Skip to main content
SpringerLink
Log in

Normal Multimodal Logics with Interaction Axioms

A Tableau Calculus and some (Un)Decidability Results

  • Chapter
Labelled Deduction

Part of the book series: Applied Logic Series ((APLS,volume 17))

Abstract

In this paper we present a prefixed analytic tableau calculus for a wide class of normal multimodal logics; the calculus can deal in a uniform way with any logic in this class. To achieve this goal, we use a prefixed tableau calculus à la Fitting, where we explicitly represent accessibility relations between worlds by means of a graph and we use the characterizing axioms as rewriting rules.Such rules create new paths among worlds in the counter-model construction. The prefixed tableau method is, then, used to prove (un)decidability results about certain classes of multimodal logics.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. A. Artosi, P. Benassi, G. Governatori, and A. Rotolo. Labelled Proofs for Quantified Modal Logic. InLogics in Artificial Intelligence JELIA ‘86, volume 1126 ofLNAI, pages 70–86. Springer-Verlag, 1996.

    Google Scholar 

  2. Y. Auffray and P. Enjalbert. Modal Theorem Proving: An equational viewpoint.Journal of Logic and Computation, 2(3):247–297, 1992.

    Article  Google Scholar 

  3. M. Baldoni.Normal Multimodal Logics: Automatic Deduction and Logic Programming Extension.PhD thesis, Dipartimento di Informatica, Università degli Studi di Torino, Italy, 1998. Available athttp://www.di.unito.it/Tbaldoni .

    Google Scholar 

  4. M. Baldoni. Normal Multimodal Logics with Interaction Axioms: extended version. Technical report, Dipartimento di Informatica, Università degli Studi di Torino, 1999.

    Google Scholar 

  5. M. Baldoni, L. Giordano, and A. Martelli. A Modal Extention of Logic Programming: Modularity, Beliefs and Hypothetical Reasoning. Journal of Logic and Computation, 6(5):597–635, 1998.

    Article  Google Scholar 

  6. M. Baldoni, L. Giordano, and A. Martelli. A Tableau Calculus for Multimodal Logics and Some (Un)Decidability Results. In H. de Swart, editorProc. of the International Conference on Analytic Tableaux and Related Methods TABLEAUX’98, volume 1397 ofLNAI, pages 44–59. Springer-Verlag, 1998.

    Chapter  Google Scholar 

  7. M. Baldoni, L. Giordano, A. Martelli, and V. Patti. A Modal Programming Language for Representing Complex Actions. In A. Bonner, B. Freitag, and L. Giordano, editorsProc. 1998 JICSLP’98 Post-Conference Workshop on Transactions and Change in Logic Databases DYNAMICS’98, pages 1–15, Manchester, UK, June 1998.

    Google Scholar 

  8. D. Basin, S. Matthews, and L. Viganó. A Topography of Labelled Model Logics. In F. Baader and K. U. Schulz, editorsFrontiers of Combining Systems — Proc. of the International Workshop, volume 3 ofApplied Logic Series, pages 75–92. Kluwer Academic Publisher, 1996.

    Google Scholar 

  9. D. Basin, S. Matthews, and L. Viganó. Labelled Propositional Modal Logics: Theory and Practice.Journal of Logic and Computation, 7(6):685–717, 1997.

    Article  Google Scholar 

  10. D. Basin, S. Matthews, and L. Viganó. Natural Deduction for Non-Classical Logics.Studia Logica, 60(1):119–160, 1998. Special issue onNatural Deductionedited by F. Pfenning and W. Sieg.

    Article  Google Scholar 

  11. B. Beckert and R. Goré. Free Variable Tableaux for Propositional Modal Logics. In D. Galmiche, editorProc. of the International Conference on Automatic Reasoning with Analytic Tableaux and Related Methods TABLEAUX’97, volume 1227 ofLNAI, pages 91–106. Springer-Verlag, 1997.

    Google Scholar 

  12. M. A. Castilho, L. Faniñas del Cerro, O. Gasquet, and A. Herzig. Modal Tableaux with Propagation Rules and Structural Rules.Fundamenta Informaticae, 32(3/4):281–297, 1998.

    Google Scholar 

  13. L. Catach. Normal Multimodal Logics. InProc. of the 7th National Conference on Artificial Intelligence, AAAI ‘88, volume 2, pages 491–495. Morgan Kaufmann, 1988.

    Google Scholar 

  14. L. Catach. Les Logiques Multimodales. PhD thesis, Universite Pierre et Marie Curie (Paris 6), Paris, 1989.

    Google Scholar 

  15. L. Catach. TABLEAUX: A General Theorem Prover for Modal Logics.Journal of Automated Reasoning, 7(4):489–510, 1991.

    Article  Google Scholar 

  16. A. Chagrov and M. Zakharyaschev.Modal Logic.Clarendon Press, Oxford, 1997.

    Google Scholar 

  17. B.F. Chellas. Modal Logic: an Introduction. Cambridge University Press, 1980.

    Book  Google Scholar 

  18. J. Cunningham and J. Pitt. Distributed Modal Theorem Proving with KE. InProc. of the 5th International Workshop on Theorem Proving with Analytic Tableaux and Related Methods TABLEAUX ‘86, volume 1071 ofLNAI, pages 160–176. Springer-Verlag, 1996.

    Google Scholar 

  19. M. D’Agostino and D. M. Gabbay. A generalization of analytic deduction via labelled systems. Part I: basic structural logics.Journal of Automated Reasoning,13:243–281, 1994.

    Article  Google Scholar 

  20. M. D’Agostino, D. M. Gabbay, R. Hähnle, and J. Posegga, editors. Handbook of Tableau Methods.Kluwer Academic Publishers, 1999.

    Google Scholar 

  21. M. D’Agostino and M. Mondadori. The Taming of the Cut.Journal of Logic and Computation, 4:285–319, 1994.

    Article  Google Scholar 

  22. G. De Giacomo and F. Massacci. Tableaux and Algorithms for Propositional Dynamic Logic with Converse. In W. McCune, editorAutomated Deduction — CADE-15, volume 1249 ofLNAI, pages 613–627. Springer, 1996.

    Google Scholar 

  23. P. Enjalbert and L. Fariñas del Cerro. Modal Resolution in Clausal Form.Theoretical Computer Science, 65(1):1–33, 1989.

    Article  Google Scholar 

  24. L. Fariñas del Cerro. MOLOG: A System that extends Prolog with Modal Logic.New Generation Computing, 4(1):35–50, 1986.

    Article  Google Scholar 

  25. L. Fariñas del Cerro and A. Herzig. Modal Deduction with Applications in Epistemic and Temporal Logics. InHandbook of Logic in Artificial Intelligence and Logic Programming, volume 4 pages 499–594. Oxford Science Publications, 1995.

    Google Scholar 

  26. L. Fariñas del Cerro and M. Penttonen. Grammar Logics. Logique et Analyse, 121–122:123–134, 1988.

    Google Scholar 

  27. M. J. Fischer and R. E. Ladner. Propositional Dynamic Logic of Regular Programs.Journal of Computer and System Sciences, 18(2):194–211, 1979.

    Article  Google Scholar 

  28. M. Fisher and R. Owens. An Introduction to Executable Modal and Temporal Logics. InProc. of theIJCAI’93 Workshop on Executable Modal and Temporal Logics, volume 897 ofLNAI, pages 1–20. Springer-Verlag, 1993.

    Google Scholar 

  29. M. Fitting.Proof Methods for Modal and Intuitionistic Logics, volume 169 ofSynthese library.D. Reidel, Dordrecht, Holland, 1983.

    Google Scholar 

  30. D. M. Gabbay.LDS - Labelled Deductive Systems (Volume 1 - Foundations).Clarendon Press, Oxford, 1996.

    Google Scholar 

  31. O. Gasquet. Optimization of deduction for multi-modal logics. InApplied Logic: How What and Why?Kluwer Academic Publishers, 1993.

    Google Scholar 

  32. M. Genesereth and N. Nilsson.Logical Foundations of Artificial Intelligence.Morgan Kaufmann, 1987.

    Google Scholar 

  33. R. A. Goré. Tableaux Methods for Modal and Temporal Logics. In D’Agostino et al. [20], pages 297–396.

    Google Scholar 

  34. G. Governatori. Labelled Tableaux for Multi-Modal Logics. InProc. of the 4th Workshop on Theorem Proving with Analytic Tableaux and Related Methods TABLEAUX ‘85, volume 918 ofLNAI, pages 79–94. Springer-Verlag, 1995.

    Google Scholar 

  35. J. Y. Halpern and Y. Moses. A Guide to Completeness and Complexity for Modal Logics of Knowledge and Belief.Artificial Intelligence, 54:319–379, 1992.

    Article  Google Scholar 

  36. D. Harel. Dynamic Logic. InHandbook of Philosophical Logic, volume 11, pages 497–604. D. Reidel Publishing Company, 1984.

    Google Scholar 

  37. J. E. Hoperoft and J. D. Ullman. Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Company, 1979.

    Google Scholar 

  38. G. E. Hughes and M. J. Cresswell. A New Introduction to Modal Logic. Routledge, 1996.

    Book  Google Scholar 

  39. M. Kracht. Highway to the Danger Zone.Journal of Logic and Computation, 5(1):93–109, 1995.

    Article  Google Scholar 

  40. F. Massacci. Strongly Analytic Tableaux for Normal Modal Logics. InProc. of the CADE’94, volume 814 ofLNAI, pages 723–737. Springer-Verlag, 1994.

    Google Scholar 

  41. R. Moddux. The equational theory of CA3 is undecidable.Journal of Symbolic Logic, 45(2):311–316, 1980.

    Article  Google Scholar 

  42. A. Nerode. Some Lectures on Modal Logic. In F. L. Bauer, editorLogic Algebra and Computation, volume 79 ofNATO ASI Series.Springer-Verlag, 1989.

    Google Scholar 

  43. A. Nonnengart. First-Order Modal Logic Theorem Proving and Functional Simulation. InProc. of International Joint Conference on Artificial Intelligence IJCAI’93, pages 80–85. Morgan Kaufmann, 1993.

    Google Scholar 

  44. A. Nonnengart. How to use Modalities and Sorts in Prolog. In C. MacNish, D. Pearce, and L. M. Pereira, editorsProc. of the JELIA’94: Logics in Artificial Intelligence, volume 838 ofLNAI, pages 365–378, York, UK, 1994. Springer-Verlag.

    Article  Google Scholar 

  45. H. J. Ohlbach. Semantics-Based Translation Methods for Modal Logics.Journal of Logic and Computation, 1(5):691–746, 1991.

    Article  Google Scholar 

  46. H. J. Ohlbach. Optimized Translation of Multi Modal Logic into Predicate Logic. In A. Voronkov, editorProc. of the Logic Programming and Automated Reasoning, volume 822 ofLNAI, pages 253–264. Springer-Verlag, 1993.

    Google Scholar 

  47. H. J. Ohlbach. Translation methods for non-classical logics: An overview.Bull. of the IGPL, 1(1):69–89, 1993.

    Article  Google Scholar 

  48. M. A. Orgun and W. Ma. An overview of temporal and modal logic programming. In D. M. Gabbay and H. J. Ohlbach, editorsProc. of the First International Conference on Temporal Logic, volume 827 ofLNAI, pages 445–479. Springer-Verlag, 1994.

    Google Scholar 

  49. A. Russo., Generalizing Propositional Modal Logics using Labelled Deductive Systems. In F. Baader and K. U. Schulz, editorsFrontiers of Combining Systems — Proc. of the International Workshop, volume 3 ofApplied Logic Series, pages 57–74. Kluwer Academic Publisher, 1996.

    Google Scholar 

  50. C. Stirling and D. Walker. Local model checking in the modal mu-calculus.Theoretical Computer Science, 89:161–177, 1991.

    Article  Google Scholar 

  51. J. Van Benthem.Modal Logic and Classical Logic. Bibliopolis, Naples, 1982.

    Google Scholar 

  52. M. Wooldridge and N. R. Jennings. Agent Theories, Architectures, and Languages: A survey. InProc. of the ECAI-94 Workshop on Agent Theories, volume 890 ofLNAI, pages 1–39. Springer-Verlag, 1995.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica, Università degli Studi di Torino, Corso Svizzera , 185, I-10149, Torino, Italy

    Matteo Baldoni

Authors
  1. Matteo Baldoni
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Albert-Ludwigs-Universität Freiburg, Germany

    David Basin  & Luca Viganò  & 

  2. Università di Ferrara, Italy

    Marcello D’Agostino

  3. Department of Computer Science, King’s College, London, UK

    Dov M. Gabbay

  4. Logical GmbH, Frankfurt am Main, Germany

    Seán Matthews

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Baldoni, M. (2000). Normal Multimodal Logics with Interaction Axioms. In: Basin, D., D’Agostino, M., Gabbay, D.M., Matthews, S., Viganò, L. (eds) Labelled Deduction. Applied Logic Series, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-4040-9_2

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-4040-9_2

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5781-3

  • Online ISBN: 978-94-011-4040-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation