An Abstract Interpretation Perspective on Linear vs. Branching Time

It is known that the branching time language ACTL and the linear time language ∀LTL of universally quantified formulae of LTL have incomparable expressive powers, i.e., Sem(ACTL) and Sem(∀LTL) are incomparable sets. Within a standard abstract interpretation framework, ACTL can be viewed as an abstract interpretation LTL∀ of LTL where the universal path quantifier ∀ abstracts each linear temporal operator of LTL to a corresponding branching state temporal operator of ACTL. In abstract interpretation terms, it turns out that the universal path quantifier abstraction of LTL is incomplete. In this paper we reason on a generic abstraction α over a domain A of a generic linear time language L. This approach induces both a language αL of α-abstracted formulae of L and an abstract language L whose operators are the best correct abstractions in A of the linear operators of L. When the abstraction α is complete for the operators in L it turns out that αL and L have the same expressive power, so that trace-based model checking of αL can be reduced with no lack of precision to Abased model checking of L. This abstract interpretation-based approach allows to compare temporal languages at different levels of abstraction and to view the standard linear vs. branching time comparison as a particular instance.