Skip to main content
Springer Nature Link
Log in

And-Or Tableaux for Fixpoint Logics with Converse: LTL, CTL, PDL and CPDL

  • Conference paper
Automated Reasoning (IJCAR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8562))

Included in the following conference series:

Abstract

Over the last forty years, computer scientists have invented or borrowed numerous logics for reasoning about digital systems. Here, I would like to concentrate on three of them: Linear Time Temporal Logic (LTL), branching time Computation Tree temporal Logic (CTL), and Propositional Dynamic Logic (PDL), with and without converse. More specifically, I would like to present results and techniques on how to solve the satisfiability problem in these logics, with global assumptions, using the tableau method. The issues that arise are the typical tensions between computational complexity, practicality and scalability. This is joint work with Linh Anh Nguyen, Pietro Abate, Linda Postniece, Florian Widmann and Jimmy Thomson.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Vardi, M.Y.: From philosophical to industrial logics. In: Ramanujam, R., Sarukkai, S. (eds.) Logic and Its Applications. LNCS (LNAI), vol. 5378, pp. 89–115. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  2. Goré, R., Nguyen, L.A.: EXPTIME tableaux for ALC using sound global caching. In: Proc. of the International Workshop on Description Logics, DL 2007 (2007)

    Google Scholar 

  3. GorĂ©, R.P., Nguyen, L.A.: EXPTIME tableaux with global caching for description logics with transitive roles, inverse roles and role hierarchies. In: Olivetti, N. (ed.) TABLEAUX 2007. LNCS (LNAI), vol. 4548, pp. 133–148. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  4. GorĂ©, R.P., Postniece, L.: An experimental evaluation of global caching for ALC (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 299–305. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  5. GorĂ©, R., Widmann, F.: Sound global state caching for ALC with inverse roles. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS, vol. 5607, pp. 205–219. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  6. GorĂ©, R., Widmann, F.: An optimal on-the-fly tableau-based decision procedure for PDL-satisfiability. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 437–452. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  7. GorĂ©, R., Widmann, F.: Optimal and cut-free tableaux for propositional dynamic logic with converse. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 225–239. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  8. Widmann, F.: Tableaux-based Decision Procedures for Fixpoint Logics. PhD thesis, The Australian National University, Australia (2010)

    Google Scholar 

  9. Goré, R.: Tableau methods for modal and temporal logics. In: D’Agostino, M., Gabbay, D.M., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods, pp. 297–396. Kluwer (1999)

    Google Scholar 

  10. Baader, F., Calvanese, D., McGuinness, D.L., Nardi, D., Patel-Schneider, P. (eds.): The Description Logic Handbook: Theory, Implementation and Applications. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  11. Pratt, V.R.: A near-optimal method for reasoning about action. Journal of Computer and System Sciences 20(2), 231–254 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  12. Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. Journal of Computer and Systems Science 18, 194–211 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  13. Beth, E.: On Padoa’s method in the theory of definition. Indag. Math. 15, 330–339 (1953)

    MathSciNet  Google Scholar 

  14. Goré, R., Nguyen, L.: ExpTime tableaux for \(\mathcal{ALC}\) using sound global caching. In: C., D., et al. (eds.) Proc. DL 2007, pp. 299–306 (2007)

    Google Scholar 

  15. Wolper, P.: Temporal logic can be more expressive. Information and Control 56, 72–99 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ben-Ari, M., Manna, Z., Pnueli, A.: The temporal logic of branching time. In: Proceedings of Principle of Programming Langauages (1981)

    Google Scholar 

  17. Emerson, E.A., Halpern, J.Y.: Decision procedures and expressiveness in the temporal logic of branching time. Journal of Computer and System Sciences 30(1), 1–24 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. Horrocks, I., Sattler, U.: A tableaux decision procedure for SHOIQ. In: IJCAI, pp. 448–453 (2005)

    Google Scholar 

  19. Kesten, Y., Manna, Z., McGuire, H., Pnueli, A.: A decision algorithm for full propositional temporal logic. In: Courcoubetis, C. (ed.) CAV 1993. LNCS, vol. 697, pp. 97–109. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  20. Schwendimann, S.: A new one-pass tableau calculus for PLTL. In: de Swart, H. (ed.) TABLEAUX 1998. LNCS (LNAI), vol. 1397, pp. 277–291. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  21. Janssen, G.: Logics for Digital Circuit Verication: Theory, Algorithms, and Applications. PhD thesis, Eindhoven University of Technology, The Netherlands (1999)

    Google Scholar 

  22. Hustadt, U., Konev, B.: TRP++: A temporal resolution prover. In: Baaz, M., Makowsky, J., Voronkov, A. (eds.) Collegium Logicum, pp. 65–79. Kurt Gödel Society (2004)

    Google Scholar 

  23. Baader, F.: Augmenting concept languages by transitive closure of roles: An alternative to terminological cycles. In: Proc. IJCAI 1991, pp. 446–451 (1991)

    Google Scholar 

  24. Abate, P., GorĂ©, R., Widmann, F.: One-pass tableaux for computation tree logic. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 32–46. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  25. Schuppan, V., Darmawan, L.: Evaluating LTL satisfiability solvers. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 397–413. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  26. Abate, P., Goré, R., Widmann, F.: Cut-free single-pass tableaux for the logic of common knowledge. In: Workshop on Agents and Deduction at TABLEAUX (2007)

    Google Scholar 

  27. Nguyen, L.A., Golinska-Pilarek, J.: An exptime tableau method for dealing with nominals and quantified number restrictions in deciding the description logic SHOQ. In: CS&P, pp. 296–308 (2013)

    Google Scholar 

  28. Nguyen, L.A.: A tableau method with optimal complexity for deciding the description logic SHIQ. In: Nguyen, N.T., van Do, T., Thi, H.A. (eds.) ICCSAMA 2013. SCI, vol. 479, pp. 331–342. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  29. Suda, M., Weidenbach, C.: A pltl-prover based on labelled superposition with partial model guidance. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 537–543. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  30. Jungteerapanich, N.: A tableau system for the modal μ-calculus. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 220–234. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  31. Friedmann, O., Latte, M., Lange, M.: A decision procedure for CTL* based on tableaux and automata. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 331–345. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  32. Friedmann, O., Lange, M.: A solver for modal fixpoint logics. Electr. Notes Theor. Comput. Sci. 262, 99–111 (2010)

    Article  MathSciNet  Google Scholar 

  33. Reynolds, M.: A faster tableau for CTL*. In: GandALF, pp. 50–63 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Logic and Computation Group, Research School of Computer Science, The Australian National University, Canberra, Australia

    Rajeev Goré

Authors
  1. Rajeev Goré
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. CNRS,, New York University &amp, USA

    Stéphane Demri

  2. Department of Computer Science, University of New Mexico, 87131-0001, Albuquerque, NM, USA

    Deepak Kapur

  3. Max Planck Institute for Informatics, Campus E1 4, 66123, SaarbrĂ¼cken, Germany

    Christoph Weidenbach

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Goré, R. (2014). And-Or Tableaux for Fixpoint Logics with Converse: LTL, CTL, PDL and CPDL. In: Demri, S., Kapur, D., Weidenbach, C. (eds) Automated Reasoning. IJCAR 2014. Lecture Notes in Computer Science(), vol 8562. Springer, Cham. https://doi.org/10.1007/978-3-319-08587-6_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08587-6_3

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08586-9

  • Online ISBN: 978-3-319-08587-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics