HYPNO: Theorem Proving with Hypersequent Calculi for Non-normal Modal Logics (System Description)

Abstract

We present HYPNO (HYpersequent Prover for NOn-normal modal logics), a Prolog-based theorem prover and countermodel generator for non-normal modal logics. HYPNO implements some hypersequent calculi recently introduced for the basic system E and its extensions with axioms M, N, and C. It is inspired by the methodology of leanT A P , so that it does not make use of any ad-hoc control mechanism. Given a formula, HYPNO provides either a proof in the calculus or a countermodel, directly built from an open saturated hypersequent. Preliminary experimental results show that the performances of HYPNO are very promising with respect to other theorem provers for the same class of logics.

On Free Description Logics with Definite Descriptions

“…Besides the open decidability problems discussed above, future research directions include the extension of our results to more expressive monodic fragments (Gabbay et al 2003;Hodkinson, Wolter, and Zakharyaschev 2002), automated support for the construction of definite descriptions and referring expressions (Artale et al 2021;Kurucz, Wolter, and Zakharyaschev 2023), the design of 'practical' reasoning algorithms for the languages considered here, and the extension of our results to modal DLs with hybrid (Braüner 2014;Indrzejczak and Zawidzki 2023), branchingtime (Hodkinson, Wolter, and Zakharyaschev 2002;Gutiérrez-Basulto, Jung, and Lutz 2012), dynamic (Harel 1979), or non-normal operators (Dalmonte et al 2023).…”

“…As future work, we intend to strengthen our results, as well as deepen the connections between free DLs with definite descriptions, on the one hand, and modal operators, on the other. On the epistemic side, we are interested in: (i) considering frames for the propositional modal logics K4, T, S4, or KD45, to model different doxastic or epistemic attitudes that might satisfy or not the so-called factivity and introspection principles [28]; (ii) investigating of non-rigid descriptions and names in the context of non-normal modal DLs, such as the ones obtained from the systems E, M, C, and N [34,35,36], to avoid the logical omniscience problem (i.e., an agent knows all the logical truths and all the consequences of their background knowledge), which affects all the systems extending K [37,38]; (iii) addressing less expressive DL languages, such as ℰℒ𝒪 𝜄 𝑢 , in an epistemic setting, and connect them with the recently investigated standpoint DL family [39,40].…”

Hypersequent calculi for non-normal modal and deontic logics: Countermodels and optimal complexity

“…The procedures reduce validity/satisfiability in each modal logic to a set of SAT problems, to be handled by a SAT solver; despite their efficiency, the procedures provide neither "proofs", nor countermodels, whence having a different aim from the calculi of this work. Our hypersequent calculi have nonetheless an interest for automated reasoning: for systems within the classical cube, they have been implemented in the theorem prover HYPNO [8].…”