Index set expressions can represent temporal logic formulas

Abstract

In Temporal Logic (TL), a well-formed formula is generally formed by applying rules of its syntax finitely many times. However, under some circumstances, although formulas such as ones expressed by index set expressions, are constructed via applying rules of the syntax infinitely many times, they are possibly still well-formed since their equivalent concise syntax formulas can be found. With this motivation, this paper investigates the relationship between formulas specified by index set expressions and concise syntax by means of fixed-point approach. Firstly, we present two kinds of formulas, namely ⋁i∈N0◯iQ and ⋁i∈N0Qi, and prove they are indeed well-formed by proving they are equivalent to formulas ◇Q and Q⁎ respectively. Further, we generalize ⋁i∈N0◯iQ to ⋁i∈N0P(i)∧◯iQ and explore the least and greatest fixed-points of an abstract equation X≡Q∨P∧◯X. Based on these, some well-formed special instances of ⋁i∈N0P(i)∧◯iQ are obtained. Besides, with the index set expression technique, we equivalently represent ‘U’ (strong until) and ‘W’ (weak until) constructs of propositional Linear Temporal Logic (LTL) within Propositional Projection Temporal Logic (PPTL).