A functional approach for temporal \(\times\) modal logics (Q1403342)

A new approach in the context of the combination of modal and temporal logics is introduced. These logics have shown its usefulness in philosophy, linguistics and computation. The goal of the paper is twofold: on the one hand, definitions are sought for the basic properties of the functions (injectivity, \dots) by means of a multimodal language; on the other hand, the authors claim that the approach has also a practical use in computation (multiagent systems, parallel processes, etc.) The paper introduces a new type of frames (the functional frames) to handle flows of time which are connected by functions, instead of using equivalence relations as occurring with Kamp-frames and \(T\times W\)-frames [see \textit{R. Thomason}, ``Combinations of tense and modality'', in: D. Gabbay and F. Guenthner (eds.), Handbook of philosophical logic. Vol. II: Extensions of classical logic. Dordrecht: Reidel. Synth. Libr. 165, 135-165 (1984; Zbl 0875.03047)]. As a previous step before attempting to solve the problem of definability, the authors introduce an interesting and purely algebraic characterization of the properties of functions. Later, a number of minimal axiomatic systems for temporal \(\times\) modal logics are presented, for instance, there are minimal systems for total functions and for a special class of partial functions which are called uniform domain functions. Some completeness results are presented, the proofs of completeness follow a Henkin style and are straightforward applications of the well-known step by step method [see \textit{J. P. Burgess}, ``Basic tense logic'', ibid., 89-133 (1984; Zbl 0875.03046)]. Also, an incompleteness result is presented: the system corresponding to the total and injective functions.