Abstract. Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules from Gentzen's original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system G3K of Negri for the modal logic K and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in G3K corresponds to a bipole-a pair of a positive and a negative phases-in LKF. Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.
The paper is a contribution both to the theoretical foundations and to the actual construction of efficient automatizable proof procedures for non-classical logics. We focus here on the case of finite-valued logics, and exhibit: (i) a mechanism for producing a classic-like description of them in terms of an effective variety of bivalent semantics; (ii) a mechanism for extracting, from the bivalent semantics so obtained, uniform (classically-labeled) cut-free standard analytic tableaux with possibly branching invertible rules and paired with proof strategies designed to guarantee termination of the associated proof procedure; (iii) a mechanism to also provide, for the same logics, uniform cut-based tableau systems with linear rules. The latter tableau systems are shown to be adequate even when restricted to analytic cuts, and they are also shown to polynomially simulate truth-tables, a feature that is not enjoyed by the former standard type of tableau systems (not even in the 2-valued case). The results are based on useful generalizations of the notions of analyticity and compositionality, and illustrate a theory that applies to many other classes of non-classical logics.
Different theorem provers tend to produce proof objects in different formats
and this is especially the case for modal logics, where several deductive
formalisms (and provers based on them) have been presented. This work falls
within the general project of establishing a common specification language in
order to certify proofs given in a wide range of deductive formalisms. In
particular, by using a translation from the modal language into a first-order
polarized language and a checker whose small kernel is based on a classical
focused sequent calculus, we are able to certify modal proofs given in labeled
sequent calculi, prefixed tableaux and free-variable prefixed tableaux. We
describe the general method for the logic K, present its implementation in a
prolog-like language, provide some examples and discuss how to extend the
approach to other normal modal logicsComment: In Proceedings GandALF 2016, arXiv:1609.0364
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