Skip to main content

Exploiting Forwardness: Satisfiability and Query-Entailment in Forward Guarded Fragment

  • 332 Accesses

Part of the Lecture Notes in Computer Science book series (LNAI,volume 12678)

Abstract

We study the complexity of two standard reasoning problems for Forward Guarded Logic (\(\mathcal {FGF}\)), obtained as a restriction of the Guarded Fragment in which variables appear in atoms only in the order of their quantification. We show that \(\mathcal {FGF}\) enjoys the higher-arity-forest-model property, which results in \(\textsc {ExpTime}\)-completeness of its (finite and unrestricted) knowledge-base satisfiability problem. Moreover, we show that \(\mathcal {FGF}\) is well-suited for knowledge representation. By employing a generalisation of Lutz’s spoiler technique, we prove that the conjunctive query entailment problem for \(\mathcal {FGF}\) remains in \(\textsc {ExpTime}\).

We find that our results are quite unusual as \(\mathcal {FGF}\) is, up to our knowledge, the first decidable fragment of First-Order Logic, extending standard description logics like \(\mathcal {ALC}\), that offers unboundedly many variables and higher-arity relations while keeping its complexity surprisingly low.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-030-75775-5_13
  • Chapter length: 15 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-75775-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Fig. 1.
Fig. 2.
Fig. 3.

References

  1. Andréka, H., Németi, I., van Benthem, J.: Modal languages and bounded fragments of predicate logic. J. Philos. Logic (1998)

    Google Scholar 

  2. Baader, F., Horrocks, I., Lutz, C., Sattler, U.: An Introduction to Description Logic. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  3. Beeri, C., Vardi, M.Y.: The Implication Problem for Data Dependencies. In: ICALP (1981)

    Google Scholar 

  4. Calí, A., Gottlob, G., Kifer, M.: Taming the infinite chase: query answering under expressive relational constraints. J. Artif. Intell. Res. (2013)

    Google Scholar 

  5. Chandra, A.K., Kozen, D., Stockmeyer, L.J.: Alternation. J. ACM (1981)

    Google Scholar 

  6. Figueira, D., Figueira, S., Baque, E.P.: Finite Controllability for Ontology-Mediated Query Answering of CRPQ. KR (2020)

    Google Scholar 

  7. Gottlob, G., Pieris, A., Tendera, L.: Querying the Guarded Fragment with Transitivity. In: ICALP (2013)

    Google Scholar 

  8. Grädel, E.: Description Logics and Guarded Fragments of First Order Logic. DL (1998)

    Google Scholar 

  9. Grädel, E.: On the restraining power of guards. J. Symb. Log. (1999)

    Google Scholar 

  10. Grädel, E., Otto, M.: The Freedoms of (Guarded) Bisimulation (2013)

    Google Scholar 

  11. Herzig, A.: A new decidable fragment of first order logic. In: Third Logical Biennial, Summer School and Conference in Honour of S. C. Kleene (1990)

    Google Scholar 

  12. Hoogland, E., Marx, M., Otto, M.: Beth Definability for the Guarded Fragment. LPAR (1999)

    Google Scholar 

  13. Horrocks, I., Tessaris, S.: Answering Conjunctive Queries over DL ABoxes: A Preliminary Report. DL (2000)

    Google Scholar 

  14. Kieronski, E.: On the complexity of the two-variable guarded fragment with transitive guards. Inf. Comput. (2006)

    Google Scholar 

  15. Kieronski, E.: One-Dimensional Guarded Fragments. MFCS (2019)

    Google Scholar 

  16. Kieronski, E., Malinowski, A.: The triguarded fragment with transitivity. LPAR (2020)

    Google Scholar 

  17. Libkin, L.: Elements of finite model theory. In: Libkin, L. (ed.) Texts in Theoretical Computer Science. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-662-07003-1

    CrossRef  Google Scholar 

  18. Lutz, C.: Inverse Roles Make Conjunctive Queries Hard. DL (2007)

    Google Scholar 

  19. Lutz, C.: Two Upper Bounds for Conjunctive Query Answering in SHIQ. DL (2008)

    Google Scholar 

  20. Otto, M.: Elementary Proof of the van Benthem-Rosen Characterisation Theorem. Technical Report (2004)

    Google Scholar 

  21. Pratt-Hartmann, I.: Complexity of the guarded two-variable fragment with counting quantifiers. J. Log. Comput. (2007)

    Google Scholar 

  22. Pratt-Hartmann, I., Szwast, W., Tendera, L.: The fluted fragment revisited. J. Symb. Log. (2019)

    Google Scholar 

  23. Quine, W.: The Ways of Paradox and Other Essays, Revised edn. Harvard University Press, Cambridge (1976)

    Google Scholar 

  24. Rosati, R.: On the decidability and finite controllability of query processing in databases with incomplete information. PODS (2006)

    Google Scholar 

  25. Stockmeyer, L.: The Complexity of Decision Problems in Automata Theory and Logic (1974)

    Google Scholar 

Download references

Acknowledgements

The author apologises for all mistakes and grammar issues that appear in the paper. He thanks A. Karykowska and P. Witkowski for proofreading, E. Kieroński for his help with the introduction, W. Faber for deadline extension and anonymous JELIA’s reviewers for many useful comments.

This work was supported by the ERC Consolidator Grant No. 771779 (DeciGUT).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Bartosz Bednarczyk .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Bednarczyk, B. (2021). Exploiting Forwardness: Satisfiability and Query-Entailment in Forward Guarded Fragment. In: Faber, W., Friedrich, G., Gebser, M., Morak, M. (eds) Logics in Artificial Intelligence. JELIA 2021. Lecture Notes in Computer Science(), vol 12678. Springer, Cham. https://doi.org/10.1007/978-3-030-75775-5_13

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-75775-5_13

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-75774-8

  • Online ISBN: 978-3-030-75775-5

  • eBook Packages: Computer ScienceComputer Science (R0)