Skip to main content
SpringerLink
Log in

Uniform Interpolation from Cyclic Proofs: The Case of Modal Mu-Calculus

  • Conference paper
  • First Online:
Book cover Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2021)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12842))

Abstract

We show how to construct uniform interpolants in the context of the modal mu-calculus. D’Agostino and Hollenberg (2000) were the first to prove that this logic has the uniform interpolation property, employing a combination of semantic and syntactic methods. This article outlines a purely proof-theoretic approach to the problem based on insights from the cyclic proof theory of mu-calculus. We argue the approach has the potential to lend itself to other temporal and fixed point logics.

Supported by the Knut and Alice Wallenberg Foundation [2015.0179] and the Swedish Research Council [2016-03502 & 2017-05111]. The authors would like to thank the anonymous referees for their valuable comments.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This definition depends on the choice of underlying logic. For example, in the case of polymodal logic, \(\mathsf {Voc}({\varphi })\) is the set of propositional constants and modal actions occurring in \(\varphi \).

  2. 2.

    The case distinction based on the provability of \( { \varGamma _i , \pi \Rightarrow \varnothing }\) brings into question the computational cost of constructing uniform interpolants. Lemmas 1 and 3, however, provide that provability of sequents with empty consequent is implicit in the interpolation template.

  3. 3.

    We assume Definition 4 is generalised to derivations with \( \mathsf {GMod}\). No additional restrictions are necessary to accommodate this rule

  4. 4.

    A non-exhaustive list of cyclic proofs systems include: first-order logic with inductive definitions [8, 9, 11], arithmetic [7, 17, 39], linear logic [3, 4, 20], modal and dynamic logics [1, 22, 23, 28, 30, 38, 40, 41, 44], program semantics [37], automated theorem proving [10, 36, 42], higher-order logic [31] and algebras and lattices [18, 19, 32].

References

  1. Afshari, B., Leigh, G.E.: Cut-free completeness for modal mu-calculus. In: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, 20–23 June 2017, pp. 1–12. IEEE Computer Society (2017)

    Google Scholar 

  2. Afshari, B., Leigh, G.E.: Lyndon interpolation for modal mu-calculus. In: Post-Proceedings of TbiLLC 2019. (to appear)

    Google Scholar 

  3. Baelde, D., Doumane, A., Kuperberg, D., Saurin, A.: Bouncing threads for infinitary and circular proofs (2020). https://arxiv.org/abs/2005.08257

  4. Baelde, D., Doumane, A., Saurin, A.: Infinitary proof theory: the multiplicative additive case. In: Talbot, J., Regnier, L. (eds.) 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29–September 1 2016, Marseille, France. LIPIcs, vol. 62, pp. 42:1–42:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

    Google Scholar 

  5. Benedikt, M.: How can reasoners simplify database querying (and why haven’t they done it yet)? In: den Bussche, J.V., Arenas, M. (eds.) Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, Houston, TX, USA, 10–15 June 2018, pp. 1–15. ACM (2018)

    Google Scholar 

  6. Benedikt, M., ten Cate, B., Vanden Boom, M.: Interpolation with decidable fixpoint logics. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, Japan, 6–10 July 2015, pp. 378–389. IEEE Computer Society (2015)

    Google Scholar 

  7. Berardi, S., Tatsuta, M.: Equivalence of inductive definitions and cyclic proofs under arithmetic. In: 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, 20–23 June 2017, pp. 1–12. IEEE Computer Society (2017)

    Google Scholar 

  8. Berardi, S., Tatsuta, M.: Classical system of Martin-Löf’s inductive definitions is not equivalent to cyclic proofs. Log. Methods Comput. Sci. 15(3), 1:1–1:39 (2019)

    Google Scholar 

  9. Brotherston, J.: Cyclic proofs for first-order logic with inductive definitions. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 78–92. Springer, Heidelberg (2005). https://doi.org/10.1007/11554554_8

    Chapter  MATH  Google Scholar 

  10. Brotherston, J., Gorogiannis, N., Petersen, R.L.: A generic cyclic theorem prover. In: Jhala, R., Igarashi, A. (eds.) APLAS 2012. LNCS, vol. 7705, pp. 350–367. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35182-2_25

    Chapter  Google Scholar 

  11. Brotherston, J., Simpson, A.: Sequent calculi for induction and infinite descent. J. Log. Comput. 21(6), 1177–1216 (2011)

    Article  MathSciNet  Google Scholar 

  12. ten Cate, B., Franconi, E., Seylan, I.: Beth definability in expressive description logics. J. Artif. Intell. Res. 48, 347–414 (2013)

    Article  MathSciNet  Google Scholar 

  13. D’Agostino, G.: Interpolation in non-classical logics. Synthese 164(3), 421–435 (2008)

    Article  MathSciNet  Google Scholar 

  14. D’Agostino, G., Hollenberg, M.: Logical questions concerning the \(\mu \)-calculus. J. Symb. Log. 65(1), 310–332 (2000)

    Article  Google Scholar 

  15. D’Agostino, G., Lenzi, G.: An axiomatization of bisimulation quantifiers via the mu-calculus. Theor. Comput. Sci. 338(1–3), 64–95 (2005)

    Article  Google Scholar 

  16. D’Agostino, G., Lenzi, G.: Bisimulation quantifiers and uniform interpolation for guarded first order logic. Theor. Comput. Sci. 563, 75–85 (2015)

    Article  MathSciNet  Google Scholar 

  17. Das, A.: On the logical complexity of cyclic arithmetic. Log. Methods Comput. Sci. 16(1), 10:1–10:25 (2020)

    Google Scholar 

  18. Das, A., Pous, D.: A cut-free cyclic proof system for Kleene algebra. In: Schmidt, R.A., Nalon, C. (eds.) TABLEAUX 2017. LNCS (LNAI), vol. 10501, pp. 261–277. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66902-1_16

    Chapter  Google Scholar 

  19. Das, A., Pous, D.: Non-wellfounded proof theory for (kleene+action)(algebras+lattices). In: Ghica, D.R., Jung, A. (eds.) 27th EACSL Annual Conference on Computer Science Logic, CSL 2018, 4–7 September 2018, Birmingham, UK. LIPIcs, vol. 119, pp. 19:1–19:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

    Google Scholar 

  20. De, A., Saurin, A.: Infinets: the parallel syntax for non-wellfounded proof-theory. In: Cerrito, S., Popescu, A. (eds.) TABLEAUX 2019. LNCS (LNAI), vol. 11714, pp. 297–316. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29026-9_17

    Chapter  Google Scholar 

  21. Demri, S., Goranko, V., Lange, M.: Temporal Logics in Computer Science: Finite-State Systems, vol. 58. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  22. Docherty, S., Rowe, R.N.S.: A non-wellfounded, labelled proof system for propositional dynamic logic. In: Cerrito, S., Popescu, A. (eds.) TABLEAUX 2019. LNCS (LNAI), vol. 11714, pp. 335–352. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-29026-9_19

    Chapter  Google Scholar 

  23. Enqvist, S., Hansen, H.H., Kupke, C., Marti, J., Venema, Y.: Completeness for game logic. In: 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2019, Vancouver, BC, Canada, 24–27 June 2019, pp. 1–13. IEEE (2019)

    Google Scholar 

  24. Gabbay, D.M., Maksimova, L.: Interpolation and Definability. Modal and Intuitionistic Logic. Oxford University Press, Oxford (2005) Logic. Oxford University Press, Oxford (2005)

    Google Scholar 

  25. Iemhoff, R.: Uniform interpolation and sequent calculi in modal logic. Arch. Math. Log. 58(1–2), 155–181 (2019)

    Article  MathSciNet  Google Scholar 

  26. Iemhoff, R.: Uniform interpolation and the existence of sequent calculi. Ann. Pure Appl. Log. 170(11), 102711 (2019)

    Google Scholar 

  27. Janin, D., Walukiewicz, I.: Automata for the modal \(\mu \)-calculus and related results. In: Wiedermann, J., Hájek, P. (eds.) MFCS 1995. LNCS, vol. 969, pp. 552–562. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-60246-1_160

    Chapter  Google Scholar 

  28. Jungteerapanich, N.: A tableau system for the modal \(\mu \)-calculus. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 220–234. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02716-1_17

    Chapter  Google Scholar 

  29. Jungteerapanich, N.: Tableau systems for the modal \(\mu \)-calculus. Ph.D. thesis, University of Edinburgh (2010)

    Google Scholar 

  30. Kokkinis, I., Studer, T.: Cyclic proofs for linear temporal logic. In: Probst, D., Schuster, P. (eds.) Concepts of Proof in Mathematics, Philosophy, and Computer Science, Ontos Mathematical Logic, vol. 6, pp. 171–192. De Gruyter (2016)

    Google Scholar 

  31. Kori, M., Tsukada, T., Kobayashi, N.: A cyclic proof system for \(\text{HFL}\_\mathbb{N}\). In: Baier, C., Goubault-Larrecq, J. (eds.) 29th EACSL Annual Conference on Computer Science Logic, CSL 2021, 25–28 January 2021, Ljubljana, Slovenia (Virtual Conference). LIPIcs, vol. 183, pp. 29:1–29:22. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

    Google Scholar 

  32. Kuznetsov, S.: Half a way towards circular proofs for Kleene lattices (2019). circularity in Syntax and Semantics

    Google Scholar 

  33. Lutz, C., Wolter, F.: Foundations for uniform interpolation and forgetting in expressive description logics. In: Walsh, T. (ed.) IJCAI 2011, Proceedings of the 22nd International Joint Conference on Artificial Intelligence, Barcelona, Catalonia, Spain, 16–22 July 2011, pp. 989–995. IJCAI/AAAI (2011)

    Google Scholar 

  34. Niwinski, D., Walukiewicz, I.: Games for the mu-calculus. Theor. Comput. Sci. 163(1 & 2), 99–116 (1996)

    Article  Google Scholar 

  35. Pitts, A.M.: On an interpretation of second order quantification in first order intuitionistic propositional logic. J. Symb. Log. 57(1), 33–52 (1992)

    Article  MathSciNet  Google Scholar 

  36. Rowe, R.N.S., Brotherston, J.: Realizability in cyclic proof: extracting ordering information for infinite descent. In: Schmidt, R.A., Nalon, C. (eds.) TABLEAUX 2017. LNCS (LNAI), vol. 10501, pp. 295–310. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66902-1_18

    Chapter  Google Scholar 

  37. Santocanale, L.: A calculus of circular proofs and its categorical semantics. In: Nielsen, M., Engberg, U. (eds.) FoSSaCS 2002. LNCS, vol. 2303, pp. 357–371. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45931-6_25

    Chapter  MATH  Google Scholar 

  38. Shamkanov, D.: Circular proofs for the Gödel-Löb provability logic. Math. Notes 96, 575–585 (2014)

    Article  MathSciNet  Google Scholar 

  39. Simpson, A.: Cyclic arithmetic is equivalent to Peano Arithmetic. In: Esparza, J., Murawski, A.S. (eds.) FoSSaCS 2017. LNCS, vol. 10203, pp. 283–300. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54458-7_17

    Chapter  Google Scholar 

  40. Sprenger, C., Dam, M.: On the structure of inductive reasoning: circular and tree-shaped proofs in the \(\mu \)-calculus. In: Gordon, A.D. (ed.) FoSSaCS 2003. LNCS, vol. 2620, pp. 425–440. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36576-1_27

    Chapter  MATH  Google Scholar 

  41. Stirling, C.: A tableau proof system with names for modal mu-calculus. In: Voronkov, A., Korovina, M.V. (eds.) HOWARD-60: A Festschrift on the Occasion of Howard Barringer’s 60th Birthday, EPiC Series in Computing, vol. 42, pp. 306–318. EasyChair (2014)

    Google Scholar 

  42. Tellez, G., Brotherston, J.: Automatically verifying temporal properties of pointer programs with cyclic proof. J. Autom. Reason. 64(3), 555–578 (2020)

    Article  MathSciNet  Google Scholar 

  43. Visser, A.: Bisimulations, model descriptions and propositional quantifiers. Logic Group Preprint Series no. 161, Utrecht (1996)

    Google Scholar 

  44. Visser, A.: Cyclic Henkin logic (2021). https://arxiv.org/abs/2101.11462v1

Download references

Author information

Authors and Affiliations

  1. Institute for Logic, Language and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Bahareh Afshari & Guillermo Menéndez Turata

  2. Department of Philosophy, Linguistics and Theory of Science, University of Gothenburg, Gothenburg, Sweden

    Bahareh Afshari & Graham E. Leigh

Authors
  1. Bahareh Afshari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Graham E. Leigh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guillermo Menéndez Turata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bahareh Afshari .

Editor information

Editors and Affiliations

  1. University of Birmingham, Birmingham, UK

    Anupam Das

  2. University of Genoa, Genoa, Italy

    Sara Negri

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Afshari, B., Leigh, G.E., Menéndez Turata, G. (2021). Uniform Interpolation from Cyclic Proofs: The Case of Modal Mu-Calculus. In: Das, A., Negri, S. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2021. Lecture Notes in Computer Science(), vol 12842. Springer, Cham. https://doi.org/10.1007/978-3-030-86059-2_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-86059-2_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-86058-5

  • Online ISBN: 978-3-030-86059-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation