Revised Selected Papers, volume 10184 of Lecture Notes in Computer Science. Springer, 2016 pp. 328–344. 10.1007/978-3-319-63139-4_19.
  2. Baader F, Snyder W. Unification Theory. In: Robinson JA, Voronkov A (eds.), Handbook of Automated Reasoning, pp. 445–532. Elsevier and MIT Press, 2001.
  3. Huet GP. A Unification Algorithm for Typed lambda-Calculus. Theor. Comput. Sci., 1975. 1(1):27–57. 10.1016/0304-3975(75)90011-0.
  4. Goldfarb WD. The Undecidability of the Second-Order Unification Problem. Theor. Comput. Sci., 1981. 13:225–230. 10.1016/0304-3975(81)90040-2.
  5. Levy J, Veanes M. On the Undecidability of Second-Order Unification. Inf. Comput., 2000. 159(1-2):125–150. 10.1006/inco.2000.2877.
  6. Nominal Unification. In: 17th CSL, 12th EACSL, and 8th KGC, volume 2803 of LNCS. Springer, 2003 pp. 513–527. 10.1007/978-3-540-45220-1_41.
  7. Nominal unification. Theor. Comput. Sci., 2004. 323(1–3):473–497. 10.1016/j.tcs.2004.06.016.
  8. Calvès C, Fernández M. A polynomial nominal unification algorithm. Theor. Comput. Sci., 2008. 403(2-3):285–306. 10.1016/j.tcs.2008.05.012.
  9. Levy J, Villaret M. Nominal Unification from a Higher-Order Perspective. ACM Trans. Comput. Log., 2012. 13(2):10. 10.1145/2159531.2159532.
  10. Miller D. A Logic Programming Language with Lambda-Abstraction, Function Variables, and Simple Unification. J. Log. Comput., 1991. 1(4):497–536. 10.1093/logcom/1.4.497.
  11. Levy J, Villaret M. An Efficient Nominal Unification Algorithm. In: Lynch C (ed.), Proc. 21st RTA, volume 6 of LIPIcs. Schloss Dagstuhl, 2010 pp. 209–226. 10.4230/LIPIcs.RTA.2010.209.
  12. Completeness in PVS of a Nominal Unification Algorithm. ENTCS, 2016. 323(3):57–74. 10.1016/j.entcs.2016.06.005.
  13. A Formalisation of Nominal α𝛼\alphaitalic_α-equivalence with A and AC Function Symbols. Electr. Notes Theor. Comput. Sci., 2017. 332:21–38. 10.1016/j.entcs.2017.04.003.
  14. Nominal Narrowing. In: Pientka B, Kesner D (eds.), Proc. first FSCD, LIPIcs. 2016 pp. 11:1–11:17. 10.4230/LIPIcs.FSCD.2016.11.
  15. Cheney J. Equivariant Unification. J. Autom. Reasoning, 2010. 45(3):267–300. 10.1007/s10817-009-9164-3.
  16. Cheney J. Nominal Logic Programming. Ph.D. thesis, Cornell University, Ithaca, New York, U.S.A., 2004.
  17. Aoto T, Kikuchi K. A Rule-Based Procedure for Equivariant Nominal Unification. In: Informal proceedings HOR. 2016 p. 5.
  18. Erratic Fudgets: A semantic theory for an embedded coordination language. In: Coordination ’99, volume 1594 of LNCS. Springer-Verlag, 1999 pp. 85–102. 10.1007/3-540-48919-3_8.
  19. Safety of Nöcker’s Strictness Analysis. J. Funct. Programming, 2008. 18(04):503–551. 10.1017/S0956796807006624.
  20. Ariola ZM, Klop JW. Cyclic Lambda Graph Rewriting. In: Proc. IEEE LICS. IEEE Press, 1994 pp. 416–425. 10.1109/LICS.1994.316066.
  21. Marlow S (ed.). Haskell 2010 – Language Report. 2010. URL https://www.haskell.org.
  22. Cheney J. Toward a General Theory of Names: Binding and Scope. In: MERLIN 2005. ACM, 2005 pp. 33–40. 10.1145/1088454.1088459.
  23. Urban C, Kaliszyk C. General Bindings and Alpha-Equivalence in Nominal Isabelle. Log. Methods Comput. Sci., 2012. 8(2). 10.2168/LMCS-8(2:14)2012.
  24. Coinductive Logic Programming. In: Etalle S, Truszczynski M (eds.), 22nd ICLP, LNCS. 2006 pp. 330–345. 10.1007/11799573_25.
  25. CoCaml: Functional Programming with Regular Coinductive Types. Fundam. Inform., 2017. 150(3-4):347–377. 10.3233/FI-2017-1473.
  26. Martelli A, Montanari U. An efficient unification algorithm. ACM Trans. Program. Lang. Syst., 1982. 4(2):258–282. 10.1145/357162.357169.
  27. Nominal unification with atom-variables. J. Symb. Comput., 2019. 90:42–64. 10.1016/j.jsc.2018.04.003.
  28. Schmidt-Schauß M, Sabel D. Nominal Unification with Atom and Context Variables. In: Kirchner [49], 2018 pp. 28:1–28:20. 10.4230/LIPIcs.FSCD.2018.28.
  29. Nominal C-Unification. In: Fioravanti F, Gallagher JP (eds.), 27th LOPSTR, Revised Selected Papers, volume 10855 of LNCS. Springer, 2017 pp. 235–251. 10.1007/978-3-319-94460-9_14.
  30. Fixed-Point Constraints for Nominal Equational Unification. In: Kirchner [49], 2018 pp. 7:1–7:16. 10.4230/LIPIcs.FSCD.2018.7.
  31. Nominal Unification of Higher Order Expressions with Recursive Let. RISC Report Series 16-03, RISC, Johannes Kepler University Linz, Austria, 2016.
  32. Fernández M, Gabbay M. Nominal rewriting. Inf. Comput., 2007. 205(6):917–965. 10.1016/j.ic.2006.12.002.
  33. A rewriting calculus for cyclic higher-order term graphs. Mathematical Structures in Computer Science, 2007. 17(3):363–406. 10.1017/S0960129507006093.
  34. Rau C, Schmidt-Schauß M. A Unification Algorithm to Compute Overlaps in a Call-by-Need Lambda-Calculus with Variable-Binding Chains. In: Proc. 25th UNIF. 2011 pp. 35–41.
  35. Rau C, Schmidt-Schauß M. Towards Correctness of Program Transformations Through Unification and Critical Pair Computation. In: Proc. 24th UNIF, volume 42 of EPTCS. 2010 pp. 39–54. 10.4204/EPTCS.42.4.
  36. Schmidt-Schauß M, Sabel D. Unification of program expressions with recursive bindings. In: Cheney J, Vidal G (eds.), 18th PPDP. ACM, 2016 pp. 160–173. 10.1145/2967973.2968603.
  37. Permissive nominal terms and their unification: an infinite, co-infinite approach to nominal techniques. Log. J. IGPL, 2010. 18(6):769–822. 10.1093/jigpal/jzq006.
  38. Algorithms for Extended Alpha-Equivalence and Complexity. In: van Raamsdonk F (ed.), 24th RTA 2013, volume 21 of LIPIcs. Schloss Dagstuhl, 2013 pp. 255–270. doi:10.4230/LIPIcs.RTA.2013255.
  39. Luks EM. Permutation Groups and Polynomial-Time Computation. In: Finkelstein L, Kantor WM (eds.), Groups And Computation, volume 11 of DIMACS. DIMACS/AMS, 1991 pp. 139–176.
  40. Polynomial-Time Algorithms for Permutation Groups. In: 21st FoCS. IEEE Computer Society, 1980 pp. 36–41. 10.1109/SFCS.1980.34.
  41. Picouleau C. Complexity of the Hamiltonian Cycle in Regular Graph Problem. Theor. Comput. Sci., 1994. 131(2):463–473. 10.1016/0304-3975(94)90185-6.
  42. The Planar Hamiltonian Circuit Problem is NP-Complete. SIAM J. Comput., 1976. 5(4):704–714. 10.1137/0205049.
  43. A call-by-need lambda calculus. In: POPL’95. ACM Press, San Francisco, CA, 1995 pp. 233–246. 10.1145/199448.199507.
  44. Schöning U. Graph Isomorphism is in the Low Hierarchy. J. Comput. Syst. Sci., 1988. 37(3):312–323. 10.1016/0022-0000(88)90010-4.
  45. Babai L. Graph Isomorphism in Quasipolynomial Time. http://arxiv.org/abs/1512.03547v2, 2016.
  46. Booth KS. Isomorphism Testing for Graphs, Semigroups, and Finite Automata Are Polynomially Equivalent Problems. SIAM J. Comput., 1978. 7(3):273–279. 10.1137/0207023.
  47. Parameter reduction and automata evaluation for grammar-compressed trees. J. Comput. Syst. Sci., 2012. 78(5):1651–1669. 10.1016/j.jcss.2012.03.003.
  48. Unification and matching on compressed terms. ACM Trans. Comput. Log., 2011. 12(4):26:1–26:37. 10.1145/1970398.1970402.
  49. Kirchner H (ed.). 3rd International Conference on Formal Structures for Computation and Deduction, FSCD 2018, July 9-12, 2018, Oxford, UK, volume 108 of LIPIcs. Schloss Dagstuhl, 2018.
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