Proof-search and countermodel generation for non-normal modal logics: The theorem prover PRONOM

In this work we present PRONOM, a theorem prover and countermodel generator for non-normal modal logics. PRONOM implements some labelled sequent calculi recently introduced for the basic system E and its extensions with axioms M, N, and C based on bi-neighbourhood semantics. PRONOM is inspired by the methodology of leanTAP and is implemented in Prolog. When a modal formula is valid, then PRONOM computes a proof (a closed tree) in the labelled calculi having a sequent with an empty left-hand side and containing only that formula on the right-hand side as a root, otherwise PRONOM is able to extract a model falsifying it from an open, saturated branch. The paper shows some experimental results, witnessing that the performances of PRONOM are promising.

Tableau-based translation from first-order logic to modal logic

We define a procedure for translating a given first-order formula to an equivalent modal formula, if one exists, by using tableau-based bisimulation invariance test. A previously developed tableau procedure tests bisimulation invariance of a given first-order formula, and therefore tests whether that formula is equivalent to the standard translation of some modal formula. Using a closed tableau as the starting point, we show how an equivalent modal formula can be effectively obtained.

Reasoning About Applicable Law in Private International Law in Logic Programming1

We formalized renvoi in private international law in JURIX 2019 in terms of modal logic fragment. In this demonstration paper, we show an implementation of the formalism by translating modal formula into a logic program.

Disjunctive Normal Form for Multi-Agent Modal Logics Based on Logical Separability

Modal logics are primary formalisms for multi-agent systems but major reasoning tasks in such logics are intractable, which impedes applications of multi-agent modal logics such as automatic planning. One technique of tackling the intractability is to identify a fragment called a normal form of multiagent logics such that it is expressive but tractable for reasoning tasks such as entailment checking, bounded conjunction transformation and forgetting. For instance, DNF of propositional logic is tractable for these reasoning tasks. In this paper, we first introduce a notion of logical separability and then define a novel disjunctive normal form SDNF for the multiagent logic Kn, which overcomes some shortcomings of existing approaches. In particular, we show that every modal formula in Kn can be equivalently casted as a formula in SDNF, major reasoning tasks tractable in propositional DNF are also tractable in SDNF, and moreover, formulas in SDNF enjoy the property of logical separability. To demonstrate the usefulness of our approach, we apply SDNF in multi-agent epistemic planning. Finally, we extend these results to three more complex multi-agent logics Dn, K45n and KD45n.

Modal Operators

Notwithstanding her rejection of quantification over an absolutely comprehensive domain, a relativist about quantifiers may still be tempted to seek other means to generalize. This chapter concerns relativist-friendly modal operators. By modalizing her quantifiers, the relativist has a systematic way to attain absolute generality, which permits her to regiment her view with a single modal formula, and to frame an attractive modal axiomatization of the iterative conception of set. In addition to the immediate cost of admitting the relevant modality into her ideology, however, this approach leads to a hybrid version of relativism, which has some significant commonalities with absolutism about quantifiers.

Chromosomal similarities and differences among three sibling species of the Acalles echinatus group (Coleoptera, Curculionidae, Cryptorhynchinae)

In order to clarify the taxonomic position of three sibling species of weevils from the Acalles echinatus group, A. echinatus, A. fallax and A. petryszaki, cytogenetic relationships are investigated by studying the mitotic and meiotic chromosomes, including the localisation of heterochromatin by C-banding, as well as the localisation of NORs by silver impregnation. These sources of data are congruent and strongly support that the examined species are closely related. All examined species are characterised by a karyotype of the same chromosome number and sex determination system but with different morphology of chromosomes. All the analysed features, such as the centromeric index, relative length, Cbands and NORs, show that the structure of the karyotype of A. echinatus is more similar to that of A. petryszaki, whereas the karyotype of A. fallax is divergent. The higher chromosome number (2n = 30) in relation to the modal formula in Curculionidae (2n = 22) suggests that karyotype evolution in these species could have occurred by centric fissions of metacentric elements leading to acrocentry.

Density of truth in modal logics

International audience The aim of this paper is counting the probability that a random modal formula is a tautology. We examine $\{ \to,\Box \}$ fragment of two modal logics $\mathbf{S5}$ and $\mathbf{S4}$ over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic $\mathbf{S5}$ and this is the key to count the proportion of tautologies of $\mathbf{S5}$ among all formulas. Although the logic $\mathbf{S4}$ does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.

The complexity of the modal predicate logic of “true in every transitive model of ZF”

Robert Solovay [8] investigated the version of the modal sentential calculus one gets by taking “□ϕ” to mean “ϕ is true in every transitive model of Zermelo-Fraenkel set theory (ZF).” Defining an interpretation to be a function * taking formulas of the modal sentential calculus to sentences of the language of set theory that commutes with the Boolean connectives and sets (□ϕ)* equal to the statement that ϕ* is true in every transitive model of ZF, and stipulating that a modal formula ϕ is valid if and only if, for every interpretation *, ϕ* is true in every transitive model of ZF, Solovay obtained a complete and decidable set of axioms.In this paper, we stifle the hope that we might continue Solovay's program by getting an analogous set of axioms for the modal predicate calculus. The set of valid formulas of the modal predicate calculus is not axiomatizable; indeed, it is complete .We also look at a variant notion of validity according to which a formula ϕ counts as valid if and only if, for every interpretation *, ϕ* is true. For this alternative conception of validity, we shall obtain a lower bound of complexity: every set which is in the set of sentences of the language of set theory true in the constructible universe will be 1-reducible to the set of valid modal formulas.

Canonical formulas for K4. Part II: Cofinal subframe logics

This paper is a continuation of Zakharyaschev [25], where the following basic results on modal logics with transitive frames were obtained:• With every finite rooted transitive frame and every set of antichains (which were called closed domains) in two formulas α (, , ⊥) and α(, ) were associated. We called them the canonical and negation free canonical formulas, respectively, and proved the Refutability Criterion characterizing the constitution of their refutation general frames in terms of subreduction (alias partial p-morphism), the cofinality condition and the closed domain condition.• We proved also the Completeness Theorem for the canonical formulas providing us with an algorithm which, given a modal formula φ, returns canonical formulas α(i, i), ⊥), for i = 1,…, n, such thatif φ is negation free then the algorithm instead of α(i, i, ⊥) can use the negation free canonical formulas α(i, i). Thus, every normal modal logic containing K4 can be axiomatized by a set of canonical formulas.In this Part we apply the apparatus of the canonical formulas for establishing a number of results on the decidability, finite model property, elementarity and some other properties of modal logics within the field of K4.Our attention will be focused on the class of logics which can be axiomatized by canonical formulas without closed domains, i.e., on the logics of the formAdapting the terminology of Fine [11], we call them the cofinal subframe logics and denote this class by . As was shown in Part I, almost all standard modal logics are in .