Automated Reasoning with Analytic Tableaux and Related by V. Michele Abrusci (auth.), Marta Cialdea Mayer, Fiora Pirri

By V. Michele Abrusci (auth.), Marta Cialdea Mayer, Fiora Pirri (eds.)

This ebook constitutes the refereed lawsuits of the foreign convention on computerized Reasoning with Analytic Tableaux and comparable tools, TABLEAUX 2003, held in Rome, Italy in September 2003.

The 20 revised complete papers offered have been rigorously reviewed and chosen for inclusion within the ebook. All present matters surrounding the mechanization of logical reasoning with tableaux and related tools are addressed within the context of a wide number of good judgment calculi.

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Extra info for Automated Reasoning with Analytic Tableaux and Related Methods : International Conference, TABLEAUX 2003, Rome, Italy, September 2003. Proceedings

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The next step is to show that applications of (cut) on atomic cut formulae can be eliminated. Proposition 8. If G|Γ, λq Δ and G |Π Σ, λq are derivable in GLBn for q atomic, then G|G |Γ, Π Δ, Σ is derivable in GLBn . Proof. We follow exactly the proof of Proposition 5 and prove the more general result that: If Q = Γ1 , λ1 q Δ1 | . . |Γk , λk q Δk and Qi = Gi |Πi Σi , μi q for i = 1, . . , k are derivable in GLBn for q atomic and μi ≥ λi ≥ 0 for i = 1, . . , k, then 46 Agata Ciabattoni and George Metcalfe Q = G1 | .

Hence, applying the induction hypothesis twice to Q4 : G|Γ Δ|Γ, Γ Δ, Δ is derivable in GL. The claim now follows using (EC) and (S). ✷ Corollary 2. Cut-elimination holds for GL + (cut). Proof. By Theorem 5 and the derivability of (cut) in GL + (gencut). ✷ 5 Hypersequent Calculi for Bounded Lukasiewicz Logics We now return our attention to bounded Lukasiewicz logics, defining the validity of a hypersequent in LBn in terms of its validity in the k-valued Lukasiewicz logics for k = 2, . . , n. Definition 11 (Interpretation of Hypersequents for Lk ).

C. Chang. Algebraic analysis of many valued logics. Trans. Amer. Math. , 88:467–490, 1958. 32, 35 [3] A. Ciabattoni. Bounded contraction in systems with linearity. In Proceedings of TABLEAUX 99, volume 1617 of LNAI, pages 113–127. Springer, 1999. 33 Bounded Lukasiewicz Logics 47 [4] F. Cicalese. Reliable Computation with Unreliable Information. PhD thesis, University of Salerno, 2001. html. 33, 37 [5] R. L. O. Cignoli, D. Mundici, and I. M. L. D’Ottaviano. Algebraic Foundations of Many-valued Reasoning.

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