On interpolation in next (kb.alt(2))

We prove that there is infinitely many tabular modal logics extending KB.Alt(2) which have interpolation

On Halldén Completeness of Modal Logics Determined by Homogeneous Kripke Frames

Halldén complete modal logics are defined semantically. They have a nice characterization as they are determined by homogeneous Kripke frames.Supported by the NCN, research grant DEC-2013/09/B/HS1/00701

Polyatomic Logics and Generalised Blok-Esakia Theory

This paper presents a novel concept of a Polyatomic Logic and initiates its systematic study. This approach, inspired by Inquisitive semantics, is obtained by taking a variant of a given logic, obtained by looking at the fragment covered by a selector term. We introduce an algebraic semantics for these logics and prove algebraic completeness. These logics are then related to translations, through the introduction of a number of classes of translations involving selector terms, which are noted to be ubiquitous in algebraic logic. In this setting, we also introduce a generalised Blok-Esakia theory which can be developed for special classes of translations. We conclude by showing some systematic connections between the theory of Polyatomic Logics and the general Blok-Esakia theory for a wide class of interesting translations.Comment: 48 pages, 2 figure

Extensions of modal logic KTB and other topics

This thesis covers four topics. They are the extensions of the modal logic KTB, the use of normal forms in modal logic, automated reasoning in the modal logic S4 and the problem of unavoidable words. Extensions of KTB: The modal logic KTB is the logic of reflexive and symmetric frames. Dually, KTB-algebras have a unary (normal) operator f that satisfies the identities f (x){u2265}x and {u231D}x{u2264}f ({u231D}f(x)). Extensions of KTB are subvarieties of the algebra KTB. Both of these form a lattice, and we investigate the structure of the bottom of the lattice of subvarieties. The unique atom is known to correspond to the modal logic whose frame is a single reflexive point. Yutaka demonstrated that this atom has a unique cover, corresponding to the frame of the two element chain. We construct covers of this element, and so demonstrate that there are a continuum of such covers. Normal Forms in Modal Logic: Fine proposed the use of normal forms as an alternative to traditional methods of determining Kripke completeness. We expand on this paper and demonstrate the application of normal forms to a number of traditional modal logics, and define new terms needed to apply normal forms in this situation. Automated reasoning in 84: History based methods for automated reasoning are well understood and accepted. Pliu{u0161}kevi{u010D}ius & Pliu{u0161}kevi{u010D}ien{u0117} propose a new, potentially revolutionary method of applying marks and indices to sequents. We show that the method is flawed, and empirically compare a different mark/index based method to the traditional methods instead. Unavoidable words: The unavoidable words problem is concerned with repetition in strings of symbols. There are two main ways to identify a word as unavoidable, one based on generalised pattern matching and one from an algorithm. Both methods are in NP, but do not appear to be in P. We define the simple unavoidable words as a subset of the standard unavoidable words that can be identified by the algorithm in P-time. We define depth separating IX x homomorphisms as an easy way to generate a subset of the unavoidable words using the pattern matching method. We then show that the two simpler problems are equivalent to each other

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book