Abstract
We revisit decidability results for resource-bounded logics and use decision problems on vector addition systems with states (VASS) in order to establish complexity characterisations of (decidable) model checking problems. We show that the model checking problem for the logic RB±ATL is 2exptime-complete by using recent results on alternating VASS (and in exptime when the number of resources is bounded). Moreover, we establish that the model checking problem for RBTL is expspace-complete. The problem is decidable and of the same complexity for RBTL⁎, proving a new decidability result as a by-product of the approach. When the number of resources is bounded, the problem is in pspace. We also establish that the model checking problem for RB±ATL⁎, the extension of RB±ATL with arbitrary path formulae, is decidable by a reduction to parity games for single-sided VASS (a variant of alternating VASS). Furthermore, we are able to synthesise values for resource parameters. Hence, the paper establishes formal correspondences between model checking problems for resource-bounded logics advocated in the AI literature and decision problems on alternating VASS, paving the way for more applications and cross-fertilizations.
Highlights
Resource-bounded logics [11,10,29,3,2,12] extend alternating-time temporal logic (ATL) [5] by adding transitions that produce and consume resources to the models
As shown in [2], the introduction of implicit counters in the models and the ability to quantify over strategies for a given set of agents can lead to undecidability, or decidability with a very high worst-case upper bound on the complexity of the model checking problem
Temporal logics on vector addition systems with states (VASS) often lead to undecidable model-checking problems, see e.g. [17,18], and this is more common with branching-time temporal logics such as CTL [18], or when the atomic formulae can state properties about the counter values [22]
Summary
Resource-bounded logics [11,10,29,3,2,12] extend alternating-time temporal logic (ATL) [5] by adding transitions that produce and consume resources to the models. We establish formal relationships between model-checking problems for resource-bounded logics and decision problems for alternating VASS ( known as single-sided VASS). We use these relationships to show new results for the decidability and complexity of model-checking resource-bounded logics. The 2EXPTIME lower bound is obtained by a reduction from the state reachability problem for alternating VASS (AVASS) [13], whereas the upper bound is established by a reduction to the state reachability and the termination problems for AVASS (both problems are needed) These results are obtained by using formal relationships between strategies in concurrent game structures and proofs in AVASS, and the key observation is that only asymmetric VASS are needed. We show that the modelchecking problems for RBTL [10] and its extension RBTL∗ are EXPSPACE-complete, and that RBTL restricted to a bounded number of resources is in PSPACE
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