Skip to main content
SpringerLink
Log in

Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi

  • Conference paper
  • First Online:
Logics in Artificial Intelligence (JELIA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10021))

Included in the following conference series:

Abstract

Interpolation is a fundamental logical property with applications in mathematics, computer science, and artificial intelligence. In this paper, we develop a general method of translating a semantic description of modal logics via Kripke models into a constructive proof of the Lyndon interpolation property (LIP) via labelled sequents. Using this method we demonstrate that all frame conditions representable as Horn formulas imply the LIP and that all 15 logics of the modal cube, as well as the infinite family of transitive Geach logics, enjoy the LIP.

This material is based upon work supported by the Austrian Science Fund (FWF) Lise Meitner Grant M 1770-N25.

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.

    Unlike internal formalisms, external ones cannot generally be translated into formulas, typically because of the essential use of semantic elements, e.g., Kripke worlds.

  2. 2.

    The method also works for sequence- and set-based sequents.

  3. 3.

    NNF is used here to simplify the notation rather than out of necessity and means that negation is restricted to propositional atoms, creating two literals P and \(\overline{P}\) for each atom. Primary connectives are \(\wedge \), \(\vee \), \(\Box \), and \(\Diamond \). Negation \(\overline{A}\) is a function of a formula A defined via De Morgan laws. \(A \rightarrow B := \overline{A}\vee B\).

  4. 4.

    It also holds for \(l_i=0\): the empty conjunction is \(\top \) and \(\mathcal {M}, \left[ \!\left[ \mathsf {w}\right] \!\right] \Vdash \Diamond \top \).

  5. 5.

    For each eigenvariable \(\mathsf {y}_j\) we have collected all formulas labelled with \(\mathsf {y}_j\) within each disjunct into one labelled formula by transforming into if more than one formula has this label or by adding if no formula has.

References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bílková, M.: A note on uniform interpolation proofs in modal deep inference calculi. In: Bezhanishvili, N., Löbner, S., Schwabe, K., Spada, L. (eds.) TbiLLC 2009. LNCS (LNAI), vol. 6618, pp. 30–45. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22303-7_3

    Chapter  Google Scholar 

  3. Brotherston, J., Goré, R.: Craig interpolation in displayable logics. In: Brünnler, K., Metcalfe, G. (eds.) TABLEAUX 2011. LNCS (LNAI), vol. 6793, pp. 88–103. Springer, Heidelberg (2011). doi:10.1007/978-3-642-22119-4_9

    Chapter  Google Scholar 

  4. ten Cate, B., Franconi, E., Seylan, İ.: Beth definability in expressive description logics. J. Arti. Intell. Res. 48(1), 347–414 (2013)

    MathSciNet  MATH  Google Scholar 

  5. Dyckhoff, R., Negri, S.: Geometrisation of first-order logic. Bull. Symbolic Logic 21(2), 123–163 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fitting, M., Kuznets, R.: Modal interpolation via nested sequents. Ann. Pure Appl. Logic 166(3), 274–305 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garson, J.: Modal logic. In: Zalta, E.N., (ed.) The Stanford Encyclopedia of Philosophy (2016). http://plato.stanford.edu/entries/logic-modal/

  8. Goré, R., Ramanayake, R.: Labelled tree sequents, tree hypersequents and nested (deep) sequents. In: Bolander, T., Braüner, T., Ghilardi, S., Moss, L. (eds.) Advances in Modal Logic, vol. 9, pp. 279–299. College Publications (2012)

    Google Scholar 

  9. Herzig, A., Mengin, J.: Uniform interpolation by resolution in modal logic. In: Hölldobler, S., Lutz, C., Wansing, H. (eds.) JELIA 2008. LNCS (LNAI), vol. 5293, pp. 219–231. Springer, Heidelberg (2008). doi:10.1007/978-3-540-87803-2_19

    Chapter  Google Scholar 

  10. Iemhoff, R.: Uniform interpolation and sequent calculi in modal logic. Preprint 325, Logic Group Preprint Series (2015)

    Google Scholar 

  11. Kripke, S.A.: Semantical analysis of modal logic I: normal modal propositional calculi. Z. Math. Logik Grundlagen Math. 9(5–6), 67–96 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kuznets, R.: Craig interpolation via hypersequents. In: Probst, D., Schuster, P. (eds.) Concepts of Proof in Mathematics, Philosophy, and Computer Science. Ontos Mathematical Logic, vol. 6, pp. 193–214. De Gruyter, Berlin (2016)

    Google Scholar 

  13. Kuznets, R.: Interpolation method for multicomponent sequent calculi. In: Artemov, S., Nerode, A. (eds.) LFCS 2016. LNCS, vol. 9537, pp. 202–218. Springer, Heidelberg (2016). doi:10.1007/978-3-319-27683-0_15

    Chapter  Google Scholar 

  14. Maksimova, L.L.: Absence of the interpolation property in the consistent normal modal extensions of the Dummett logic. Algebra Logic 21(6), 460–463 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Marx, M., Venema, Y.: Multi-dimensional Modal Logic. Applied Logic Series, vol. 4. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  16. Negri, S., von Plato, J.: Proof Analysis: A Contribution to Hilbert’s Last Problem. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  17. Pattinson, D.: The logic of exact covers: completeness and uniform interpolation. In: 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 418–427. IEEE (2013)

    Google Scholar 

  18. Thomason, R.: Logic and artificial intelligence. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2014). http://plato.stanford.edu/entries/logic-ai/

Download references

Acknowledgments

I am grateful to M. Fitting, whose idea started this interpolation project. I thank S. Negri for encouragement, V. Sikimić for procuring a source not available online, Y. Venema and M. Marx for valuable information on the non-constructive method. I am deeply indebted to B. Lellmann, who is always ready to listen and has provided many inspiring suggestions for improving this paper. I thank the anonymous reviewers for the suggestions on terminology.

Author information

Authors and Affiliations

  1. Institut für Computersprachen, TU Wien, Vienna, Austria

    Roman Kuznets

Authors
  1. Roman Kuznets
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman Kuznets .

Editor information

Editors and Affiliations

  1. Open University of Cyprus, Nicosia, Cyprus

    Loizos Michael

  2. University of Cyprus, Nicosia, Cyprus

    Antonis Kakas

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Kuznets, R. (2016). Proving Craig and Lyndon Interpolation Using Labelled Sequent Calculi. In: Michael, L., Kakas, A. (eds) Logics in Artificial Intelligence. JELIA 2016. Lecture Notes in Computer Science(), vol 10021. Springer, Cham. https://doi.org/10.1007/978-3-319-48758-8_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-48758-8_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-48757-1

  • Online ISBN: 978-3-319-48758-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation