Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
132 tokens/sec
GPT-4o
28 tokens/sec
Gemini 2.5 Pro Pro
42 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Left-Linear Completion with AC Axioms (2405.17109v2)

Published 27 May 2024 in cs.LO

Abstract: We revisit completion modulo equational theories for left-linear term rewrite systems where unification modulo the theory is avoided and the normal rewrite relation can be used in order to decide validity questions. To that end, we give a new correctness proof for finite runs and establish a simulation result between the two inference systems known from the literature. Given a concrete reduction order, novel canonicity results show that the resulting complete systems are unique up to the representation of their rules' right-hand sides. Furthermore, we show how left-linear AC completion can be simulated by general AC completion. In particular, this result allows us to switch from the former to the latter at any point during a completion process.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (36)
  1. Jürgen Avenhaus. Reduktionssysteme. Springer Berlin Heidelberg, 1995. In German. doi:10.1007/978-3-642-79351-6.
  2. Leo Bachmair. Canonical Equational Proofs. Progress in Theoretical Computer Science. Birkhäuser Boston, 1991. doi:10.1007/978-1-4684-7118-2.
  3. Ahlem Ben Cherifa and Pierre Lescanne. Termination of rewriting systems by polynomial interpretations and its implementation. Science of Computer Programming, 9(2):137–159, 1987. doi:10.1016/0167-6423(87)90030-X.
  4. Term Rewriting and All That. Cambridge University Press, 1998. doi:10.1017/CBO9781139172752.
  5. Evelyne Contejean. A certified AC matching algorithm. In Vincent van Oostrom, editor, Proc. 15th International Conference on Rewriting Techniques and Applications, volume 3091 of Lecture Notes in Computer Science, pages 70–84, 2004. doi:10.1007/978-3-540-25979-4_5.
  6. Nachum Dershowitz. A note on simplification orderings. Information Processing Letters, 9(5):212–215, 1979. doi:10.1016/0020-0190(79)90071-1.
  7. Group-like structures in general categories I muliplications and comultiplications. Mathematische Annalen, 145(3):227–255, 1962. doi:10.1007/BF01451367.
  8. A Haskell library for term rewriting. In Proc. 1st International Workshop on Haskell and Rewriting Techniques, 2013. doi:10.48550/ARXIV.1307.2328.
  9. Bertram Felgenhauer and Vincent van Oostrom. Proof orders for decreasing diagrams. In Femke van Raamsdonk, editor, Proc. 24th International Conference on Rewriting Techniques and Applications, volume 21 of Leibniz International Proceedings in Informatics, pages 174–189, 2013. doi:10.4230/LIPIcs.RTA.2013.174.
  10. MU-TERM: Verify termination properties automatically (system description). In Nicolas Peltier and Viorica Sofronie-Stokkermans, editors, Proc. 10th International Joint Conference on Automated Reasoning, volume 12167 of Lecture Notes in Artificial Intelligence, pages 436–447, 2020. doi:10.1007/978-3-030-51054-1_28.
  11. Nao Hirokawa. Completion and reduction orders. In Naoki Kobayashi, editor, Proc. 6th International Conference on Formal Structures for Computation and Deduction, volume 195 of Leibniz International Proceedings in Informatics, pages 2:1–2:9, 2021. doi:10.4230/LIPIcs.FSCD.2021.2.
  12. Abstract completion, formalized. Logical Methods in Computer Science, 15(3):19:1–19:42, 2019. doi:10.23638/LMCS-15(3:19)2019.
  13. Gérard Huet. Confluent reductions: Abstract properties and applications to term rewriting systems. Journal of the ACM, 27(4):797–821, 1980. doi:10.1145/322217.322230.
  14. Completion of a set of rules modulo a set of equations. SIAM Journal on Computing, 15(4):1155–1194, 1986. doi:10.1137/0215084.
  15. Simple word problems in universal algebras. In John Leech, editor, Computational Problems in Abstract Algebra, pages 263–297. Pergamon Press, 1970. doi:10.1016/B978-0-08-012975-4.50028-X.
  16. Maximal completion. In Manfred Schmidt-Schauß, editor, Proc. 22nd International Conference on Rewriting Techniques and Applications, volume 10 of Leibniz International Proceedings in Informatics, pages 71–80, 2011. doi:10.4230/LIPIcs.RTA.2011.71.
  17. Only prime superpositions need be considered in the Knuth–Bendix completion procedure. Journal of Symbolic Computation, 6(1):19–36, 1988. doi:10.1016/S0747-7171(88)80019-1.
  18. Tyrolean Termination Tool 2. In Ralf Treinen, editor, Proc. 20th International Conference on Rewriting Techniques and Applications, volume 5595 of Lecture Notes in Computer Science, pages 295–304, 2009. doi:10.1007/978-3-642-02348-4_21.
  19. Claude Marché. Normalized rewriting: An alternative to rewriting modulo a set of equations. Journal of Symbolic Computation, 21(3):253–288, 1996. doi:10.1006/jsco.1996.0011.
  20. José Meseguer. Strict coherence of conditional rewriting modulo axioms. Theoretical Computer Science, 672:1–35, 2017. doi:10.1016/j.tcs.2016.12.026.
  21. Church–Rosser modulo for left-linear TRSs revisited. In Cyrille Chenavier and Sarah Winkler, editors, Proc. 12th International Workshop on Confluence, pages 14–19, 2023.
  22. Left-linear completion with AC axioms. In Brigitte Pientka and Cesare Tinelli, editors, Proc. 29th International Conference on Automated Deduction, volume 14132 of Lecture Notes in Artificial Intelligence, pages 401–418, 2023. doi:10.1007/978-3-031-38499-8_23.
  23. Enno Ohlebusch. Church–Rosser theorems for abstract reduction modulo an equivalence relation. In Tobias Nipkow, editor, Proc. 9th International Conference on Rewriting Techniques and Applications, volume 1379 of Lecture Notes in Computer Science, pages 17–31, 1998. doi:10.1007/BFb0052358.
  24. Complete sets of reductions for some equational theories. Journal of the ACM, 28(2):233–264, 1981. doi:10.1145/322248.322251.
  25. Andrea Sattler-Klein. About changing the ordering during Knuth–Bendix completion. In Patrice Enjalbert, Ernst W. Mayr, and Klaus W. Wagner, editors, Proc. 11th Annual Symposium on Theoretical Aspects of Computer Science, volume 775 of Lecture Notes in Computer Science, pages 175–186, 1994. doi:10.1007/3-540-57785-8_140.
  26. Formalizing Knuth–Bendix orders and Knuth–Bendix completion. In Femke van Raamsdonk, editor, Proc. 24th International Conference on Rewriting Techniques and Applications, volume 21 of Leibniz International Proceedings in Informatics, pages 286–301, 2013. doi:10.4230/LIPIcs.RTA.2013.287.
  27. Encoding dependency pair techniques and control strategies for maximal completion. In Amy P. Felty and Aart Middeldorp, editors, Proc. 25th International Conference on Automated Deduction, volume 9195 of Lecture Notes in Computer Science, pages 152–162, 2015. doi:10.1007/978-3-319-21401-6_10.
  28. KBCV – Knuth–Bendix completion visualizer. In Bernhard Gramlich, Dale Miller, and Uli Sattler, editors, Proc. 6th International Joint Conference on Automated Reasoning, volume 7364 of Lecture Notes in Artificial Intelligence, pages 530–536, 2012. doi:10.1007/978-3-642-31365-3_41.
  29. Vincent van Oostrom. Confluence by decreasing diagrams. Theoretical Computer Science, 126(2):259–280, 1994. doi:10.1016/0304-3975(92)00023-K.
  30. Sarah Winkler. Termination Tools in Automated Reasoning. PhD thesis, University of Innsbruck, 2013.
  31. Sarah Winkler. Extending maximal completion. In Herman Geuvers, editor, Proc. 4th International Conference on Formal Structures for Computation and Deduction, volume 131 of Leibniz International Proceedings in Informatics, pages 3:1–3:15, 2019. doi:10.4230/LIPIcs.FSCD.2019.3.
  32. AC completion with termination tools. In Nikolaj Bjørner and Viorica Sofronie-Stokkermans, editors, Proc. 23rd International Conference on Automated Deduction, volume 6803 of Lecture Notes in Artificial Intelligence, pages 492–498, 2011. doi:10.1007/978-3-642-22438-6_37.
  33. Normalized completion revisited. In Femke van Raamsdonk, editor, Proc. 24th International Conference on Rewriting Techniques and Applications, volume 21 of Leibniz International Proceedings in Informatics, pages 318–333, 2013. doi:10.4230/LIPIcs.RTA.2013.319.
  34. Multi-completion with termination tools. Journal of Automated Reasoning, 50(3):317–354, 2013. doi:10.1007/s10817-012-9249-2.
  35. Slothrop: Knuth–Bendix completion with a modern termination checker. In Frank Pfenning, editor, Proc. 17th International Conference on Rewriting Techniques and Applications, volume 4098 of Lecture Notes in Computer Science, pages 287–296, 2006. doi:10.1007/11805618_22.
  36. Nagoya Termination Tool. In Gilles Dowek, editor, Proc. 25th International Conference on Rewriting Techniques and Applications and 12th International Conference on Typed Lambda Calculi and Applications, volume 8560 of Lecture Notes in Computer Science, pages 466–475, 2014. doi:10.1007/978-3-319-08918-8_32.

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now
X Twitter Logo Streamline Icon: https://streamlinehq.com

Tweets