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

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

This e-book constitutes the refereed lawsuits of the foreign convention on automatic Reasoning with Analytic Tableaux and similar equipment, TABLEAUX 2003, held in Rome, Italy in September 2003.

The 20 revised complete papers awarded have been rigorously reviewed and chosen for inclusion within the e-book. All present concerns surrounding the mechanization of logical reasoning with tableaux and related tools are addressed within the context of a huge number of good judgment calculi.

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

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OL. More precisely, one has: ∀ ϕ ∈ LI,Ψ , OL ϕ ⇔ ∅ [ϕ]ε Proof The ⇒-implication can be easily proved by checking that all axioms of OL can be derived in the sequent-calculus formalism. In particular, for axiom L, one has to use the fact that i i·i for any index i ∈ I. t. the following translation in terms of representation systems: [γ1 ]Λ1 . . [γn ]Λn [ϕ]Λ ∀ i, [[KΛj γj ]]i ⊆ [[KΛ ϕ]]i j It follows from this that one has: ∀ ϕ, OL ϕ ⇒ ∅ [ϕ]ε ⇒ |=S ϕ ⇒ OL ϕ Now, in order to use this sequent calculus efficiently for the search of proofs, two rules deserve attention: the Cut-rule which may introduce new formulas, and the Univ-rule, which introduces new indices.

The following translation in terms of representation systems: [γ1 ]Λ1 . . [γn ]Λn [ϕ]Λ ∀ i, [[KΛj γj ]]i ⊆ [[KΛ ϕ]]i j It follows from this that one has: ∀ ϕ, OL ϕ ⇒ ∅ [ϕ]ε ⇒ |=S ϕ ⇒ OL ϕ Now, in order to use this sequent calculus efficiently for the search of proofs, two rules deserve attention: the Cut-rule which may introduce new formulas, and the Univ-rule, which introduces new indices. In the following section, we will investigate some proofs manipulations, and show that the Cut-rule can always be eliminated, and that the Univ-rule can be used at most once.

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. 3 Axiomatization We provide an axiomatization of this satisfaction predicate |=S for representation systems by defining the logic OL, as the intuitionistic logic [18, 19, 20] together with the modal axioms and rules listed in figure 2.

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