Skip to main content
Springer Nature Link
Log in

Craig Interpolation for Linear Temporal Languages

  • Conference paper
Computer Science Logic (CSL 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5771))

Included in the following conference series:

Abstract

We study Craig interpolation for fragments and extensions of propositional linear temporal logic (PLTL). We consider various fragments of PLTLobtained by restricting the set of temporal connectives and, for each of these fragments, we identify its smallest extension that has Craig interpolation. Depending on the underlying set of temporal operators, this extension turns out to be one of the following three logics: the fragment of PLTLhaving only the Next operator; the extension of PLTLwith a fixpoint operator μ (known as linear time μ-calculus); the fixpoint extension of the “Until-only” fragment of PLTL.

We are grateful to Alexandru Baltag for helpful comments and to Frank Wolter for first raising the question. The first author was supported by a GLoRiClass fellowship of the European Commission (Research Training Fellowship MEST-CT-2005-020841) and the second author by the Netherlands Organization for Scientific Research (NWO) grant 639.021.508 and by ERC Advanced Grant Webdam on Foundation of Web data management.

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

Access this chapter

Log in via an institution

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. Amir, E., McIlraith, S.A.: Partition-based logical reasoning for first-order and propositional theories. Artif. Intell. 162(1-2), 49–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. Areces, C., de Rijke, M.: Interpolation and bisimulation in temporal logic. In: Guerra, R.J., de Queiroz, B., Finger, M. (eds.) Proceedings of WoLLIC 1998, pp. 15–21 (1998)

    Google Scholar 

  3. Arnold, A., Niwinski, D.: Rudiments of μ-Calculus. Studies in Logic and Foundations of Mathematics, vol. 146. North-Holland, Amsterdam (2001)

    MATH  Google Scholar 

  4. Barwise, J., Feferman, S.: Model-theoretic logics. Springer, New York (1985)

    MATH  Google Scholar 

  5. Bruyère, V., Carton, O.: Automata on linear orderings. J. Comput. System Sci. 73(1), 1–24 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. ten Cate, B.: Model Theory for Extended Modal Languages. PhD thesis, University of Amsterdam, ILLC Dissertation Series DS-2005-01 (2005)

    Google Scholar 

  7. Craig, W.: Three Uses of the Herbrand-Gentzen Theorem in Relating Model Theory and Proof Theory. Journal of Symbolic Logic 22(3), 269–285 (1957)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. D’Agostino, G., Hollenberg, M.: Logical Questions Concerning the μ-Calculus: Interpolation, Lyndon and Lös-Tarski. Journal of Symbolic Logic 65(1), 310–332 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerard, R., de Lavalette, R.: Interpolation in computing science: the semantics of modularization. Synthese 164(3), 437–450 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 995–1072. Elsevier, Amsterdam (1990)

    Google Scholar 

  12. Etessami, K.: Stutter-Invariant Languages, ω-Automata, and Temporal Logic. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 236–248. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  13. Gross, R.: Invariance under stuttering in branching-time temporal logic. Master’s thesis, Israel Institute of Technology, Haifa (2008)

    Google Scholar 

  14. Kaivola, R.: Using Automata to Characterise Fixed Point Temporal Logics. PhD thesis, University of Edinburgh (1997)

    Google Scholar 

  15. Kamp, H.: Tense Logic and the Theory of Linear Order. PhD thesis, UCLA, Los Angeles (1968)

    Google Scholar 

  16. Lamport, L.: What Good is Temporal Logic? In: Mason, R.E.A. (ed.) Proceedings of the IFIP 9th World Computer Congress, pp. 657–668. North-Holland/IFIP (1983)

    Google Scholar 

  17. Maksimova, L.: Temporal logics with “the next” operator do not have interpolation or the Beth property. Siberian Mathematical Journal 32(6), 989–993 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Niwinski, D.: Fixed Points vs. Infinite Generation. In: Proceedings of LICS, pp. 402–409 (1988)

    Google Scholar 

  19. Peled, D., Wilke, T.: Stutter-invariant temporal properties are expressible without the next-time operator. Inf. Process. Lett. 63(5), 243–246 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Thomas, W.: Languages, automata, and logic. In: Handbook of formal Languages. Beyond Words, vol. 3, pp. 389–455. Springer, New York (1997)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ILLC, Universiteit van Amsterdam, Netherlands

    Amélie Gheerbrant

  2. INRIA and ENS Cachan, France

    Balder ten Cate

Authors
  1. Amélie Gheerbrant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Balder ten Cate
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Mathematische Grundlagen der Informatik, RWTH Aachen, 52056, Aachen, Germany

    Erich Grädel

  2. CENTRIA and Departamento de Matemática, Universidade Nova de Lisboa, 2829-516, Caparica, Portugal

    Reinhard Kahle

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gheerbrant, A., ten Cate, B. (2009). Craig Interpolation for Linear Temporal Languages. In: Grädel, E., Kahle, R. (eds) Computer Science Logic. CSL 2009. Lecture Notes in Computer Science, vol 5771. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04027-6_22

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04027-6_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04026-9

  • Online ISBN: 978-3-642-04027-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics