Temporal logic of surjective bounded morphisms between finite linear processes

Abstract

In this paper we study a temporal logic for finite linear structures and a surjective bounded morphism between them. We give a modal axiomatization of such structures and also show that any such structure can be uniquely characterised by a temporal formula, up to an isomorphism. As a main theorem we prove Kripke completeness of the proposed axiomatization. Finite linear structures, i.e., finite sets with a strict linear ordering, naturally arise as representations of a discrete, bounded time flow. Many domains of our everyday practice including time series [2], linear planning [4], [7], scene analysis [1], [9], chain-of-responsibility design pattern in programming [3], [8], etc. involve a finite linear structure to represent a sequence of consecutive steps. In many such scenarios, the processes (F,<) comes with a natural partition ⋃ i∈I Fi of F into convex equivalence classes Fi. We call Fi convex if for each a, b ∈ Fi with a < b and each t with a < t < b, we have t ∈ Fi. In such a case, the index set I naturally “inherits” a linear ordering from (F,<). Mathematically, such a process ( ⋃ i∈I Fi, <) can be represented as two temporal linear structures F and I related by means of a bounded morphism. An example of such a structure comes from the analysis of video data where the linear sequence of image frames is partitioned into intervals (grouped in some way e.g. by homogeneous sound moments, or by each interval representing an episode, or a scene). In the area of computer vision, deep learning (DL) methods usually process a video stream as a black box, without looking into the temporal structure or content [6]. By contrast, we aim to represent a high-level knowledge about frames, scenes and their temporal interrelationships and to develop formal languages capable of reasoning about resulting structures [5]. To flesh out this approach a little more, let us consider a conceptual representation of a movie. The raw video data of the movie can simply be represented as a sequence of frames. On a slightly higher level of conceptualization, the same raw data can be understood as a sequence of scenes, where a scene is a subset of logically related consecutive frames. If one also “remembers” which frame belongs to which scene, the following structure emerges: