Abstract We show that the monadic fragment of the modal predicate logic of a single Kripke frame with finitely many possible worlds, but possibly infinite domains, is decidable. This holds true even for multimodal logics with equality, regardless of whether equality is interpreted as identity or as congruence. By the Gödel–Tarski translation, similar results follow for superintuitionistic predicate logics, with or without equality. Using these observations, we establish upper algorithmic bounds, which match the known lower bounds, for monadic fragments of some modal predicate logics. In particular, we prove that, if $L$ is a propositional modal logic contained in $\textbf{S5}$, $\textbf{GL.3}$ or $\textbf{Grz.3}$ and the class of finite Kripke frames validating $L$ is recursively enumerable, then the monadic fragment with equality of the predicate logic of finite Kripke frames validating $L$ is $\varPi ^{0}_{1}$-complete; this, in particular, holds if $L$ is one of the following propositional logics: $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{KTB}$, $\textbf{K4}$, $\textbf{K4.3}$, $\textbf{S4}$, $\textbf{S4.3}$, $\textbf{GL}$, $\textbf{Grz}$, $\textbf{K5}$, $\textbf{K45}$ and $\textbf{S5}$. We also prove that monadic fragments with equality of logics $\textbf{QAlt}^=_{n}$ and $\textbf{QTAlt}^=_{n}$ are decidable. The obtained results are easily extendable to the multimodal versions of the predicate logics we consider and to logics with the Barcan formula.

Abstract We prove that propositional intuitionistic modal logics $\textbf{FS}$ (also known as $\textbf{IK}$) and $\textbf{MIPC}$ (also known as $\textbf{IS5}$) are polynomial-time embeddable into, and hence polynomial-time equivalent to, their own one-variable fragments. It follows that the one-variable fragment of $\textbf{MIPC}$ is coNEXPTIME-complete. The method of proof applies to a wide range of intuitionistic modal logics characterizable by two-dimensional frames, among them intuitionistic analogues of such classical modal logics as $\textbf{K4}$ and $\textbf{S4}$.

Abstract We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

Abstract We show that products of propositional modal logics where at least one factor is one of the monomodal logics $\textbf {K}$, $\textbf {KT}$, $\textbf {KB}$ and $\textbf {KTB}$ are polynomial-time embeddable into their single-variable fragments. Consequently, we obtain results about the computational complexity of single-variable fragments of logics belonging to intervals bounded by such products. We generalize our embeddability results to expanding relativized products and to products with polymodal logics.

Abstract We show that products and expanding relativized products of propositional modal logics where one component is the minimal monomodal logic K are polynomial-time reducible to their single-variable fragments. Therefore, the known lower-bound complexity and undecidability results for such logics are extended to their single-variable fragments. Similar results are obtained for products where one component is a polymodal logic with a K-style modality; these include products with propositional dynamic logics.

Abstract We consider the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate superintuitionistic logics of finite Kripke frames on the logics’ algorithmic properties. By a finite frame we mean a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that positive fragments of many predicate superintuitionistic logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter; this applies to the logics of finite frames of the predicate counterparts of propositional logics lying between the intuitionistic logic and the logic of the weak law of the excluded middle.

Abstract We study the algorithmic properties of first-order monomodal logics of frames $\langle {\textrm{I}\!\textrm{N}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{N}}, < \rangle $, $\langle \mathbb {Q}, \leqslant \rangle $, $\langle \mathbb {Q}, < \rangle $, $\langle {\textrm{I}\!\textrm{R}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{R}}, < \rangle $, as well as some related logics, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. We show that the logics of frames based on $ {\textrm{I}\!\textrm{N}}$ are $\varPi ^1_1$-hard—thus, not recursively enumerable—in languages with two individual variables, one monadic predicate letter and one proposition letter. We also show that the logics of frames based on $\mathbb {Q}$ and ${\textrm{I}\!\textrm{R}}$ are $\varSigma ^0_1$-hard in languages with the same restrictions. Similar results are obtained for a number of related logics.

Abstract We investigate the relationship between recursive enumerability and elementary frame definability in first-order predicate modal logic. On one hand, it is well known that every first-order predicate modal logic complete with respect to an elementary class of Kripke frames, i.e. a class of frames definable by a classical first-order formula, is recursively enumerable. On the other, numerous examples are known of predicate modal logics, based on ‘natural’ propositional modal logics with essentially second-order Kripke semantics, that are either not recursively enumerable or Kripke incomplete. This raises the question of whether every Kripke complete, recursively enumerable predicate modal logic can be characterized by an elementary class of Kripke frames. We answer this question in the negative, by constructing a normal predicate modal logic which is Kripke complete, recursively enumerable, but not complete with respect to an elementary class of frames. We also present an example of a normal predicate modal logic that is recursively enumerable, Kripke complete, and not complete with respect to an elementary class of rooted frames, but is complete with respect to an elementary class of frames that are not rooted.

Abstract We study the effect of restricting the number of individual variables, as well as the number and arity of predicate letters, in languages of first-order predicate modal logics of finite Kripke frames on the logics’ algorithmic properties. A finite frame is a frame with a finite set of possible worlds. The languages we consider have no constants, function symbols or the equality symbol. We show that most predicate modal logics of natural classes of finite Kripke frames are not recursively enumerable—more precisely, $\varPi ^0_1$-hard—in languages with three individual variables and a single monadic predicate letter. This applies to the logics of finite frames of the predicate extensions of the sublogics of propositional modal logics $\textbf{GL}$, $\textbf{Grz}$ and $\textbf{KTB}$—among them, $\textbf{K}$, $\textbf{T}$, $\textbf{D}$, $\textbf{KB}$, $\textbf{K4}$ and $\textbf{S4}$.

We develop an incremental tableau-based decision procedure for the alternating-time temporal logic ATL and some of its variants. While running within the theoretically established complexity upper bound, we believe that our tableaux are 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.

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 agents of agents, as well as operators of distributed and common knowledge among all agents in agents. 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.