Certification of Prefixed Tableau Proofs for Modal Logic

Abstract

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

A general proof certification framework for modal logic

“…According to this specification, which can be found in [Libal and Volpe 2016], each decide step is completely determined by the proof evidence. 4.3.2.…”

“…The treatment of labeled systems (LS) [Negri 2005] was already implemented in the previous version of Checkers, which is described in [Libal and Volpe 2016]. In order to get emulation of LS, we require a very simple use of the framework LM F * , where at each sequent the present corresponds to the set of all the labels occurring in the proof, no use of multi-focusing is required and the future of a formula is set, in the case of ♦-formulas, to the index of the corresponding -formula.…”

“…The idea behind it -called the "de Bruijn Criterion" [de Bruijn 1970] -is that small and simple kernels offer higher trust. The kernel employed in this paper, based on LKF, consists in 93 lines of λProlog code and is the same used in other ProofCert publications (e.g., [Chihani et al 2015, Libal and Volpe 2016, Chihani et al 2017]) † . The problem of the great variety of different proof formalisms and proof systems to be considered, when dealing with proof checking, is especially apparent in the case of modal logics, whose proof theory is notoriously non-trivial.…”

“…The idea behind it -called the 'de Bruijn Criterion' (de Bruijn 1970) -is that small and simple kernels offer higher trust. The kernel employed in this paper, based on LKF, consists in 93 lines of λProlog code and is the same used in other ProofCert publications (e.g., Chihani et al 2015Chihani et al , 2017Libal and Volpe 2016) † .…”

Translating Between Implicit and Explicit Versions of Proof

“…Instead of starting with dependently typed λ-calculus, the FPC framework is based on Gentzen's more low-level notion of sequent calculus proof. FPC definitions have been formulated for resolution refutations [46], expansion trees [38] (a generalization of Herbrand disjunctions), Frege proof systems, matings [2], simply typed and dependently typed λ-terms, equality reasoning [15], tableau proofs for some modal logics [30,31,37], and decision procedures based on conjunctive normal forms, truth table evaluation, and the G4ip calculus [21,50]. Additionally, FPCs have been used to formalize proof outlines [8] and have been applied to model checking [28].…”