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A tableau method for checking rule admissibility in S4. (English) Zbl 1345.03033
Bolander, Thomas (ed.) et al., Proceedings of the 6th workshop on methods for modalities (M4M-6 2009), Copenhagen, Denmark, November 12–14, 2009. Amsterdam: Elsevier. Electronic Notes in Theoretical Computer Science 262, 17-32 (2010).
Summary: Rules that are admissible can be used in any derivations in any axiomatic system of a logic. In this paper we introduce a method for checking the admissibility of rules in the modal logic S4. Our method is based on a standard semantic ground tableau approach. In particular, we reduce rule admissibility in S4 to satisfiability of a formula in a logic that extends S4. The extended logic is characterised by a class of models that satisfy a variant of the co-cover property. The class of models can be formalised by a well-defined first-order specification. Using a recently introduced framework for synthesising tableau decision procedures this can be turned into a sound, complete and terminating tableau calculus for the extended logic, and gives a tableau-based method for determining the admissibility of rules.
For the entire collection see [Zbl 1281.03003].

##### MSC:
 03B45 Modal logic (including the logic of norms)
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##### References:
 [1] Babenyshev, S., The decidability of admissibility problems for modal logics S4.2 and S4.2grz and superintuitionistic logic KC, Algebra logic, 31, 4, 205-216, (1992) · Zbl 0795.03017 [2] Friedman, H., One hundred and two problems in mathematical logic, J. symb. log., 40, 3, 113-130, (1975) · Zbl 0318.02002 [3] Ghilardi, S., Unification in intuitionistic logic, J. symb. log., 64, 2, 859-880, (1999) · Zbl 0930.03009 [4] Ghilardi, S., Best solving modal equations, Ann. pure appl. logic, 102, 3, 183-198, (2000) · Zbl 0949.03010 [5] Ghilardi, S., A resolution/tableaux algorithm for projective approximations in IPC, Log. J. IGPL, 10, 3, 229-243, (2002) · Zbl 1005.03504 [6] Ghilardi, S.; Sacchetti, L., Filtering unification and most general unifiers in modal logic, J. symb. log., 69, 3, 879-906, (2004) · Zbl 1069.03011 [7] Harrop, R., Concerning formulas of the types $$a \rightarrow b \vee c$$, $$a \rightarrow \exists x b(x)$$ in intuitionistic formal system, J. symb. log., 25, 27-32, (1960) · Zbl 0098.24201 [8] Iemhoff, R., On the admissible rules of intuitionistic propositional logic, J. symb. log., 66, 1, 281-294, (2001) · Zbl 0986.03013 [9] Jeřábek, E., Admissible rules of modal logics, J. log. comput., 15, 4, 411-431, (2005) · Zbl 1077.03011 [10] Jeřábek, E., Complexity of admissible rules, Arch. math. logic, 46, 2, 73-92, (2007) · Zbl 1115.03010 [11] Kiyatkin, V.R.; Rybakov, V.V.; Oner, T., On finite model property for admissible rules, Mathematical logic quarterly, 45, 505-520, (1999) · Zbl 0938.03033 [12] Lorenzen, P., Einführung in die operative logik und Mathematik, (1955), Springer · Zbl 0066.24802 [13] Mints, G., Derivability of admissible rules, Journal of soviet mathematics, 6, 4, 417-421, (1976) · Zbl 0375.02014 [14] Roziere, P., Admissible and derivable rules, Mathematical structures in computer science, 3, 129-136, (1993) · Zbl 0797.03001 [15] Rybakov, V.V., A criterion for admissibility of rules in modal system S4 and the intuitionistic logic, Algebra logic, 23, 5, 369-384, (1984) · Zbl 0598.03013 [16] Rybakov, V.V., Semantic admissibility criteria for deduction rules in S4 and int, Mat. zametki, 50, 1, 84-91, (1991) · Zbl 0729.03013 [17] Rybakov, V.V., Rules of inference with parameters for intuitionistic logic, J. symb. log., 57, 3, 912-923, (1992) · Zbl 0788.03007 [18] Rybakov, V.V., Admissibility of logical inference rules, Studies in logic and the foundations of mathematics, vol. 136, (1997), Elsevier · Zbl 0872.03002 [19] Rybakov, V.V., Logics with universal modality and admissible consecutions, J. appl. non-classical log., 17, 3, 381-394, (2007) [20] Schmidt, R.A.; Tishkovsky, D., Using tableau to decide expressive description logics with role negation, (), 438-451 [21] Schmidt, R.A.; Tishkovsky, D., A general tableau method for deciding description logics, modal logics and related first-order fragments, (), 194-209 · Zbl 1165.03319 [22] Schmidt, R.A.; Tishkovsky, D., Automated synthesis of tableau calculi, (), 310-324 · Zbl 1218.03011 [23] Wolter, F.; Zakharyaschev, M., Undecidability of the unification and admissibility problems for modal and description logics, ACM transdactions on computational logic, 9, 4, 1-20, (2008) · Zbl 1367.03026 [24] D. Zucchelli. Studio e realizzazione di algoritmi per l’unificazione nelle logiche modali. Laurea specialistica in informatica (Masters Thesis), Università degli Studi di Milano, 2004. In Italian. Supervisor: Silvio Ghilardi
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