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Journal of Logic and Computation Advance Access published online on January 8, 2008
Journal of Logic and Computation, doi:10.1093/logcom/exm078
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© The Author, 2007. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Original papers

Multi-modal and Temporal Logics with Universal Formula---Reduction of Admissibility to Validity and Unification*

V Rybakov

Department of Computing and Mathematics, Manchester Metropolitan University, Manchester M1 5GD, UK.

E-mail: v.rybakov{at}mmu.ac.uk

The article studies multi-modal (in particular temporal, tense, logics) possessing universal formulas. We prove (Theorems 11 and 12) that the admissibility problem for inference rules (inference rules with parameters) is decidable for all decidable multi-modal (in particular, temporal) logics possessing an universal formula and decidable w.r.t. unification (unification with parameters). These theorems use characterizations of admissible rules in terms of valid rules and unifiable formulas. Results are applied to wide range of multi-modal logics, including all linear transitive temporal logics, all temporal logics with bounded zigzag, all multi-modal logics with explicit universal modality. As consequence, we show that the admissibility problem for all such logics is decidable (e.g. for all logics of the series S4nU, nisin N).

Keywords: Decidability; algorithms; logical consecutions; inference rules; temporal logic; linear temporal logic; admissible rules

*This research is supported by Engineering and Physical Sciences Research Council (EPSRC), U.K., grant EP/F014406/1



References

  1. Areces G, Blackburn P, Marx M. The computational complexity of Hybrid Temporal Logics. Logic Journal of IGPL (2000) 8:653–679.[Abstract/Free Full Text]
  2. Bull RA. An algebraic study of tense logics with linear time. The Journal of Symbolic Logic (1968) 33:27–38.[CrossRef]
  3. Bull RA. Note on a paper in tense logic. The Journal of Symbolic Logic (1969) 34:215–218.[CrossRef]
  4. Friedman H. One hundred and two problems in mathematical logic. Journal of Symbolic Logic (1975) 40:113–130.[CrossRef][ISI]
  5. Chagrov A. A decidable logic with undecidable admissibility problem for inference rulres. Algebra and Logic (1992) 31:53–55.[CrossRef]
  6. Gabbay DM, Hodkinson IM, Reynolds MA. Temporal logic. In: Mathematical Foundations and Computational Aspects (1994) Vol. 1. Oxford: Clarendon Press.
  7. Gabbay DM, Hodkinson IM. An axiomatisation of the temporal logic with Until and Since over the real numbers. Journal of Logic and Computation (1990) 1:229–260.[Abstract/Free Full Text]
  8. Ghilardi S. Unification in intuitionistic logic. Journal of Symbolic Logic (1999) 64:859–880.[CrossRef][ISI]
  9. Ghilardi S. Best solving modal equations. Annals of Pure and Applied Logic (2000) 102:183–198.[CrossRef][ISI]
  10. Ghilardi S, Sacchetti L. Filtering unification and most general unifiers in modal logic. Journal of Symbolic Logic (2004) 69:879–906.[CrossRef][ISI]
  11. Goldblatt R. Logics of time and computation. In: CSLI Lecture Notes (1992) Stanford University. No. 7.
  12. Goldblatt R. Mathematical modal logic: a view of its evolution. Journal of Applied Logic (2003) 1:309–392.[CrossRef]
  13. Goranko V, Passy S. Using the universal modality: gains and questions. Journal of Logic Computation (1992) 2:5–30.[CrossRef]
  14. Harrop R. Concerning formulas of the types A -> B {vee} C, A -> exist x B(x) in intuitionistic formal system. Journal of Symbolic Logic (1960) 25:27–32.[CrossRef]
  15. Iemhoff R. On the admissible rules of intuitionistic propositional logic. Journal of symbolic logic (2001) 66:281–294.[CrossRef][ISI]
  16. Jerábek E. Admissible rules of modal logics. Journal of Logic and Computation (2005) 15:411–431.[Abstract/Free Full Text]
  17. Kapron BM. Modal sequents and definability. Journal of Symbolic Logic (1987) 52:756–765.[CrossRef][ISI]
  18. Litak T, Wolter F. All finitely axiomatizable tense logics of linear time flows are coNP-complete. Studia Logica (2005) 81:153–165.[CrossRef]
  19. Lorenzen P. Einführung in die operative Logik und Mathematik (1955) Berlin-Göttingen: Heidelberg, Springer-Verlag.
  20. Mints GE. Derivability of admissible rules. Journal of Soviet Mathematics (1976) 6:417–421.[CrossRef]
  21. Roziere P. Admissible and derivable rules. Mathematical Structures in Computer Science (1993) 3:129–136.
  22. Rybakov VV. A criterion for admissibility of rules in the modal system $S4$ and the intuitionistic logic. Algebra and Logic (1984) 23:369–384. (English translation).[CrossRef]
  23. Rybakov VV. The bases for admissible rules of logics S4 and int. Algebra and Logic (1985) 24:55–68. (English translation).[CrossRef]
  24. Rybakov VV. Equations in free Topoboolean algebra. Algebra and Logic (1986) 25:172–204. (in Russian).
  25. Rybakov VV. Logical equations and admissible rules with parameters in modal logics of provability. Studia Logica (1990) XLIX:215–239.
  26. Rybakov VV. Rules of inference with parameters for intuitionistic logic. Journal of Symbolic Logic (1992) 57:912–923.[CrossRef][ISI]
  27. Rybakov VV. Admissible logical inference rules. In: Series: Studies in Logic and the Foundations of Mathematics (1997) Vol. 136. North-Holland: Elsevier Sci. Publ.
  28. Rybakov VV, Kiyatkin, Oner T. On finite model property for admissible rules. Mathematical Logic Quarterly (1999) 45:505–520.[ISI]
  29. Rybakov VV. Construction of an explicit basis for rules admissible in modal system S4. Mathematical Logic Quarterly (2001) 47:441–451.[CrossRef][ISI]
  30. Rybakov VV. Logical consecutions in discrete linear temporal logic. Journal of Symbolic Logic (2005) 70:1137–1149.[CrossRef][ISI]
  31. Rybakov VV. Logical consecutions in intransitive temporal linear logic of finite intervals. Journal of Logic Computation (2005) 15:633–657.
  32. Rybakov VV. Linear temporal logic with until and before on integer numbers, deciding algorithms. Vol. 3967. New York, Heidelberg: Springer Publishing Company. 322–334. Computer science -- theory and applications, Lecture Notes in Computer Science.
  33. Segerberg K. Modal logics with linear alternative relations. Theoria (1970) 36:301–322.[ISI]
  34. van Benthem J. The logic of time. Synthese Library (1983) Vol. 156. Reidel, Dordrecht.
  35. Wolter F. Tense logics without tense operators. Mathematocal Logic Quaterley (1996) 42:145–171.[CrossRef]
  36. Wolter F, Zakharyaschev M. Undecidability of the unification and admissibility problems for modal and description logics. ACM Transdactions on Computational Logic, 2006. in press.

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