Kleene Algebra (KA) is a useful tool for proving that two programs are equivalent. Because KA's equational theory is decidable, it integrates well with interactive theorem provers. This raises the question: which equations can we (not) prove using the laws of KA? Moreover, which models of KA are complete, in the sense that they satisfy exactly the provable equations? Kozen (1994) answered these questions by characterizing KA in terms of its language model. Concretely, equivalences provable in KA are exactly those that hold for regular expressions. Pratt (1980) observed that KA is complete w.r.t. relational models, i.e., that its provable equations are those that hold for any relational interpretation. A less known result due to Palka (2005) says that finite models are complete for KA, i.e., that provable equivalences coincide with equations satisfied by all finite KAs. Phrased contrapositively, the latter is a finite model property (FMP): any unprovable equation is falsified by a finite KA. Both results can be argued using Kozen's theorem, but the implication is mutual: given that KA is complete w.r.t. finite (resp. relational) models, Palka's (resp. Pratt's) arguments show that it is complete w.r.t. the language model. We embark on a study of the different complete models of KA, and the connections between them. This yields a novel result subsuming those of Palka and Pratt, namely that KA is complete w.r.t. finite relational models. Next, we put an algebraic spin on Palka's techniques, which yield a new elementary proof of the finite model property, and by extension, of Kozen's and Pratt's theorems. In contrast with earlier approaches, this proof relies not on minimality or bisimilarity of automata, but rather on representing the regular expressions involved in terms of transformation automata.

We continue to explore the multi-agent logic of computational trees relative to the relational Kripke semantics of possible worlds: we investigate the question of logical solvability, the complexity of model construction, feasibility testing, and correctness. For the semantics introduced earlier, we proved a strong finite model property, and obtained polynomial estimates of the dimension of minimal models for an arbitrary formulas. We proved the recursive enumerability of finite frames of logic and proposed an effective algorithm for checking the feasibility of formulas, which makes it possible to conclude the solvability of logic. The obtained polynomial estimates are within the framework of theoretical expectations, which makes it possible to perceive the logic under study as an effective tool for analyzing multi-agent distributed systems and practical modelchecking, and to count on a positive resolution of issues of unification and description of the admissibility for inference rules.

The selection method is one of the methods to prove that various modal logics have the finite model property. For a given formula that is satisfiable in some model, we select a finite tree-like submodel, while preserving the satisfiability of the observed formula. In this paper, we adapt the selection method for the inquisitive modal logic InqML. We first define a tree-like model in the inquisitive setting and show that each satisfiable formula is satisfiable in a tree-like model. Then, using the notions of n-bisimulation and characteristic formulas, we show that InqML has the finite tree model property, i.e., each satisfiable formula is satisfiable in a finite tree-like model. Furthermore, we show analogous results for the inquisitive modal logic InqML⇒, and as a consequence we obtain the decidability of InqML⇒.

We consider a modal logic that can formalise statements about uncertainty and beliefs such as I think that my wallet is in the drawer rather than elsewhere or I am confused whether my appointment is on Monday or Tuesday.To do that, we expand Gdel modal logic KG with the involutive negation defined as v( , w) = 1v( , w).We provide semantics with the finite model property for our new logic that we call KG inv and show its equivalence to the standard semantics over [0, 1]-valued Kripke models.Namely, we show that is valid in the standard semantics of KG inv iff it is valid in the new semantics.Using this new semantics, we construct a constraint tableaux calculus for KG inv that allows for an explicit extraction of countermodels from complete open branches and then employ the tableaux calculus to obtain the PSPACE-completeness of the validity in KG inv .

Abstract We introduce the adjacent fragment $\mathcal{AF}$ of first-order logic, obtained by restricting the sequences of variables occurring as arguments in atomic formulas. The adjacent fragment generalizes (after a routine renaming) the two-variable fragment of first-order logic as well as the so-called fluted fragment. We show that the adjacent fragment has the finite model property, and that the satisfiability problem for its $k$-variable sub-fragment is in $(k{-}1)$-$\text{NExpTime}$. Using known results on the fluted fragment, it follows that the satisfiability problem for the whole adjacent fragment is $\text{Tower}$-complete. We additionally consider the effect of the adjacency requirement on the well-known guarded fragment of first-order logic, whose satisfiability problem is $2\text{ExpTime}$-complete. We show that the satisfiability problem for the intersection of the adjacent and guarded adjacent fragments remains $2\text{ExpTime}$-hard. Finally, we show that any relaxation of the adjacency condition on the allowed order of variables in argument sequences yields a logic whose satisfiability and finite satisfiability problems are undecidable.

Abstract We use high girth, high chromatic number hypergraphs to show that there are finite models of the equational theory of the semiring of non‐negative integers whose equational theory has no finite axiomatisation, and show this also holds if factorial, fixed base exponentiation and operations for binomial coefficients are adjoined. We also derive the decidability of the equational logical entailment operator for antecedents true on by way of a form of the finite model property. Two appendices contain additional basic development of combinatorial operations. Amongst the observations are an eventual dominance well‐ordering of combinatorial functions and consequent representation of the ordinal in terms of factorial functions; the equivalence of the equational logic of combinatorial algebra over the natural numbers and over the positive reals; and a candidate list of elementary axioms.

A classic result in modal logic, known as the Blok Dichotomy Theorem, states that the degree of incompleteness of a normal extension of the basic modal logic [Formula: see text] is [Formula: see text] or [Formula: see text]. It is a long-standing open problem whether Blok Dichotomy holds for normal extensions of other prominent modal logics (such as [Formula: see text] or [Formula: see text]) or for extensions of the intuitionistic propositional calculus [Formula: see text] (see Problem 10.5 of Chagrov and Zakharyaschev’s “Modal Logic”). In this paper, we introduce the notion of the degree of finite model property (fmp), which is a natural variation of the degree of incompleteness. It is a consequence of the Blok Dichotomy Theorem that the degree of fmp of a normal extension of [Formula: see text] remains [Formula: see text] or [Formula: see text]. In contrast, our main result establishes the following Antidichotomy Theorem for the degree of fmp for extensions of [Formula: see text]: each nonzero cardinal [Formula: see text] such that [Formula: see text] or [Formula: see text] is realized as the degree of fmp of some extension of [Formula: see text]. We then use the Blok–Esakia theorem to establish the same Antidichotomy Theorem for normal extensions of [Formula: see text] and [Formula: see text]. This provides a solution of the reformulation of Problem. 10.5 of Chagrov and Zakharyaschev’ “Modal Logic” for the degree of fmp.

Layered algebras are introduced and used to express layered graphs. Layered graphs are considered to be a highly effective abstract tool to manage the difficulty in conceptualizing and reasoning regarding complex systems related to coding in email exchange and access control in security. In the present paper, we study the varieties of several classes of lattice-based layer algebras and show that all these varieties have decidable equational theory via a finite model property.

Particular reasoning enables the deductive proof of existential properties, such as the satisfiability/consistency of a set of formulas. In this work, we consider particular reasoning in the context of a theory of a given logic. The logic is presented by a semantic constraint specification. From this specification, we induce a particular calculus for the logic at hand. In this calculus we define what is a particular derivation in the context of a theory and show how to extract a model of the theory that satisfies the assertions within the derivation. We demonstrate that the induced particular calculus is both sound and complete with regard to the intended semantics. Our results are applicable to logics with a strong finite model property, including classical, intuitionistic, certain modal logics, and Nelson’s N4 logic, among others.

The uniform one-dimensional fragment of first-order logic was introduced a few years ago as a generalization of the two-variable fragment to contexts involving relations of arity greater than two. Quantifiers in this logic are used in blocks, each block consisting only of existential quantifiers or only of universal quantifiers. In this paper we consider the possibility of mixing both types of quantifiers in blocks. We show the finite (exponential) model property and NExpTime-completeness of the satisfiability problem for two restrictions of the resulting formalism: in the first we require that every block of quantifiers is either purely universal or ends with the existential quantifier, in the second we restrict the number of variables to three; in both equality is not allowed. We also extend the second variation to a rich subfragment of the three-variable fragment (without equality) that still has the finite model property and decidable, NExpTime-complete satisfiability.Comment: arXiv admin note: text overlap with arXiv:2310.00994

We explore an inquisitive modal logic designed to reason about neighborhood models. This logic is based on an inquisitive strict conditional operator Rrightarrow , which quantifies over neighborhoods, and which can be applied to both statements and questions. In terms of this operator we also define two unary modalities boxplus and , which act respectively as a universal and existential quantifier over neighborhoods. We prove that the expressive power of this logic matches the natural notion of bisimilarity in neighborhood models. We show that certain fragments of the language are invariant under certain modifications of the set of neighborhoods, and use this to show that Rrightarrow is not definable from boxplus and , and that questions embedded under Rrightarrow are indispensable. We provide a sound and complete axiomatization of our logic, both in general and in restriction to some salient frame classes, establish decidability via the finite model property, and discuss the relations between our logic and other modal logics interpreted over neighborhood models.

We study intransitive temporal multi-agent logic with agents’ multi-valuations for letters and formulas. In previous wide accepted research the time and knowledge primarily were modeled by Kripke models with structure looking as simply a single time cluster with multi-relations for agents’ accessibility relations. Here we develop this approach and use Kripke models with linear intransitive time and states represented by arbitrary time clusters for agents accessibility multi-relations. This logic is defined in a semantic way, as a set of formulas, which are true at linear models with multi-valued variables by agents’ and clusters of states. We propose a background for such approach and a technique for computation truth values of formulas. Main result concerns decidability problem. We prove that the resulting logic is decidable and obtain a sort of finite model property.

Abstract It is a classic result of Segerberg and Maksimova that a variety of $\mathsf {S4}$ -algebras is locally finite iff it is of finite depth. Since the logic $\mathsf {MS4}$ (monadic $\mathsf {S4}$ ) axiomatizes the one-variable fragment of $\mathsf {QS4}$ (predicate $\mathsf {S4}$ ), it is natural to try to generalize the Segerberg–Maksimova theorem to this setting. We obtain several results in this direction. Our positive results include the identification of the largest semisimple variety of $\mathsf {MS4}$ -algebras. We prove that the corresponding logic $\mathsf {MS4_S}$ has the finite model property. We show that both $\mathsf {S5}^2$ and $\mathsf {S4}_u$ are proper extensions of $\mathsf {MS4_S}$ , and that a direct generalization of the Segerberg–Maksimova theorem holds for a family of varieties containing the variety of $\mathsf {S4}_u$ -algebras. Our negative results include a translation of varieties of $\mathsf {S5}_2$ -algebras into varieties of $\mathsf {MS4_S}$ -algebras of depth 2, which preserves and reflects local finiteness. This, in particular, shows that the problem of characterizing locally finite varieties of $\mathsf {MS4}$ -algebras (even of $\mathsf {MS4_S}$ -algebras) is at least as hard as that of characterizing locally finite varieties of $\mathsf {S5}_2$ -algebras—a problem that remains wide open.

In this work, we study Hilbert-style proof systems for logics based on the data-aware language CoreDataXPath ( ↓ ) where the comparison relation between nodes is not necessarily an equivalence relation. We give a sound and complete axiomatization of the class of tree-like Kripke frames endowed with a general comparison relation between nodes. Modular extensions of this axiomatization are also discussed, including cases where the comparison relation is reflexive, symmetric, transitive and an equivalence. A notable highlight that we recover an axiomatization for CoreDataXPath ( ↓ ) over any data-tree when the comparison is by ‘equal data’. Moreover, we prove that all these systems are decidable by leveraging the finite model property of their corresponding frame classes.

Gödel–Dummett linear temporal logic

A sequent calculus S for the variety tqBa of all topological quasi-Boolean algebras is established. Using a construction of syntactic finite algebraic model, the finite model property of S is shown, and thus the decidability of S is obtained. We also introduce two non-distributive variants of topological quasi-Boolean algebras. For the variety TDM 5 of all topological De Morgan lattices with the axiom 5, we establish a sequent calculus S 5 and prove that the cut elimination holds for it. Consequently the decidability of S 5 is established. Furthermore, this proof-theoretic method is applied to some subvarieties of TDM 5 .

This paper examines the logic of conditional obligation, which originates from the works of Hansson, Lewis, and others. Some weakened forms of transitivity of the betterness relation are studied. These are quasi-transitivity, Suzumura consistency, acyclicity and the interval order condition. The first three do not change the logic. The axiomatic system is the same whether or not they are introduced. This holds true under a rule of interpretation in terms of maximality and strong maximality. The interval order condition gives rise to a new axiom. Depending on the rule of interpretation, this one changes. With the rule of maximality, one obtains the principle known as disjunctive rationality. With the rule of strong maximality, one obtains the Spohn axiom (also known as the principle of rational monotony, or Lewis’ axiom CV). A completeness theorem further substantiates these observations. For interval order, this yields the finite model property and decidability of the calculus.

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.

In this article, we extend the research programme in algebraic proof theory from axiomatic extensions of the full Lambek calculus to logics algebraically captured by certain varieties of normal lattice expansions (normal LE-logics). Specifically, we generalize the residuated frames in Reference [ 34 ] to arbitrary signatures of normal lattice expansions (LE). Such a generalization provides a valuable tool for proving important properties of LE-logics in full uniformity. We prove semantic cut elimination for the display calculi \(\mathrm{D.LE}\) associated with the basic normal LE-logics and their axiomatic extensions with analytic inductive axioms. We also prove the finite model property (FMP) for each such calculus \(\mathrm{D.LE}\) , as well as for its extensions with analytic structural rules satisfying certain additional properties.

Setting the basic rules of inference is fundamental to logic. The most general variant of possible inference rules are admissible inference rules:in logic L, a rule of inference is admissible if the set of theorems L is closed with respect to this rule. The study of admissible inference rules was stimulated by the formulation of problems about decidability by admissibility (Friedman) and the presence of a finite basis of admissible rules (Kuznetsov) in Int logic. In the early 2000s, for most basic non-classical logics and some tabular logics, the Fridman-Kuznetsov problem was solved by describing an explicit basis for admissible rules. The next stage in the study of admissible inference rules for non-classical logics can be considered the concept of a globally admissible inference rule. Globally admissible rules in the logic L are those inference rules that are admissible simultaneously in all (with finite model property) extensions of the given logic. Such rules develop and generalize the concept of an admissible inference rule. The presented work is devoted to the study of bases for globally admissible rules of logic S4. An algorithm for constructing a set of inference rules in a reduced form was described, forming the basis for globally admissible inference rules in S4 logic.