On the Mosaic Method for Many-Dimensional Modal Logics: A Case Study Combining Tense and Modal Operators

Abstract

Abstract. We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an "orthogonal" S5 -like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system.

Decidability and Complexity via Mosaics of the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders

“…It has also been applied for proving completeness and decidability of temporal logics over linear flows of time. See [5, section 6.4], or [7,16,17,20]. For their use, specialized systems such as temporal logics must be combined with each other.…”

About the Temporal Logic of the Lexicographic Products of Unbounded Dense Linear Orders: A New Study of Its Computability

“…The mosaic method has been applied for proving completeness and decidability of temporal logics over multifarious linear flows of time [9,19,20,22]. The operation of lexicographic product of Kripke frames has been introduced as a variant of the more classical operation of Cartesian product [12,14,17,23].…”