Trace consistency

Two runs of a distributed system can be considered equivalent if they represent diierent interleavings of the same run. Formulas of the propositional temporal logic of linear time (PTL) are said to be trace consistent when they cannot distinguish between equivalent runs. Determining whether a formula is trace consistent is decidable. In this paper, we consider a closely related notion: a PTL formula is said to be inevitable, if for every system run, there is at least one equivalent run (that is, one interleaving of that run) which satisses the formula. We show that determining whether a formula is inevitable is undecid-able. However, for a subclass of formulas termed coherent (propositions in them can be aaected by either of a pair of independent operations but not both), inevitability becomes decidable. 1 A synopsis The Propositional Temporal Logic of Linear Time (PTL) is used to describe the temporal evolution of global states of distributed systems. Reasoning in PTL about safety, liveness and fairness properties of systems is well understood by now. With the development of both automata-theoretic model checking techniques VW86] and theorem proving techniques MP91], automatic or semi-automatic veriication of temporal properties has seen considerable progress. A number of researchers ((GW94], KP92a], V90]) have noted that there is a natural equivalence relation on runs of systems which equates runs obtained by diierent interleavings of the same run. For instance, if a 1 and a 2 happen to be causally independent operations and a run is of the form s 0 a1 !s 1 a2 !s 2 , we would consider another run of the form s 0 a2 !s 0 1 a1 !s 2 to be equivalent to. These constitute two diierent ways of interleaving the concurrent operations a 1 and a 2. If a 1 and a 2 are actions on distinct variables, the sequence of state changes that each variable is subject to is identical in both runs. Often, a system is deened to be truly distributed (or in some contexts, fair) only when it is closed with respect to this equivalence, that is, given any system run, all other interleavings of concurrent operations along that run are also admitted as runs.