Linear, Branching Time and Joint Closure Semantics for Temporal Logic

Temporal logic can be used to describe processes: their behaviour is characterized by a set of temporal models axiomatized by a temporal theory. Two types of models are most often used for this purpose: linear and branching time models. In this paper a third approach, based on socalled joint closure models, is studied using models which incorporate all possible behaviour in one model. Relations between this approach and the other two are studied. In order to define constructions needed to relate branching time models, appropriate algebraic notions are defined (in a category theoretical manner) and exploited. In particular, the notion of joint closure is used to construct one model subsuming a set of models. Using this universal algebraic construction we show that a set of linear models can be merged to a unique branching time model. Logical properties of the described algebraic constructions are studied. The proposed approach has been successfully aplied to obtain an appropriate semantics for nonmonotonic reasoning processes based on default logic. References are discussed that show the details of these applications.