A general proof certification framework for modal logic

Abstract

One of the main issues in proof certification is that different theorem provers, even when designed for the same logic, tend to use different proof formalisms and produce outputs in different formats. The project ProofCert promotes the usage of a common specification language and of a small and trusted kernel in order to check proofs coming from different sources and for different logics. By relying on that idea and by using a classical focused sequent calculus as a kernel, we propose here a general framework for checking modal proofs. We present the implementation of the framework in a Prolog-like language and show how it is possible to specialize it in a simple and modular way in order to cover different proof formalisms, such as labeled systems, tableaux, sequent calculi and nested sequent calculi. We illustrate the method for the logic K by providing several examples and discuss how to further extend the approach. † When calculating the size of the program, we placed each atomic predicate on a new line. ‡ In fact, it is possible to show that every modal formula can be translated into a formula in the fragment of first-order logic which uses only two variables [Blackburn and Van Benthem 2007]. By the decidability of such a fragment, an easy proof of the decidability of the modal logic K follows.

Proof Search and Certificates for Evidential Transactions

“…Defining LKF or LJF FPCs for cyberlogic is challenging due to the modalities : and kb, and digital evidences. LKF has been used to check proofs in modal logics [19], but the translation of modal formulas into LK formulas used the…”

“…However, the existing work did not consider proof certificates for attestation logics. Closer to our objective is the work of Libal and Volpe [19], which define proof certificates for modal logics by encoding (the semantics of) these logics in LKF. Our work instead proposes proof certificates directly in Cyberlogic.…”