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Confluence by Critical Pair Analysis Revisited

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Automated Deduction – CADE 27 (CADE 2019)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11716))

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Abstract

We present two methods for proving confluence of left-linear term rewrite systems. One is hot-decreasingness, combining the parallel/development closedness theorems with rule labelling based on a terminating subsystem. The other is critical-pair-closing system, allowing to boil down the confluence problem to confluence of a special subsystem whose duplicating rules are relatively terminating.

Supported by JSPS KAKENHI Grant Number 17K00011 and Core to Core Program.

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Notes

  1. 1.

    Modelled in various ways, via e.g.: tree homomorphisms (tree automata [7]), term-operations (algebra), context-variables, labelling (rippling [5]), to name a few.

  2. 2.

    For space reasons we have omitted the proof by decreasing diagrams of Theorem 3.

  3. 3.

    \(\mathsf {src}\) can be viewed as tree homomorphism [7], or as a term algebra \({\varrho }^{\mathcal {Lhs}}(\varvec{{t}}) ={\ell }^{[\varvec{{x}}\mathbin {{:}{=}}\varvec{{t}}]}\).

  4. 4.

    Here convex means that for each pair of positions \(p\),\(q\) in the set, all positions on the shortest path from \(p\) to \(q\) in the term tree are also in the set, cf. [32, Definition 8.6.21].

  5. 5.

    This fails for, e.g., connected graphs; these may fall apart into non-connected ones.

  6. 6.

    For the amount of overlap for redexes in parallel reduction , see e.g. [1, 17, 24].

  7. 7.

    We exclude neither overlays of a rule with itself nor pairs obtained by symmetry.

  8. 8.

    This does not create ambiguity with joins of multipatterns since if , then unless the let-bindings of both are empty, so both are bottom.

  9. 9.

    \(C\) is a prefix of the left-hand side \({\ell }\) of \({\varrho }\). For instance, for a peak from f(g(a)) between and , is mapped by to and by to .

  10. 10.

    For this to be a valid 2nd-order substitution, the 1st-order variables of \(\hat{{\varOmega }}\) (\(\hat{{L}}\)) must be contained in those of \(\hat{{t}}\), which we may assume by Lemma 7(1).

  11. 11.

    See problem 127 of http://cops.uibk.ac.at/results/?y=2019-full-run&c=TRS.

  12. 12.

    By the Finite Developments Theorem lengths of such developments are finite [32].

  13. 13.

    Detailed data are available from: http://www.jaist.ac.jp/project/saigawa/19cade/.

References

  1. Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)

    Book  Google Scholar 

  2. Barendregt, H.: The Lambda Calculus: Its Syntax and Semantics, Studies in Logic and the Foundations of Mathematics, vol. 103. North-Holland (1985)

    Google Scholar 

  3. Bechet, D., de Groote, P., Retoré, C.: A complete axiomatisation for the inclusion of series-parallel partial orders. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 230–240. Springer, Heidelberg (1997). https://doi.org/10.1007/3-540-62950-5_74

    Chapter  Google Scholar 

  4. Boudol, G.: Computational semantics of term rewriting systems. In: Nivat, M., Reynolds, J. (eds.) Algebraic Methods in Semantics, pp. 169–236. Cambridge University Press (1985)

    Google Scholar 

  5. Bundy, A., Basin, D., Hutter, D., Ireland, A.: Rippling: meta-level guidance for mathematical reasoning. In: Cambridge Tracts in Theoretical Computer Science, Cambridge University Press (2005). https://doi.org/10.1017/CBO9780511543326

  6. Church, A., Rosser, J.: Some properties of conversion. Transact. Am. Math. Soc. 39, 472–482 (1936)

    Article  MathSciNet  Google Scholar 

  7. Comon, H., et al.: Tree Automata Techniques and Applications (2007). http://www.grappa.univ-lille3.fr/tata

  8. Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  9. Dershowitz, N., Jouannaud, J.P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, Formal Models and Semantics, pp. 243–320. Elsevier (1990)

    Google Scholar 

  10. Dutertre, B.: Yices 2.2. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 737–744. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_49

    Chapter  Google Scholar 

  11. Endrullis, J., Klop, J., Overbeek, R.: Decreasing diagrams with two labels are complete for confluence of countable systems. In: Proceedings of 3rd FSCD. LIPIcs, vol. 108, pp. 14:1–14:15 (2018). https://doi.org/10.4230/LIPIcs.FSCD.2018.14

  12. Felgenhauer, B., Middeldorp, A., Zankl, H., van Oostrom, V.: Layer systems for proving confluence. ACM Transact. Computat. Logic 16, 1–32 (2015)

    Article  MathSciNet  Google Scholar 

  13. Felgenhauer, B.: Labeling multi-steps for confluence of left-linear term rewrite systems. In: Tiwari, A., Aoto, T. (eds.) Proceedings of 4th IWC, pp. 33–37 (2015)

    Google Scholar 

  14. Hirokawa, H., Klein, D.: Saigawa: A confluence tool. In: Proceedings of 1st IWC, p. 49 (2012). http://www.jaist.ac.jp/project/saigawa/

  15. Hirokawa, N., Nagele, J., Middeldorp, A.: Cops and CoCoWeb: infrastructure for confluence tools. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 346–353. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_23

    Chapter  Google Scholar 

  16. Hirokawa, N., Middeldorp, A.: Decreasing diagrams and relative termination. J. Autom. Reasoning 47(4), 481–501 (2011)

    Article  MathSciNet  Google Scholar 

  17. Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. J. ACM 27(4), 797–821 (1980)

    Article  MathSciNet  Google Scholar 

  18. Huet, G., Lévy, J.J.: Computations in orthogonal rewriting systems, I. In: Lassez, J.L., Plotkin, G. (eds.) Computational Logic: Essays in Honor of Alan Robinson, chap. 11. The MIT Press (1991)

    Google Scholar 

  19. Knuth, D.E., Bendix, P.B.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, Proceedings of a Conference held at Oxford under the Auspices of the Science Research Council Atlas Computer Laboratory, 29 August–2 September 1967. pp. 263–297 (1970)

    Google Scholar 

  20. Liu, J.L.: Propriétés de Confluence des Règles de Réécriture par des Diagrammes Décroissants. Ph.D. thesis, Tsinghua University and l’Université Paris-Saclay préparée à l’École Polytechnique (2016)

    Google Scholar 

  21. Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. Theoret. Comput. Sci. 192(1), 3–29 (1998). https://doi.org/10.1016/S0304-3975(97)00143-6

    Article  MathSciNet  MATH  Google Scholar 

  22. Meseguer, J.: Conditional rewriting logic as a unified model of concurrency. Theoret. Comput. Sci. 96, 73–155 (1992)

    Article  MathSciNet  Google Scholar 

  23. Métivier, Y.: About the rewriting systems produced by the Knuth–Bendix completion algorithm. Inf. Process. Lett. 16(1), 31–34 (1983)

    Article  MathSciNet  Google Scholar 

  24. Nagele, J., Middeldorp, A.: Certification of classical confluence results for left-linear term rewrite systems. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 290–306. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43144-4_18

    Chapter  Google Scholar 

  25. Newman, M.: On theories with a combinatorial definition of equivalence. Ann. Math. 43(2), 223–243 (1942)

    Article  MathSciNet  Google Scholar 

  26. Ohlebusch, E.: Advanced Topics in Term Rewriting. Springer, Heidelberg (2002). https://doi.org/10.1007/978-1-4757-3661-8

    Book  MATH  Google Scholar 

  27. Okui, S.: Simultaneous critical pairs and Church-Rosser property. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 2–16. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0052357

    Chapter  Google Scholar 

  28. van Oostrom, V.: Confluence for Abstract and Higher-Order Rewriting. Ph.D. thesis, Vrije Universiteit, Amsterdam, March 1994

    Google Scholar 

  29. van Oostrom, V.: Developing developments. Theoret. Comput. Sci. 175(1), 159–181 (1997)

    Article  MathSciNet  Google Scholar 

  30. van Oostrom, V.: Confluence by decreasing diagrams. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 306–320. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70590-1_21

    Chapter  Google Scholar 

  31. Rosen, B.: Tree-manipulating systems and Church-Rosser theorems. J. ACM 20, 160–187 (1973)

    Article  MathSciNet  Google Scholar 

  32. Terese: Term Rewriting Systems. Cambridge University Press (2003)

    Google Scholar 

  33. Toyama, Y.: On the Church-Rosser property for the direct sum of term rewriting systems. J. ACM 34(1), 128–143 (1987)

    Article  MathSciNet  Google Scholar 

  34. Yamada, A., Kusakari, K., Sakabe, T.: Nagoya termination tool. In: Dowek, G. (ed.) RTA 2014. LNCS, vol. 8560, pp. 466–475. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08918-8_32

    Chapter  Google Scholar 

  35. Zankl, H., Felgenhauer, B., Middeldorp, A.: Labelings for decreasing diagrams. J. Autom. Reasoning 54(2), 101–133 (2015)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. JAIST, Nomi, Japan

    Nao Hirokawa

  2. Queen Mary University of London, London, UK

    Julian Nagele

  3. University of Innsbruck, Innsbruck, Austria

    Vincent van Oostrom

  4. Nagoya University, Nagoya, Japan

    Michio Oyamaguchi

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  1. Nao Hirokawa
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  4. Michio Oyamaguchi
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Correspondence to Nao Hirokawa .

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  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Hirokawa, N., Nagele, J., van Oostrom, V., Oyamaguchi, M. (2019). Confluence by Critical Pair Analysis Revisited. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_19

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