Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
153 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 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

Ruitenburg's Theorem mechanized and contextualized (2402.01840v1)

Published 2 Feb 2024 in cs.LO

Abstract: In 1984, Wim Ruitenburg published a surprising result about periodic sequences in intuitionistic propositional calculus (IPC). The property established by Ruitenburg naturally generalizes local finiteness (intuitionistic logic is not locally finite, even in a single variable). However, one of the two main goals of this note is to illustrate that most "natural" non-classical logics failing local finiteness also do not enjoy the periodic sequence property; IPC is quite unique in separating these properties. The other goal of this note is to present a Coq formalization of Ruitenburg's heavily syntactic proof. Apart from ensuring its correctness, the formalization allows extraction of a program providing a certified implementation of Ruitenburg's algorithm.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (47)
  1. Johan Benthem “Modal Frame Correspondences and Fixed-Points” In Studia Logica 83.1-3, 2006, pp. 133–155
  2. P.N. Benton, Gavin M. Bierman and Valeria Paiva “Computational Types from a Logical Perspective” In J. Funct. Program. 8.2, 1998, pp. 177–193
  3. “First Steps in Synthetic Guarded Domain Theory: Step-Indexing in the Topos of Trees” In LMCS 8, 2012, pp. 1–45
  4. “Algebraizable logics” 77 (396), Memoirs AMS AMS, 1989
  5. “Undecidability of Propositional Separation Logic and Its Neighbours” In J. ACM 61.2 New York, NY, USA: ACM, 2014, pp. 14:1–14:43 DOI: 10.1145/2542667
  6. “Modal Logic”, Oxford Logic Guides 35 Oxford: Clarendon Press, 1997
  7. “Sequent Calculus in the Topos of Trees” In Proceedings of FoSSaCS 2015 9034, LNCS Springer, 2015, pp. 133–147 DOI: 10.1007/978-3-662-46678-0_9
  8. J.Michael Dunn “Algebraic Completeness Results for R-Mingle and Its Extensions” In The Journal of Symbolic Logic 35.1 Association for Symbolic Logic, 1970, pp. 1–13 URL: http://www.jstor.org/stable/2271149
  9. Roy Dyckhoff “Contraction-free sequent calculi for intuitionistic logic” In Journal of Symbolic Logic 57, 1992, pp. 795–807
  10. “Propositional Lax Logic” In Inf. Comput. 137.1, 1997, pp. 1–33
  11. “Formalizing and Computing Propositional Quantifiers” In Proceedings of CPP 2023 ACM, 2023, pp. 148–158 DOI: 10.1145/3573105.3575668
  12. Josep Maria Font, Ramon Jansana and Don Pigozzi “A Survey of Abstract Algebraic Logic” In Studia Logica 74.1-2, 2003, pp. 13–97
  13. “Residuated Lattices: An Algebraic Glimpse at Substructural Logics”, Studies in Logic and the Foundations of Mathematics 151 Elsevier, 2007
  14. “Sheaves, Games and Model Completions” 14, Trends In Logic, Studia Logica Library Dordrecht: Kluwer, 2002
  15. Silvio Ghilardi, Maria João Gouveia and Luigi Santocanale “Fixed-Point Elimination in the Intuitionistic Propositional Calculus” In Proceedings of FoSSaCS 2016 Berlin, Heidelberg: Springer Berlin Heidelberg, 2016, pp. 126–141 DOI: 10.1007/978-3-662-49630-5_8
  16. Silvio Ghilardi, Maria João Gouveia and Luigi Santocanale “Fixed-point Elimination in the Intuitionistic Propositional Calculus” In ACM Trans. Comput. Log. 21.1, 2020, pp. 4:1–4:37 DOI: 10.1145/3359669
  17. “Free Heyting algebra endomorphisms: Ruitenburg’s Theorem and beyond” In Math. Struct. Comput. Sci. 30.6, 2020, pp. 572–596 DOI: 10.1017/S0960129519000203
  18. “Ruitenburg’s Theorem via Duality and Bounded Bisimulations” In Proceedings of AiML 2018 College Publications, 2018, pp. 277–290 URL: http://www.aiml.net/volumes/volume12/Ghilardi-Santocanale.pdf
  19. “A sheaf representation and duality for finitely presented Heyting algebras” In Journal of Symbolic Logic 60, 1995, pp. 911–939
  20. José Gil-Férez, Peter Jipsen and George Metcalfe “Structure theorems for idempotent residuated lattices” In Algebra universalis 81.2, 2020, pp. 28 DOI: 10.1007/s00012-020-00659-5
  21. Rajeev Goré, Revantha Ramanayake and Ian Shillito “Cut-Elimination for Provability Logic by Terminating Proof-Search: Formalised and Deconstructed Using Coq” In Proceedings of TABLEAUX 2021 12842, Lecture Notes in Computer Science Springer, 2021, pp. 299–313 DOI: 10.1007/978-3-030-86059-2\_18
  22. Jörg Hudelmaier “Bounds for cut elimination in intuitionistic propositional logic”, Ph.D. Thesis Tübingen: University of Tübingen, 1989
  23. Lloyd Humberstone “The Connectives” MIT Press, 2011
  24. Tadeusz Litak “Constructive modalities with provability smack” https://arxiv.org/abs/1708.05607 In Leo Esakia on duality in modal and intuitionistic logics 4, Outstanding Contributions to Logic Springer, 2014 DOI: 10.1007/978-94-017-8860-1_7
  25. Tadeusz Litak, Miriam Polzer and Ulrich Rabenstein “Negative Translations and Normal Modality” In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017) 84, Leibniz International Proceedings in Informatics (LIPIcs) Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2017, pp. 27:1–27:18 DOI: 10.4230/LIPIcs.FSCD.2017.27
  26. Sergey I. Mardaev “Convergence of positive schemes in S4 and Int” In Algebra and Logic 33.2, 1994, pp. 95–101 DOI: 10.1007/BF00739995
  27. Sergey I. Mardaev “Definable fixed points in modal and temporal logics—a survey” In Journal of Applied Non-Classical Logics 17.3 Taylor & Francis, 2007, pp. 317–346
  28. Sergey I. Mardaev “Fixed points of modal negative operators” In Bull. Sect. Log., Univ. Lodz, Dep. Log 26, 1997, pp. 135–138
  29. Sergey I. Mardaev “Least fixed points in Grzegorczyk’s logic and in the intuitionistic propositional logic” In Algebra and Logic 32.5, 1993, pp. 279–288 DOI: 10.1007/BF02261708
  30. Sergey I. Mardaev “Negative modal schemes” In Algebra and Logic 37.3, 1998, pp. 187–191 DOI: 10.1007/BF02671590
  31. “Applicative programming with effects” In J. Funct. Program. 18.1, 2008, pp. 1–13
  32. Iwao Nishimura “On Formulas of One Variable in Intuitionistic Propositional Calculus” In The Journal of Symbolic Logic 25.4 Association for Symbolic Logic, 1960, pp. 327–331 URL: http://www.jstor.org/stable/2963526
  33. Peter W. O’Hearn and David J. Pym “The Logic of Bunched Implications” In The Bulletin of Symbolic Logic 5.2 Association for Symbolic Logic, 1999, pp. 215–244 URL: http://www.jstor.org/stable/421090
  34. Andrew M. Pitts “On an interpretation of second order quantification in first order intuitionistic propositional logic” In The Journal of Symbolic Logic 57, 1992, pp. 33–52 DOI: 10.2307/2275175
  35. David J. Pym, Peter W. O’Hearn and Hongseok Yang “Possible worlds and resources: the semantics of BI” Mathematical Foundations of Programming Semantics In Theoretical Computer Science 315.1, 2004, pp. 257–305 DOI: http://dx.doi.org/10.1016/j.tcs.2003.11.020
  36. James Raftery “Representable idempotent commutative residuated lattices” In Trans. Am. Math. Soc. 359, 2007 DOI: 10.1090/S0002-9947-07-04235-3
  37. John C. Reynolds “Separation Logic: A Logic for Shared Mutable Data Structures” In Proceedings of LiCS 2002 IEEE Computer Society, 2002, pp. 55–74 DOI: 10.1109/LICS.2002.1029817
  38. Ladislav Rieger “On the lattice theory of Brouwerian propositional logic”, Acta Facultatis Rerum Naturalium Universitatis Carolinae 189 F. Řivnáč, Prague, 1949
  39. W. Ruitenburg “On the period of sequences (An⁢(p))superscript𝐴𝑛𝑝(A^{n}(p))( italic_A start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ( italic_p ) ) in intuitionistic propositional calculus” In Journal of Symbolic Logic 49, 1984, pp. 892–899
  40. Ilya B Shapirovsky “Glivenko’s theorem, finite height, and local finiteness” arXiv preprint arXiv:1806.06899 In Advances in Modal Logic 2018, 2018
  41. “A New Calculus for Intuitionistic Strong Löb Logic: Strong Termination and Cut-Elimination, Formalised” In Proceedings of TABLEAUX 2023 14278, Lecture Notes in Computer Science Springer, 2023, pp. 73–93 DOI: 10.1007/978-3-031-43513-3\_5
  42. “Direct elimination of additive-cuts in GL4ip: verified and extracted” In Proceedings of AiML 2022 College Publications, 2022, pp. 429–450
  43. Alex K. Simpson “The Proof Theory and Semantics of Intuitionistic Modal Logic”, 1994 URL: http://homepages.inf.ed.ac.uk/als/Research/thesis.ps.gz
  44. Albert Visser “Löb’s Logic Meets the μ𝜇\muitalic_μ-calculus” In Processes, Terms and Cycles: Steps on the Road to Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday 3838, Lecture Notes in Computer Science Springer, 2005, pp. 14–25
  45. Albert Visser “Uniform Interpolation and Layered Bisimulation” reprinted as Lecture Notes in Logic 6, Association of Symbolic Logic In Gödel ’96, Logical Foundations of Mathematics, Computer Science and Physics —Kurt Gödel’s Legacy Berlin: Springer, 1996, pp. 139–164
  46. “Intuitionistic Modal Logics as fragments of Classical Modal Logics” In Logic at Work, Essays in honour of Helena Rasiowa Springer–Verlag, 1998, pp. 168–186
  47. “On the relation between intuitionistic and classical modal logics” In Algebra and Logic 36, 1997, pp. 121–125

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