Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter

We prove that the positive fragment of first-order intuitionistic logic in the language with two variables and a single monadic predicate letter, without constants and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most "natural" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without constants and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia Logica,doi:10.1007/s11225-018-9815-

Tableau-based decision procedure for the multi-agent epistemic logic with operators of common and distributed knowledge

We develop an incremental-tableau-based decision procedure for the multi-agent epistemic logic MAEL(CD) (aka S5_n (CD)), whose language contains operators of individual knowledge for a finite set Ag of agents, as well as operators of distributed and common knowledge among all agents in Ag. Our tableau procedure works in (deterministic) exponential time, thus establishing an upper bound for MAEL(CD)-satisfiability that matches the (implicit) lower-bound known from earlier results, which implies ExpTime-completeness of MAEL(CD)-satisfiability. Therefore, our procedure provides a complexity-optimal algorithm for checking MAEL(CD)-satisfiability, which, however, in most cases is much more efficient. We prove soundness and completeness of the procedure, and illustrate it with an example.Comment: To appear in the Proceedings of the 6th IEEE Conference on Software Engineering and Formal Methods (SEFM 2008

Tableau-based decision procedures for logics of strategic ability in multi-agent systems

We develop an incremental tableau-based decision procedures for the Alternating-time temporal logic ATL and some of its variants. While running within the theoretically established complexity upper bound, we claim that our tableau is practically more efficient in the average case than other decision procedures for ATL known so far. Besides, the ease of its adaptation to variants of ATL demonstrates the flexibility of the proposed procedure.Comment: To appear in ACM Transactions on Computational Logic. 48 page

Modal Logics with Existential Modality, Finite-iteration Modality, and Intuitionistic Base: Decidability and Completeness

This thesis investigates some modal logics that have been found to be useful in modelling computational phenomena and, therefore, of interest to theoretical computer science---namely, modal intuitionistic logics, logics with finite-iteration modality, and logics with existential modality. We prove a number of new general results concerning these logics. In particular, in chapter 3, we prove a general decidability result for intuitionistic modal logics through embedding them into the two-variable monadic second-order guarded fragment GF mon with certain conditions imposed on relations occurring in GF mon -formulas. In chapter 4, we prove the analogue of Makinson theorem for logics with finite-iteration modality, that is that every consistent logic in this language is either a sublogic of the logic of a Kripke frame containing a single reflexive point or a sublogic of the logic of a Kripke frame containing a single irreflexive point; the by-product of the theorem is the decidability of the problem of consistency for effectively finitely axiomatizable logics with finite-iteration modality. In chapter 5, we prove completeness of Hilbert-style axiomatizations of three logics whose language contains an existential modality ###: the minimal normal logic with ###, K# ; its deterministic extension DK# ; and the logic that is CPDL (converse PDL) with a single nominal and ### (this logic is known from the literature as PDL ). Apart from the presentation of the above-mentioned results, the thesis contains, in chapter 2, an overview of background material on modal logics and guarded fragments; this overview can also be read as a concise survey of the field of guarded fragments

2003, ‘On decidability of intuitionistic modal logics

We prove a general decidability result for a class of intuitionistic modal logics. The proof is a slight modification of the Ganzinger, Meyer and Veanes [6] result on decidability of the two variable monadic guarded fragment of first order logic with constraints on the guard relations expressible in monadic second order logic.