Extending the WMSO+U Logic with Quantification over Tuples

Authors Anita Badyl, Paweł Parys



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Anita Badyl
  • Institute of Informatics, University of Warsaw, Poland
Paweł Parys
  • Institute of Informatics, University of Warsaw, Poland

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Anita Badyl and Paweł Parys. Extending the WMSO+U Logic with Quantification over Tuples. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CSL.2024.12

Abstract

We study a new extension of the weak MSO logic, talking about boundedness. Instead of a previously considered quantifier 𝖴, expressing the fact that there exist arbitrarily large finite sets satisfying a given property, we consider a generalized quantifier 𝖴, expressing the fact that there exist tuples of arbitrarily large finite sets satisfying a given property. First, we prove that the new logic WMSO+𝖴_{tup} is strictly more expressive than WMSO+𝖴. In particular, WMSO+𝖴_{tup} is able to express the so-called simultaneous unboundedness property, for which we prove that it is not expressible in WMSO+𝖴. Second, we prove that it is decidable whether the tree generated by a given higher-order recursion scheme satisfies a given sentence of WMSO+𝖴_{tup}.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Verification by model checking
  • Theory of computation → Rewrite systems
Keywords
  • Boundedness
  • logic
  • decidability
  • expressivity
  • recursion schemes

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References

  1. Alfred V. Aho. Indexed grammars - an extension of context-free grammars. J. ACM, 15(4):647-671, 1968. URL: https://doi.org/10.1145/321479.321488.
  2. David Barozzini, Lorenzo Clemente, Thomas Colcombet, and Paweł Parys. Cost automata, safe schemes, and downward closures. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference), volume 168 of LIPIcs, pages 109:1-109:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.109.
  3. Mikołaj Bojańczyk. A bounding quantifier. In Jerzy Marcinkowski and Andrzej Tarlecki, editors, Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 20-24, 2004, Proceedings, volume 3210 of Lecture Notes in Computer Science, pages 41-55. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-30124-0_7.
  4. Mikołaj Bojańczyk. Weak MSO with the unbounding quantifier. Theory Comput. Syst., 48(3):554-576, 2011. URL: https://doi.org/10.1007/s00224-010-9279-2.
  5. Mikołaj Bojańczyk. Weak MSO+U with path quantifiers over infinite trees. In Javier Esparza, Pierre Fraigniaud, Thore Husfeldt, and Elias Koutsoupias, editors, Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part II, volume 8573 of Lecture Notes in Computer Science, pages 38-49. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-43951-7_4.
  6. Mikołaj Bojańczyk and Thomas Colcombet. Bounds in ω-regularity. In 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12-15 August 2006, Seattle, WA, USA, Proceedings, pages 285-296. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/LICS.2006.17.
  7. Mikołaj Bojańczyk and Szymon Toruńczyk. Weak MSO+U over infinite trees. In Christoph Dürr and Thomas Wilke, editors, 29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France, volume 14 of LIPIcs, pages 648-660. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.STACS.2012.648.
  8. Tomás Brázdil, Krishnendu Chatterjee, Antonín Kucera, and Petr Novotný. Efficient controller synthesis for consumption games with multiple resource types. In P. Madhusudan and Sanjit A. Seshia, editors, Computer Aided Verification - 24th International Conference, CAV 2012, Berkeley, CA, USA, July 7-13, 2012 Proceedings, volume 7358 of Lecture Notes in Computer Science, pages 23-38. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-31424-7_8.
  9. Luca Breveglieri, Alessandra Cherubini, Claudio Citrini, and Stefano Crespi-Reghizzi. Multi-push-down languages and grammars. Int. J. Found. Comput. Sci., 7(3):253-292, 1996. URL: https://doi.org/10.1142/S0129054196000191.
  10. Christopher H. Broadbent, Arnaud Carayol, C.-H. Luke Ong, and Olivier Serre. Higher-order recursion schemes and collapsible pushdown automata: Logical properties. ACM Trans. Comput. Log., 22(2):12:1-12:37, 2021. URL: https://doi.org/10.1145/3452917.
  11. Julius Richard Büchi. On a decision method in restricted second order arithmetic. In Proceedings of the 1960 International Congress on Logic, Methodology and Philosophy of Science, pages 1-11. Stanford University Press, 1962. Google Scholar
  12. Claudia Carapelle, Alexander Kartzow, and Markus Lohrey. Satisfiability of CTL* with constraints. In Pedro R. D'Argenio and Hernán C. Melgratti, editors, CONCUR 2013 - Concurrency Theory - 24th International Conference, CONCUR 2013, Buenos Aires, Argentina, August 27-30, 2013. Proceedings, volume 8052 of Lecture Notes in Computer Science, pages 455-469. Springer, 2013. URL: https://doi.org/10.1007/978-3-642-40184-8_32.
  13. Lorenzo Clemente, Paweł Parys, Sylvain Salvati, and Igor Walukiewicz. Ordered tree-pushdown systems. In Prahladh Harsha and G. Ramalingam, editors, 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, volume 45 of LIPIcs, pages 163-177. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2015.163.
  14. Lorenzo Clemente, Paweł Parys, Sylvain Salvati, and Igor Walukiewicz. The diagonal problem for higher-order recursion schemes is decidable. In Martin Grohe, Eric Koskinen, and Natarajan Shankar, editors, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016, pages 96-105. ACM, 2016. URL: https://doi.org/10.1145/2933575.2934527.
  15. Thomas Colcombet. The theory of stabilisation monoids and regular cost functions. In Susanne Albers, Alberto Marchetti-Spaccamela, Yossi Matias, Sotiris E. Nikoletseas, and Wolfgang Thomas, editors, Automata, Languages and Programming, 36th Internatilonal Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Proceedings, Part II, volume 5556 of Lecture Notes in Computer Science, pages 139-150. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-02930-1_12.
  16. Wojciech Czerwiński, Wim Martens, Lorijn van Rooijen, Marc Zeitoun, and Georg Zetzsche. A characterization for decidable separability by piecewise testable languages. Discret. Math. Theor. Comput. Sci., 19(4), 2017. URL: https://doi.org/10.23638/DMTCS-19-4-1.
  17. Werner Damm. The IO- and OI-hierarchies. Theor. Comput. Sci., 20:95-207, 1982. URL: https://doi.org/10.1016/0304-3975(82)90009-3.
  18. Calvin C. Elgot. Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc., 98(1):21-51, 1961. URL: https://doi.org/10.2307/1993511.
  19. E. Allen Emerson and Charanjit S. Jutla. Tree automata, mu-calculus and determinacy (extended abstract). In 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1-4 October 1991, pages 368-377. IEEE Computer Society, 1991. URL: https://doi.org/10.1109/SFCS.1991.185392.
  20. Nathanaël Fijalkow and Martin Zimmermann. Cost-parity and cost-Streett games. In Deepak D'Souza, Telikepalli Kavitha, and Jaikumar Radhakrishnan, editors, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012, December 15-17, 2012, Hyderabad, India, volume 18 of LIPIcs, pages 124-135. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2012. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2012.124.
  21. Matthew Hague, Jonathan Kochems, and C.-H. Luke Ong. Unboundedness and downward closures of higher-order pushdown automata. In Rastislav Bodík and Rupak Majumdar, editors, Proceedings of the 43rd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2016, St. Petersburg, FL, USA, January 20 - 22, 2016, pages 151-163. ACM, 2016. URL: https://doi.org/10.1145/2837614.2837627.
  22. Matthew Hague, Andrzej S. Murawski, C.-H. Luke Ong, and Olivier Serre. Collapsible pushdown automata and recursion schemes. ACM Trans. Comput. Log., 18(3):25:1-25:42, 2017. URL: https://doi.org/10.1145/3091122.
  23. Teodor Knapik, Damian Niwiński, and Paweł Urzyczyn. Higher-order pushdown trees are easy. In Mogens Nielsen and Uffe Engberg, editors, Foundations of Software Science and Computation Structures, 5th International Conference, FOSSACS 2002. Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2002 Grenoble, France, April 8-12, 2002, Proceedings, volume 2303 of Lecture Notes in Computer Science, pages 205-222. Springer, 2002. URL: https://doi.org/10.1007/3-540-45931-6_15.
  24. Naoki Kobayashi. Model checking higher-order programs. J. ACM, 60(3):20:1-20:62, 2013. URL: https://doi.org/10.1145/2487241.2487246.
  25. Naoki Kobayashi and C.-H. Luke Ong. A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, 11-14 August 2009, Los Angeles, CA, USA, pages 179-188. IEEE Computer Society, 2009. URL: https://doi.org/10.1109/LICS.2009.29.
  26. Orna Kupferman, Nir Piterman, and Moshe Y. Vardi. From liveness to promptness. Formal Methods Syst. Des., 34(2):83-103, 2009. URL: https://doi.org/10.1007/s10703-009-0067-z.
  27. C.-H. Luke Ong. On model-checking trees generated by higher-order recursion schemes. In 21th IEEE Symposium on Logic in Computer Science (LICS 2006), 12-15 August 2006, Seattle, WA, USA, Proceedings, pages 81-90. IEEE Computer Society, 2006. URL: https://doi.org/10.1109/LICS.2006.38.
  28. Paweł Parys. Recursion schemes and the WMSO+U logic. In Rolf Niedermeier and Brigitte Vallée, editors, 35th Symposium on Theoretical Aspects of Computer Science, STACS 2018, February 28 to March 3, 2018, Caen, France, volume 96 of LIPIcs, pages 53:1-53:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.STACS.2018.53.
  29. Paweł Parys. Recursion schemes, the MSO logic, and the U quantifier. Log. Methods Comput. Sci., 16(1), 2020. URL: https://doi.org/10.23638/LMCS-16(1:20)2020.
  30. Paweł Parys. A type system describing unboundedness. Discret. Math. Theor. Comput. Sci., 22(4), 2020. URL: https://doi.org/10.23638/DMTCS-22-4-2.
  31. Michael O. Rabin. Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc., 141:1-35, 1969. URL: https://doi.org/10.2307/1995086.
  32. Sylvain Salvati and Igor Walukiewicz. Krivine machines and higher-order schemes. Inf. Comput., 239:340-355, 2014. URL: https://doi.org/10.1016/j.ic.2014.07.012.
  33. Sylvain Salvati and Igor Walukiewicz. A model for behavioural properties of higher-order programs. In Stephan Kreutzer, editor, 24th EACSL Annual Conference on Computer Science Logic, CSL 2015, September 7-10, 2015, Berlin, Germany, volume 41 of LIPIcs, pages 229-243. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.CSL.2015.229.
  34. Sylvain Salvati and Igor Walukiewicz. Simply typed fixpoint calculus and collapsible pushdown automata. Math. Struct. Comput. Sci., 26(7):1304-1350, 2016. URL: https://doi.org/10.1017/S0960129514000590.
  35. Boris Trakhtenbrot. Finite automata and the logic of monadic predicates. Doklady Akademii Nauk SSSR, 140:326-329, 1961. Google Scholar
  36. Georg Zetzsche. An approach to computing downward closures. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 440-451. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47666-6_35.