Two classical semantical approaches to studying logics which combine time and modality are the T × W-frames and Kamp-frames (see Thomason, 84). In this paper we study a new kind of frame that extends the one introduced in [Burrieza and P. de Guzmán(2002)]. The motivation is twofold: theoretical, i.e., representing properties of the basic theory of functions (definability); and practical, their use in computational applications (considering time-flows as memory of computers connected in a net, each computer with its own clock). Specifically, we present a temporal × modal (labelled) logic, whose semantics are given by indfunctional frames in which accessibility functions are used in order to interconnect time-flows. This way, we can: (i) specify to what time-flow we want to go; (ii) carry out different comparisons among worlds with different time measures, and (iii) define properties of certain kinds of functions (in particular, of total, injective, surjective, constant, increasing and decreasing functions), without the need to resort to second-order theories. In addition, we define a minimal axiomatic system and give the completeness theorem (Henkin-style).
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