On decidability of LTL model checking for process rewrite systems

Laura Bozzelli, Mojmír Křetínský, Vojtěch Řehák, Jan Strejček
2008 Acta Informatica  
We establish a decidability boundary of the model checking problem for infinite-state systems defined by Process Rewrite Systems (PRS) or weakly extended Process Rewrite Systems (wPRS), and properties described by basic fragments of action-based Linear Temporal Logic (LTL). It is known that the problem for general LTL properties is decidable for Petri nets and for pushdown processes, while it is undecidable for PA processes. As our main result, we show that the problem is decidable for wPRS if
more » ... e consider properties defined by formulae with only modalities strict eventually and strict always. Moreover, we show that the problem remains undecidable for PA processes even with respect to the LTL fragment with the only modality until or the fragment with modalities next and infinitely often. On Decidability of LTL Model Checking for Process Rewrite Systems 249 during any computation. We have proved that the reachability problem remains decidable for wPRS [KŘS04a] and that the problem called reachability Hennessy-Milner property (whether there is a reachable state satisfying a given Hennessy-Milner formula) is decidable for wPRS as well [KŘS05] . The hierarchy of all PRS and wPRS classes is depicted in Figure 1 . Concerning the model checking problem, a broad overview of (un)decidability results for subclasses of PRS and various temporal logics can be found in [May98] . Here we focus exclusively on (future) Linear Temporal Logic (LTL). It is known that LTL model checking of PDA is EXPTIME-complete [BEM97]. LTL model checking of PN is also decidable, but at least as hard as the reachability problem for PN [Esp94] (the reachability problem is EXPSPACE-hard [May84, Lip76] and no primitive recursive upper bound is known). If we consider only infinite runs, then the problem for PN is EXPSPACE-complete [Hab97, May98]. Conversely, LTL model checking is undecidable for all classes subsuming PA [BH96, May98]. So far, there are only two positive results for these classes. Bouajjani and Habermehl [BH96] have identified a fragment called simple PLTL 2 for which model checking of infinite runs is decidable for PA (strictly speaking, simple PLTL 2 is not a fragment of LTL as it can express also some non-regular properties, while LTL cannot). Only recently, we have demonstrated that model checking of infinite runs is decidable for PRS and the fragment of LTL capturing exactly fairness properties [Boz05]. Our contribution: This paper completely locates the decidability boundary of the model checking problem for all subclasses of PRS (and wPRS) and all basic LTL fragments, where a basic LTL fragment is a set of all formulae containing only a given subset of standard modalities. The boundary is depicted in Figure 2 . To locate the boundary, we show the following results.
doi:10.1007/s00236-008-0082-3 fatcat:kmgaunp5h5cq3ai7dpcbe3uaty