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International Joint Conference on Automated Reasoning

IJCAR 2018: Automated Reasoning pp 498–515Cite as

A Resolution-Based Calculus for Preferential Logics

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10900))

Abstract

The vast majority of modal theorem provers implement modal tableau, or backwards proof search in (cut-free) sequent calculi. The design of suitable calculi is highly non-trivial, and employs nested sequents, labelled sequents and/or specifically designated transitional formulae. Theorem provers for first-order logic, on the other hand, are by and large based on resolution. In this paper, we present a resolution system for preference-based modal logics, specifically Burgess’ system . Our main technical results are soundness and completeness. Conceptually, we argue that resolution-based systems are not more difficult to design than cut-free sequent calculi but their purely syntactic nature makes them much better suited for implementation in automated reasoning systems.

The first author was partially supported by FWF START Y544-N23.

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References

  1. Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. JLC 4(3), 217–247 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Beckert, B., Goré, R.: System description: leanK 2.0. In: Kirchner, C., Kirchner, H. (eds.) CADE 1998. LNCS, vol. 1421, pp. 51–55. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054247

    Chapter  Google Scholar 

  3. Burgess, J.P.: Quick completeness proofs for some logics of conditionals. Notre Dame J. Formal Log. 22(1), 76–84 (1981)

    Article  MathSciNet  Google Scholar 

  4. Chellas, B.: Modal Logic. Cambridge University Press, Cambridge (1980)

    Book  Google Scholar 

  5. David, A.: Deciding \(\sf ATL^*\) satisfiability by tableaux. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 214–228. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21401-6_14

    Chapter  Google Scholar 

  6. Friedman, N., Halpern, J.Y.: On the complexity of conditional logics. In: Doyle, J., Sandewall, E., Torasso, P. (eds.) Proceedings of KR 1994, pp. 202–213. M. Kaufmann (1994)

    Chapter  Google Scholar 

  7. Gasquet, O., Herzig, A., Longin, D., Sahade, M.: LoTREC: logical tableaux research engineering companion. In: Beckert, B. (ed.) TABLEAUX 2005. LNCS (LNAI), vol. 3702, pp. 318–322. Springer, Heidelberg (2005). https://doi.org/10.1007/11554554_25

    Chapter  MATH  Google Scholar 

  8. Giordano, L., Gliozzi, V., Olivetti, N., Pozzato, G.L.: Analytic tableaux calculi for KLM logics of nonmonotonic reasoning. ACM Trans. Comput. Log. 10(3), 18:1–18:47 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Giordano, L., Gliozzi, V., Olivetti, N., Schwind, C.: Tableau calculus for preference-based conditional logics: PCL and its extensions. ACM Trans. Comput. Log. 10(3), 21 (2009)

    MathSciNet  MATH  Google Scholar 

  10. Gorín, D., Pattinson, D., Schröder, L., Widmann, F., Wißmann, T.: Cool – a generic reasoner for coalgebraic hybrid logics (system description). In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 396–402. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08587-6_31

    Chapter  Google Scholar 

  11. Haarslev, V., Möller, R.: RACER system description. In: Goré, R., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS, vol. 2083, pp. 701–705. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45744-5_59

    Chapter  Google Scholar 

  12. Hustadt, U., Gainer, P., Dixon, C., Nalon, C., Zhang, L.: Ordered resolution for coalition logic. In: De Nivelle, H. (ed.) TABLEAUX 2015. LNCS (LNAI), vol. 9323, pp. 169–184. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24312-2_12

    Chapter  Google Scholar 

  13. Nalon, C., Hustadt, U., Dixon, C.: K\(_{{\rm S}}\)P: a resolution-based prover for multimodal K. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 406–415. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_28

    Chapter  Google Scholar 

  14. Olivetti, N., Pozzato, G.L.: Nested sequent calculi and theorem proving for normal conditional logics: the theorem prover NESCOND. Intell. Artif. 9(2), 109–125 (2015)

    Google Scholar 

  15. Olivetti, N., Pozzato, G.L., Schwind, C.: A sequent calculus and a theorem prover for standard conditional logics. ACM Trans. Comput. Log. 8(4), 22 (2007)

    Article  MathSciNet  Google Scholar 

  16. Plaisted, D.A., Greenbaum, S.A.: A structure-preserving clause form translation. JLC 2, 293–304 (1986)

    MathSciNet  MATH  Google Scholar 

  17. Riazanov, A., Voronkov, A.: The design and implementation of VAMPIRE. AI Commun. 15(2–3), 91–110 (2002)

    MATH  Google Scholar 

  18. Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12(1), 23–41 (1965)

    Article  MathSciNet  Google Scholar 

  19. Schröder, L., Pattinson, D., Hausmann, D.: Optimal tableaux for conditional logics with cautious monotonicity. In: Coelho, H., Studer, R., Wooldridge, M. (eds.) Proceedings of the ECAI 2010 Frontiers in Artificial Intelligence and Applications, vol. 215 , pp. 707–712. IOS Press (2010)

    Google Scholar 

  20. Schulz, S.: System description: E 0.81. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 223–228. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-25984-8_15

    Chapter  Google Scholar 

  21. Tsarkov, D., Horrocks, I.: FaCT++ description logic reasoner: system description. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 292–297. Springer, Heidelberg (2006). https://doi.org/10.1007/11814771_26

    Chapter  Google Scholar 

  22. Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 140–145. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02959-2_10

    Chapter  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Brasília, Brasília, DF, 70910-090, Brazil

    Cláudia Nalon

  2. Research School of Computer Science, Australian National University, Acton ACT, Canberra, 2601, Australia

    Dirk Pattinson

Authors
  1. Cláudia Nalon
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  2. Dirk Pattinson
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Corresponding author

Correspondence to Cláudia Nalon .

Editor information

Editors and Affiliations

  1. Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Didier Galmiche

  2. Baden-Wuerttemberg Cooperative State University, Stuttgart, Germany

    Stephan Schulz

  3. University of Trento, Trento, Italy

    Roberto Sebastiani

Proofs

Proofs

The next two proofs were automatically generated by a prototype prover which implements the calculus given in this paper. Only clauses needed in the refutation are shown. Also the inference rule is always applied together with , so clauses are already in simplified form. First, as part of the proof of Lemma 7, we show that for \(\varphi ,\lnot \psi \in b\), we have that \(\mathsf {Prefer}(a,b,B) \) and \(\varphi \Rightarrow \psi \in (a,A)\) is contradictory.

figure j

The following refutation is part of the proof of Lemma 8, where we show that, for \(\varphi , \psi \in b\), we have that \(\lnot ((b \vee \bigvee B) \Rightarrow \bigvee B)\) and \(\lnot (\varphi \Rightarrow \psi ) \in a\) is not -consistent

figure k

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Nalon, C., Pattinson, D. (2018). A Resolution-Based Calculus for Preferential Logics. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_33

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  • DOI: https://doi.org/10.1007/978-3-319-94205-6_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94204-9

  • Online ISBN: 978-3-319-94205-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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