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

The 20 revised complete papers provided have been conscientiously reviewed and chosen for inclusion within the booklet. All present matters surrounding the mechanization of logical reasoning with tableaux and comparable equipment are addressed within the context of a extensive number of good judgment calculi.

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

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If G|Γ derivable in GL. A ⊃ B, Δ, then G|Γ Δ and G|Γ, A B, Δ are Proof. Similar to that of Proposition 3. ✷ We now show that applications of (cut) on atomic 7 formulae are eliminable. Proposition 5. If G|Γ, λq Δ and G |Π λq, Σ are derivable in GL for q atomic and λ ≥ 0, then G|G |Γ, Π Δ, Σ is derivable in GL. Proof. We 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 GL for q atomic and μi ≥ λi ≥ 0 for i = 1, .

Given such a representation model M, the definition of [[·]]M,i is summarized in figure 1. With this interpretation function, it is possible to define a notion of satisfiability, so as to identify which propositions of LI,Ψ do properly correspond to the behavior of information in our partial description approach. Definition 2 (Satisfiability) i. A proposition ϕ ∈ LI,Ψ is satisfied by a representation model M = S, ν ∈ RM(I, Ψ ) (which we denote M |=S,I,Ψ ϕ) if and only if ∀ i, [[ϕ]]M,i = Pi . ii. A proposition ϕ ∈ LI,Ψ is satisfied by representation systems over I if and only if it is satisfied by all representation models over I: |=S,I,Ψ ϕ ⇔ ∀ M ∈ RM(I, Ψ ), M |=S,I,Ψ ϕ In the following, for readability reasons, we will drop the I, Ψ subscripts, since we will refer to only one set of indices and one set of atomic propositions.

Sequent and hypersequent calculi for Abelian and Lukasiewicz logics. Submitted. org/list/cs/0211, 2003. 33, 37, 38, 39, 44 [12] D. Mundici. The logic of Ulam’s game with lies. In C. Bicchieri and M. L. Dalla Chiara, editors, Knowledge, belief and strategic interaction, pages 275–284. Cambridge University Press, 1992. 32, 35, 36 [13] D. Mundici. Ulam’s game, Lukasiewicz logic and C∗ -algebras. Fundamenta Informaticae, 18:151–161, 1993. 32, 35, 36 [14] A. Pelc. Searching with known error probability.

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