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Refining the Process Rewrite Systems Hierarchy via Ground Tree Rewrite Systems

Stefan Göller, Anthony Widjaja Lin

2014
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ACM Transactions on Computational Logic
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In his seminal paper, R. Mayr introduced the well-known Process Rewrite Systems (PRS) hierarchy, which contains many well-studied classes of infinite systems including pushdown systems, Petri nets and PA-processes. A seperate development in the term rewriting community introduced the notion of Ground Tree Rewrite Systems (GTRS), which is a model that strictly extends pushdown systems while still enjoying desirable decidable properties. There have been striking similarities between the
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... on problems that have been shown decidable (and undecidable) over GTRS and over models in the PRS hierarchy such as PA and PAD processes. It is open to what extent PRS and GTRS are connected in terms of their expressive power. In this paper we pinpoint the exact connection between GTRS and models in the PRS hierarchy in terms of their expressive power with respect to strong, weak, and branching bisimulation. Among others, this connection allows us to give new insights into the decidability results for subclasses of PRS, e.g., simpler proofs of known decidability results of verifications problems on PAD. Rewrite Systems (PRS), which generalize PDS, PA-processes, and Petri nets, (2) PADprocesses, which unify PDS and PA-processes, and (3) PAN-processes, which unify both PA-processes and Petri nets. Mayr showed that the hierarchy is strict with respect to strong bisimulation. Despite of its expressive power PRS is not Turing-powerful since reachability is still decidable for this class. Before the PRS hierarchy was introduced, another class of infinite-state systems called Ground Tree/Term Rewrite Systems (GTRS) already emerged in the term rewriting community as a class with nice decidability properties. While extending the expressive power of PDS, GTRS still enjoys decidability of reachability (e.g. [8, 11] ), recurrent reachability [15], model checking first-order logic with reachability [12] , and model checking the fragments LTL det and LTL(F s , G s ) of LTL [24, 23] . Due to the tree structures that GTRS use in their rewrite rules, GTRS can be used to model concurrent systems with both unbounded parallelism (a new thread may be spawned at any given time) and unbounded recursions (each thread may behave as a pushdown system). When comparing the definitions of PRS (and subclasses thereof) and GTRS, one cannot help but notice their similarity. Moreover, there is a striking similarity between the problems that are decidable (and undecidable) over subclasses of PRS like PA/PADprocesses and GTRS. For example, reachability, EF model checking, and LTL(F s , G s ) and LTL det model checking are decidable for both 15, 18, 19, 23, 24 ]. Furthermore, model checking general LTL properties is undecidable for both PA-processes and GTRS [7, 24] . Despite these, the precise connection between the PRS hierarchy and GTRS is currently still open. Contributions: In this paper, we pinpoint the precise connection between the expressive powers of GTRS and models inside the PRS hierarchy with respect to strong, branching, and weak bisimulation. Bisimulations are well-known and important notions of semantic equivalences on transition systems. Among others, most properties of interests in verification (e.g. those expressible in standard modal/temporal logics) cannot distinguish two transition systems that are bisimilar. Strong/weak bisimulations are historically the most important notions of bisimulations on transition systems in verification [20] . Weak bisimulations extend strong bisimulations by distinguishing observable and non-observable (i.e. τ ) actions, and only requiring the observable behavior of two systems to agree. In this sense, weak bisimulation is a coarser notion than strong bisimulation. Branching bisimulation [25] is a notion of semantic equivalence that is strictly coarser than strong bisimulation but is strictly finer than weak bisimulation. It refines weak bisimulation equivalence by preserving the branching structure of two processes even in the presence of unobservable τ -actions; it is required that all intermediate states that are passed through during τ -transitions are related. Our results are summarized in the middle and right diagrams in Figure 1 . Our first main result is that the expressive power of GTRS with respect to branching and weak bisimulation is strictly in between PAD and PRS but incomparable with PAN. This result allows us to transfer many decidability/complexity results of model checking problems over GTRS to PA and PAD-processes. In particular, it gives a simple proof of the decidability of model checking the logic EF over PAD [19] , and decidability (with good complexity upper bounds) of model checking the common fragments LTL det and LTL(F s , G s ) of LTL over PAD (this decidability result was initially given in [7] with-

doi:10.1145/2629679
fatcat:myhh5aypjvbq3g2dy55k2onxsy