Products Of Modal Logics Research Articles

On Strictly Positive Fragments of Modal Logics with Confluence

We axiomatize strictly positive fragments of modal logics with the confluence axiom. We consider unimodal logics such as K.2, D.2, D4.2 and S4.2 with unimodal confluence ⋄□p→□⋄p as well as the products of modal logics in the set K,D,T,D4,S4, which contain bimodal confluence ⋄1□2p→□2⋄1p. We show that the impact of the unimodal confluence axiom on the axiomatisation of strictly positive fragments is rather weak. In the presence of ⊤→⋄⊤, it simply disappears and does not contribute to the axiomatisation. Without ⊤→⋄⊤ it gives rise to a weaker formula ⋄⊤→⋄⋄⊤. On the other hand, bimodal confluence gives rise to more complicated formulas such as ⋄1p∧⋄2n⊤→⋄1(p∧⋄2n⊤) (which are superfluous in a product if the corresponding factor contains ⊤→⋄⊤). We also show that bimodal confluence cannot be captured by any finite set of strictly positive implications.

Matching Topological and Frame Products of Modal Logics

The simplest combination of unimodal logics $${\mathrm{L}_1 \rm and \mathrm{L}_2}$$L1andL2 into a bimodal logic is their fusion, $${\mathrm{L}_1 \otimes \mathrm{L}_2}$$L1?L2, axiomatized by the theorems of $${\mathrm{L}_1 \rm for \square_1 \rm and of \mathrm{L}_2 \rm for \square_{2}}$$L1for?1andofL2for?2. Shehtman introduced combinations that are not only bimodal, but two-dimensional: he defined 2-d Cartesian products of 1-d Kripke frames, using these Cartesian products to define the frame product$${\mathrm{L}_1 \times \mathrm{L}_2 \rm of \mathrm{L}_1 \rm and \mathrm{L}_2}$$L1×L2ofL1andL2. Van Benthem, Bezhanishvili, ten Cate and Sarenac generalized Shehtman's idea and introduced the topological product$${\mathrm{L}_1 \times_{t}\mathrm{L}_2}$$L1×tL2, using Cartesian products of topological spaces rather than of Kripke frames. Frame products have been extensively studied, but much less is known about topological products. The goal of the current paper is to give necessary and sufficient conditions for the topological product to match the frame product, for Kripke complete extensions of $${\mathrm{S}4: \mathrm{L}_1 \times_t \mathrm{L}_2 = \mathrm{L}_1 \times \mathrm{L}_2 \rm iff \mathrm{L}_1 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_2 \supsetneq \mathrm{S}5 \rm or \mathrm{L}_1, \mathrm{L}_2 = \mathrm{S}5}$$S4:L1×tL2=L1×L2iffL1?S5orL2?S5orL1,L2=S5.

Products of modal logics and tensor products of modal algebras

One of natural combinations of Kripke complete modal logics is the product, an operation that has been extensively investigated over the last 15 years. In this paper we consider its analogue for arbitrary modal logics: to this end, we use product-like constructions on general frames and modal algebras. This operation was first introduced by Y. Hasimoto in 2000; however, his paper remained unnoticed until recently. In the present paper we quote some important Hasimoto's results, and reconstruct the product operation in an algebraic setting: the Boolean part of the resulting modal algebra is exactly the tensor product of original algebras (regarded as Boolean rings). Also, we propose a filtration technique for Kripke models based on tensor products and obtain some decidability results.

A study on multi-dimensional products of graphs and hybrid logics

In this work, we address some issues related to products of graphs and products of modal logics. Our main contribution is the presentation of a necessary and sufficient condition for a countable and connected graph to be a product, using a property called intransitivity. We then proceed to describe this property in a logical language. First, we show that intransitivity is not modally definable and also that no necessary and sufficient condition for a graph to be a product can be modally definable. Then, we exhibit a formula in a hybrid language that describes intransitivity. With this, we get a logical characterization of products of graphs of arbitrary dimensions. We then use this characterization to obtain two other interesting results. First, we determine that it is possible to test in polynomial time, using a model-checking algorithm, whether a finite connected graph is a product. This test has cubic complexity in the size of the graph and quadratic complexity in its number of dimensions. Finally, we use this characterization of countable connected products to provide sound and complete axiomatic systems for a large class of products of modal logics. This class contains the logics defined by product frames obtained from Kripke frames that satisfy connectivity, transitivity and symmetry plus any additional property that can be defined by a pure hybrid formula. Most sound and complete axiomatic systems presented in the literature are for products of a pair of modal logics, while we are able, using hybrid logics, to provide sound and complete axiomatizations for many products of arbitrary dimensions.

Axiomatization and completeness of lexicographic products of modal logics

This paper sets out a new way of combining Kripke-complete modal logics: lexicographic product. It discusses some basic properties of the lexicographic product construction and proves axiomatization/completeness results.

Axiomatization of modal logic squares with distinguished diagonal

Modal logics of squared Kripke frames with distinguished diagonal are considered. It is shown that many such logics, unlike ordinary two-dimensional products, cannot be axiomatized by formulas with finitely many variables. The method resembles that used to obtain a similar result for ≥ 3-dimensional products of modal logics. The proof uses, in particular, generalized Sahlquist formulas.

Products of modal logics, part 3: Products of modal and temporal logics .

In this paper we improve the results of [2] by proving the product f.m.p. for the product of minimal n-modal and minimal n-temporal logic. For this case we modify the finite depth method introduced in [1]. The main result is applied to identify new fragments of classical first-order logic and of the equational theory of relation algebras, that are decidable and have the finite model property.

An elementary construction for a non-elementary procedure

We consider the problem of the product finite model property for binary products of modal logics. First we give a new proof for the product finite model property of the logic of products of Kripke frames, a result due to Shehtman. Then we modify the proof to obtain the same result for logics of products of Kripke frames satisfying any combination of seriality, reflexivity and symmetry. We do not consider the transitivity condition in isolation because it leads to infinity axioms when taking products.

Products of modal logics, part 1

Products of modal logics, part 1