Download Automated Reasoning with Analytic Tableaux and Related by Piero A. Bonatti (auth.), Nicola Olivetti (eds.) PDF

By Piero A. Bonatti (auth.), Nicola Olivetti (eds.)

This e-book constitutes the refereed complaints of the sixteenth overseas convention on automatic Reasoning with Analytic Tableaux and comparable equipment, TABLEAUX 2007, held in Aix en Provence, France in July 2007.

The 14 revised learn papers awarded including procedure descriptions in addition to 3 invited talks have been conscientiously reviewed and chosen from forty three submissions. The papers disguise many issues within the wide variety of logics, from intuitionistic and substructural logics to modal logics (including temporal and dynamic logics), from many-valued logics to nonmonotonic logics, and from classical first-order good judgment to description logics. a few contributions are thinking about selection approaches, others on effective reasoning, in addition to on implementation of theorem provers. a couple of papers discover purposes resembling model-checking, verification, or wisdom engineering. furthermore, different contributions utilize tableaux as a device for theoretical research of logics.

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Additional info for Automated Reasoning with Analytic Tableaux and Related Methods: 16th International Conference, TABLEAUX 2007, Aix en Provence, France, July 3-6, 2007. Proceedings

Example text

Theorem 8. If G∀Ł G, then |=Ł G. Proof. By induction on the height of a derivation of G in G∀Ł. Since all other rules have been checked in [15], we just check that the quantifier rules preserve validity. Also note that using linearity, we can disregard the side-hypersequent G occurring in both premises and conclusion. For (⇒ ∀), suppose that for each interpretation I: (vI (C) − 1) ≤ (vI (A(a)) − 1) + C∈Γ (vI (D) − 1) D∈Δ where a does not occur in Γ , Δ, or A. Since this holds for any a, we have also: (vI (C) − 1) ≤ (vI (∀xA(x)) − 1) + C∈Γ (vI (D) − 1) D∈Δ as required.

The definition of validity for ∀Ł given above can be generalized to validity for safe MV-algebras (see [7] for definitions and a wealth of results). An MV-algebra is an algebra A = L, ⊕, ¬, 0 such that: 1. 2. 3. 4. L, ⊕, 0 is a commutative monoid. ¬¬x = x. x ⊕ ¬0 = ¬0. ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x. where 1 =def ¬0; x → y =def ¬x ⊕ y; x y =def ¬(¬x ⊕ ¬y); x ∧ y =def x (¬x ⊕ y); x ∨ y =def (x ¬y) ⊕ y; and x ≤ y iff x ∧ y = y. An MV-chain is an MV-algebra that is linearly ordered. An interpretation I for an MV-algebra A = L, ⊕, ¬, 0 is defined as for ∀Ł except that n-ary predicate symbols p are mapped by vI to functions from Dn into L and vI is extended to all formulas by: vI (⊥) = 0 vI (A → B) = vI (∀xA(x)) = vI (∃xA(x)) = vI (A) → vI (B) if vI (A) and vI (B) are defined undefined otherwise inf{vI [x ← a](A(x)) : a ∈ D} if the infimum exists undefined otherwise sup{vI [x ← a](A(x)) : a ∈ D} if the supremum exists undefined otherwise I satisfies a formula A iff vI (A) = 1, and A is valid in A iff A is satisfied by all interpretations for A.

Pe˜ naloza initial assertion is since its presence depends only on the input p, and not on any of the axioms. By applying the rule (2) using axiom ax2 , we can add the assertion s. Since the addition of this assertion depends on the presence of ax2 , it receives label ax2 . Then we can use ax3 to add q. Since this addition depends on the presence of ax3 and of the assertion s, which has label ax2 , the label of this new assertion is ax2 ∧ ax3 . There is, however, also another possibility to generate the assertion q: apply the rule (2) using axiom ax1 .

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