A product version of dynamic linear time temporal logic

Abstract

We present here a linear time temporal logic which simultaneously extends LTL, the propositional temporal logic of linear time, along two dimensions. Firstly, the until operator is strengthened by indexing it with the regular programs of propositional dynamic logic (PDL). Secondly, the core formulas of the logic are decorated with names of sequential agents drawn from fixed finite set. The resulting logic has a natural semantics in terms of the runs of a distributed program consisting of a finite set of sequential programs that communicate by performing common actions together. We show that our logic, denoted DLTL admits an exponential time decision procedure. We also show that DLTL is expressively equivalent to the so called regular product languages. Roughly speaking, this class of languages is obtained by starting with synchronized products of (ω-)regular languages and closing under boolean operations. We also sketch how the behaviours captured by our temporal logic fit into the framework of labelled partial orders known as Mazurkiewicz traces.