Skip to main content
SpringerLink
Log in

Minimisation of ATL\(^*\) Models

  • Conference paper
  • First Online:
Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

  • 491 Accesses

Abstract

The aim of this work is to provide a general method to minimize the size (number of states) of a model \(\mathcal {M}\) of an \(\mathsf {ATL^*}\) formula. Our approach is founded on the notion of alternating bisimulation: given a model \(\mathcal {M}\), it is transformed in a stepwise manner into a new model \({\mathcal {M}}\)’ minimal with respect to bisimulation. The method has been implemented and will be integrated into the prover TATL, that constructively decides satifiability of an \(\mathsf {ATL^*}\) formula by building a tableau from which, when open, models of the input formula can be extracted.

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 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
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.

    Indeed, the existence of sound, complete and terminating tableaux for \(\mathsf {ATL}\) * is a proof of the finite model property for \(\mathsf {ATL^*}\).

  2. 2.

    A similar approach might be used also for models of the \(\mu \)-calculus.

References

  1. Ågotnes, T., Goranko, V., Jamroga, W.: Alternating-time temporal logics with irrevocable strategies. In: Proceedings of the 11th Conference on Theoretical Aspects of Rationality and Knowledge (TARK-2007), Brussels, Belgium 25–27 June 2007, pp. 15–24 (2007)

    Google Scholar 

  2. Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Belardinelli, F., Condurache, R., Dima, C., Jamroga, W., Jones, A.V.: Bisimulations for verifying strategic abilities applied to voting protocols. In: Proceedings of AAMAS 2017. IFAAMAS (2017)

    Google Scholar 

  4. Bonatti, P.A., Lutz, C., Wolter, F.: The complexity of circumscription in DLs. J. Artif. Intell. Res. 35, 717–773 (2009)

    MATH  Google Scholar 

  5. Bry, F., Yahya, A.: Positive unit hyperresolution tableaux and their application to minimal model generation. J. Autom. Reason. 25(1), 35–82 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cerrito, S., David, A.: Minimisation of ATL* models: extended draft. https://www.ibisc.univ-evry.fr/~serena/MiniDraft.pdf

  7. Cerrito, S., David, A., Goranko, V.: Optimal tableaux-based decision procedure for testing satisfiability in the alternating-time temporal logic ATL+. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 277–291. Springer, Cham (2014). doi:10.1007/978-3-319-08587-6_21

    Google Scholar 

  8. David, A.: TATL: tableaux for ATL*. http://atila.ibisc.univ-evry.fr/tableau_ATL_star/index.php

  9. David, A.: Deciding \(\sf ATL^*\) satisfiability by tableaux. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS, vol. 9195, pp. 214–228. Springer, Cham (2015). doi:10.1007/978-3-319-21401-6_14

    Chapter  Google Scholar 

  10. Demri, S., Goranko, V., Lange, M.: Temporal Logics in Computer Science. Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, Cambridge (2016)

    Book  MATH  Google Scholar 

  11. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Reasoning about typicality in preferential description logics. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS, vol. 5293, pp. 192–205. Springer, Heidelberg (2008). doi:10.1007/978-3-540-87803-2_17

    Chapter  Google Scholar 

  12. Grimm, S., Hitzler, P.: A preferential tableaux calculus for circumscriptive \({\cal{A}LCO}\). In: Polleres, A., Swift, T. (eds.) RR 2009. LNCS, vol. 5837, pp. 40–54. Springer, Heidelberg (2009). doi:10.1007/978-3-642-05082-4_4

    Chapter  Google Scholar 

  13. Georgieva, L., Hustadt, U., Schmidt, R.A.: Computational space efficiency and minimal model generation for guarded formulae. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 85–99. Springer, Heidelberg (2001). doi:10.1007/3-540-45653-8_6

    Chapter  Google Scholar 

  14. Hasegawa, R., Fujita, H., Koshimura, M.: Efficient minimal model generation using branching lemmas. In: McAllester, D. (ed.) CADE 2000. LNCS, vol. 1831, pp. 184–199. Springer, Heidelberg (2000). doi:10.1007/10721959_15

    Chapter  Google Scholar 

  15. Hintikka, J.: Model minimization - an alternative to circumscription. J. Autom. Reason. 4(1), 1–13 (1988)

    Article  MathSciNet  Google Scholar 

  16. Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM 32(1), 137–161 (1985)

    Article  MATH  Google Scholar 

  17. Kanellakis, P.C., Smolka, S.A.: CCS expressions, finite state processes, and three problems of equivalence. Inf. Comput. 86(1), 43–68 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  18. Aceto, L., Ingolfsdottir, A., Jiri, S.: The algorithmics of bisimilarity. In: Sangiorgi, D., Rutten, J. (eds.) Advanced Topics in Bisimulation and Coinduction, pp. 100–171. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  19. Lorenz, S.: A tableau prover for domain minimization. J. Autom. Reason. 13(3), 375–390 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. McCarthy, J.: Circumscription: a form of non-monotonic reasoning. In: Ginsberg, M.L. (ed.) Readings in Nonmonotonic Reasoning, pp. 145–151. Kaufmann, Los Altos (1987)

    Google Scholar 

  21. Niemelä, I.: A tableau calculus for minimal model reasoning. In: Miglioli, P., Moscato, U., Mundici, D., Ornaghi, M. (eds.) TABLEAUX 1996. LNCS, vol. 1071, pp. 278–294. Springer, Heidelberg (1996). doi:10.1007/3-540-61208-4_18

    Chapter  Google Scholar 

  22. Papacchini, F., Schmidt, R.A.: Terminating minimal model generation procedures for propositional modal logics. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 381–395. Springer, Cham (2014). doi:10.1007/978-3-319-08587-6_30

    Google Scholar 

  23. Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Aknowledgements

The authors would like to thank Damien Regnault and Marta Cialdea Mayer for their careful reading of first drafts of this work and for their useful remarks. The very first ideas underlying this work rose in the context of the direction of a project of two fourth year university students at the university of Evry Val d’Essonne: Lylia Bellabiod and Théo Chelim.

Author information

Authors and Affiliations

  1. IBISC, Université Evry Val d’Essonne, Évry, France

    Serenella Cerrito

  2. Université Paris-Descartes, Paris, France

    Amélie David

Authors
  1. Serenella Cerrito
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amélie David
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serenella Cerrito .

Editor information

Editors and Affiliations

  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Cerrito, S., David, A. (2017). Minimisation of ATL\(^*\) Models. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-66902-1_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-66901-4

  • Online ISBN: 978-3-319-66902-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation