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An NP-complete fragment of fibring logic

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Abstract

The fibring method provides a semantic way to take various modal logics as arguments to produce an integrated one, and the benefit of this method is clear: a stronger expressive power. In this article, we prove the computational complexity of a class of fibring logics. Especially for a fibring logic composed of two S 5 systems, we present a novel reduction method, Fibring Structure Mapping, to settle its complexity. Then, we found a special N P -complete fragment for the fibred S 5 system. The significance of these results is that, on the one hand, the reduction method presented in this article can be generalized to settle the computational complexity problem of other fibring logics, and on the other hand, they help us to achieve a balance between the expressive power and the difficulty of computation.

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References

  1. Achilleos, A., Lampis, M., Mitsou, V.: Parameterized modal satisfiability. Algorithmica 64(1), 38–55 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chagrov, A.V., Rybakov, M.N.: How many variables does one need to prove pspace-hardness of modal logics?. In: Advances in Modal Logic 4 (AiML02) (2003)

  3. Artale, A., Franconi, E.: A survey of temporal extensions of description logics. Ann. Math. Artif. Intell. 30(1–4), 171–210 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bennett, B., Cohn, A.G., Wolter, F., Zakharyaschev, M.: Multi-dimensional modal logic as a framework for spatio-temporal reasoning. Appl. Intell. 17(3), 239–251 (2002a)

    Article  MATH  Google Scholar 

  5. Bennett, B., Dixon, C., Fisher, M., Hustadt, U., Franconi, E., Horrocks, I., Rijke, M.D.: Combinations of modal logics. Artif. Intell. Rev. 17(1), 1–20 (2002b)

    Article  MATH  Google Scholar 

  6. Beth, E.W.: The foundations of mathematics A Study in the Philosophy of Science. North-Holland (1959)

  7. Blackburn, P., Rijke, M.D.: Why combine logics Stud. Logica. 59(1), 5–27 (1997)

    Article  MATH  Google Scholar 

  8. Blackburn, P., Rijke, M.D., Venema, Y.: Modal Logic. Cambridge University Press (2002)

  9. Blackburn, P., van Benthem, J., Wolter, F.: Handbook of Modal Logic, Studies in Logic and Practical Reasoning, 1st edn., vol. 3. Elsevier, Amsterdam and Boston (2007)

    Google Scholar 

  10. Caleiro, C., Sernadas, C., Sernadas, A.: Parameterisation of logics. In: Recent Trends in Algebraic Development Techniques, pp. 48–63. Springer (1999)

  11. Caleiro, C., Carnielli, W.A., Coniglio, M.E., Sernadas, A., Sernadas, C.: Fibring non-truth-functional logics: completeness preservation. J. Log. Lang. Inf. 12(2), 183–211 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Caleiro, C., Sernadas, A., Sernadas, C.: Fibring logics: past, present and future. We Will Show Them: Essays in Honour of Dov Gabbay, vol. One, pp. 363–388 (2005)

  13. Carnielli, W., Coniglio, M.E.: Combining logics (2007)

  14. Fine, K., Schurz, G.: Transfer theorems for multimodal logics. logic and reality. essays in pure and applied logic. Memory of Arthur Prior OUP (1992)

  15. Gabbay, D.: An overview of fibred semantics and the combination of logics. Frontiers of Combining Systems 3, 1–56 (1996)

    Article  MathSciNet  Google Scholar 

  16. Gabbay, D., Shehtman, V.: Products of modal logics. part 3: products of modal and temporal logics. Stud. Logica. 72(2), 157–183 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Gabbay, D.M.: Fibring Logics, Oxford Logic Guides, vol. 38. Clarendon, Oxford and New York (1999)

    Google Scholar 

  18. Gabbay, D.M.: Many-dimensional Modal Logics: Theory and Applications, Studies in Logic and the Foundations of Mathematics, 1st edn., vol. 148. Elsevier North Holland, Amsterdam and Boston (2003)

    Google Scholar 

  19. Gabbay, D.M.: What is a logical system? an evolutionary view: 1964–2014. In: Computational Logic, Newnes, vol. 9, p. 41 (2014)

  20. Gabbay, D.M., Shehtman, V.B.: Products of modal logics, part 1. Log. J. IGPL 6(1), 73–146 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gabelaia, D., Kontchakov, R., Kurucz, A., Wolter, F., Zakharyaschev, M.: Combining spatial and temporal logics: expressiveness vs. complexity. J. Artif. Intell. Res. 23(1), 167–243 (2005)

    MATH  MathSciNet  Google Scholar 

  22. Gonçalves, R., Alferes, J.J.: Parametrized logic programming. In: Logics in Artificial Intelligence, pp. 182–194. Springer (2010)

  23. Halpern, J.Y.: The effect of bounding the number of primitive propositions and the depth of nesting on the complexity of modal logic. Artif. Intell. 75(2), 361–372 (1995)

    Article  MATH  Google Scholar 

  24. Halpern, J.Y., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artif. Intell. 54(3), 319–379 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Halpern, J.Y., Rêgo, L.C.: Characterizing the np-pspace gap in the satisfiability problem for modal logic. J. Log. Comput. 17(4), 795–806 (2007)

    Article  MATH  Google Scholar 

  26. Hirsch, R., Hodkinson, I., Kurucz, A.: On modal logics between k×k×k and s5∗s5∗s5. J. Symb. Log., 221–234 (2002)

  27. Kontchakov, R., Kurucz, A., Wolter, F., Zakharyaschev, M.: Spatial logic+ temporal logic=?. In: Handbook of Spatial Logics, pp. 497–564. Springer (2007)

  28. Kracht, M., Wolter, F.: Simulation and transfer results in modal logic–a survey. Stud. Logica. 59(2), 149–177 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kripke, S.A.: Semantical analysis of modal logic i normal modal propositional calculi. Math. Log. Q. 9(5–6), 67–96 (1963)

    Article  MATH  MathSciNet  Google Scholar 

  30. Ladner, R.E.: The computational complexity of provability in systems of modal propositional logic. SIAM J. Comput. 6(3), 467–480 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  31. Marx, M.: Complexity of products of modal logics. J. Log. Comput. 9(2), 197–214 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  32. Papadimitriou, C.H.: Computational Complexity. Wiley (2003)

  33. Rasga, J., Sernadas, A., Sernadas, C.: Fibring as biporting subsumes asymmetric combinations. Stud. Logica. 102(5), 1041–1074 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  34. Russell, S.J., Norvig, P., Davis, E.: Artificial Intelligence: A Modern Approach. Prentice Hall, Englewood Cliffs (2010)

    Google Scholar 

  35. Schurz, G.: Combinations and completeness transfer for quantified modal logics. Log. J. IGPL 19(4), 598–616 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Segerberg, K.: Two-dimensional modal logic. J. Philos. Log. 2(1), 77–96 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  37. Sernadas, A., Sernadas, C.: Combining logic systems: why, how, what for. CIM Bull. 15, 9–14 (2003)

    Google Scholar 

  38. Sernadas, A., Sernadas, C., Caleiro, C.: Synchronization of logics. Stud. Logica. 59(2), 217–247 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  39. Sernadas, A., Sernadas, C., Caleiro, C.: Fibring of logics as a categorial construction. J. Log. Comput. 9(2), 149–179 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  40. Sernadas, A., Sernadas, C., Zanardo, A.: Fibring modal first-order logics: completeness preservation. Log. J. IGPL 10(4), 413–451 (2002a)

    Article  MATH  MathSciNet  Google Scholar 

  41. Sernadas, A., Sernadas, C., Rasga, J., Coniglio, M.: On graph-theoretic fibring of logics. J. Log. Comput. 19(6), 1321–1357 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  42. Sernadas, A., Sernadas, C., Rasga, J.: On meet-combination of logics. J. Log. Comput. 22(6), 1453–1470 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Sernadas, C., Rasga, J., Carnielli, W.A.: Modulated fibring and the collapsing problem. J. Symb. Log. 67(4), 1541–1569 (2002b)

    Article  MATH  MathSciNet  Google Scholar 

  44. Shekhtman, V.B.: Two-dimensional modal logic. Mathematical notes of the Academy of Sciences of the USSR 23(5), 417–424 (1978)

    Article  MATH  Google Scholar 

  45. Spaan, E.: Complexity of modal logics. PhD thesis, Universiteit van Amsterdam (1993)

  46. Thomason, R.H.: Combinations of tense and modality. In: Handbook of Philosophical Logic, pp. 135–165. Springer (1984)

  47. Vardi, M.Y.: Why is modal logic so robustly decidable. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 31, 149–184 (1997)

    MathSciNet  Google Scholar 

  48. Wolter, F.: Fusions of modal logics revisited. Advances in Modal Logic 1, 361–379 (1998)

    MathSciNet  Google Scholar 

  49. Zanardo, A., Sernadas, A., Sernadas, C.: Fibring: completeness preservation. J. Symb. Log., 414–439 (2001)

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Authors and Affiliations

  1. Department of Cognitive Science and Technology, Xiamen University and Fujian Key Laboratory of Brain-like Intelligent Systems, Xiamen, China

    Yin Wu, Min Jiang, Zhongqiang Huang, Fei Chao & Changle Zhou

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  1. Yin Wu
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  2. Min Jiang
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  3. Zhongqiang Huang
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Correspondence to Min Jiang.

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Wu, Y., Jiang, M., Huang, Z. et al. An NP-complete fragment of fibring logic. Ann Math Artif Intell 75, 391–417 (2015). https://doi.org/10.1007/s10472-015-9468-4

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