341,132 research outputs found

    Path Checking for MTL and TPTL over Data Words

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    Metric temporal logic (MTL) and timed propositional temporal logic (TPTL) are quantitative extensions of linear temporal logic, which are prominent and widely used in the verification of real-timed systems. It was recently shown that the path checking problem for MTL, when evaluated over finite timed words, is in the parallel complexity class NC. In this paper, we derive precise complexity results for the path-checking problem for MTL and TPTL when evaluated over infinite data words over the non-negative integers. Such words may be seen as the behaviours of one-counter machines. For this setting, we give a complete analysis of the complexity of the path-checking problem depending on the number of register variables and the encoding of constraint numbers (unary or binary). As the two main results, we prove that the path-checking problem for MTL is P-complete, whereas the path-checking problem for TPTL is PSPACE-complete. The results yield the precise complexity of model checking deterministic one-counter machines against formulae of MTL and TPTL

    Model checking Branching-Time Properties of Multi-Pushdown Systems is Hard

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    We address the model checking problem for shared memory concurrent programs modeled as multi-pushdown systems. We consider here boolean programs with a finite number of threads and recursive procedures. It is well-known that the model checking problem is undecidable for this class of programs. In this paper, we investigate the decidability and the complexity of this problem under the assumption of bounded context-switching defined by Qadeer and Rehof, and of phase-boundedness proposed by La Torre et al. On the model checking of such systems against temporal logics and in particular branching time logics such as the modal μ\mu-calculus or CTL has received little attention. It is known that parity games, which are closely related to the modal μ\mu-calculus, are decidable for the class of bounded-phase systems (and hence for bounded-context switching as well), but with non-elementary complexity (Seth). A natural question is whether this high complexity is inevitable and what are the ways to get around it. This paper addresses these questions and unfortunately, and somewhat surprisingly, it shows that branching model checking for MPDSs is inherently an hard problem with no easy solution. We show that parity games on MPDS under phase-bounding restriction is non-elementary. Our main result shows that model checking a kk context bounded MPDS against a simple fragment of CTL, consisting of formulas that whose temporal operators come from the set {\EF, \EX}, has a non-elementary lower bound

    Model-checking Quantitative Alternating-time Temporal Logic on One-counter Game Models

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    We consider quantitative extensions of the alternating-time temporal logics ATL/ATLs called quantitative alternating-time temporal logics (QATL/QATLs) in which the value of a counter can be compared to constants using equality, inequality and modulo constraints. We interpret these logics in one-counter game models which are infinite duration games played on finite control graphs where each transition can increase or decrease the value of an unbounded counter. That is, the state-space of these games are, generally, infinite. We consider the model-checking problem of the logics QATL and QATLs on one-counter game models with VASS semantics for which we develop algorithms and provide matching lower bounds. Our algorithms are based on reductions of the model-checking problems to model-checking games. This approach makes it quite simple for us to deal with extensions of the logical languages as well as the infinite state spaces. The framework generalizes on one hand qualitative problems such as ATL/ATLs model-checking of finite-state systems, model-checking of the branching-time temporal logics CTL and CTLs on one-counter processes and the realizability problem of LTL specifications. On the other hand the model-checking problem for QATL/QATLs generalizes quantitative problems such as the fixed-initial credit problem for energy games (in the case of QATL) and energy parity games (in the case of QATLs). Our results are positive as we show that the generalizations are not too costly with respect to complexity. As a byproduct we obtain new results on the complexity of model-checking CTLs in one-counter processes and show that deciding the winner in one-counter games with LTL objectives is 2ExpSpace-complete.Comment: 22 pages, 12 figure

    The parameterized space complexity of model-checking bounded variable first-order logic

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    The parameterized model-checking problem for a class of first-order sentences (queries) asks to decide whether a given sentence from the class holds true in a given relational structure (database); the parameter is the length of the sentence. We study the parameterized space complexity of the model-checking problem for queries with a bounded number of variables. For each bound on the quantifier alternation rank the problem becomes complete for the corresponding level of what we call the tree hierarchy, a hierarchy of parameterized complexity classes defined via space bounded alternating machines between parameterized logarithmic space and fixed-parameter tractable time. We observe that a parameterized logarithmic space model-checker for existential bounded variable queries would allow to improve Savitch's classical simulation of nondeterministic logarithmic space in deterministic space O(log2n)O(\log^2n). Further, we define a highly space efficient model-checker for queries with a bounded number of variables and bounded quantifier alternation rank. We study its optimality under the assumption that Savitch's Theorem is optimal

    On the Complexity of ATL and ATL* Module Checking

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    Module checking has been introduced in late 1990s to verify open systems, i.e., systems whose behavior depends on the continuous interaction with the environment. Classically, module checking has been investigated with respect to specifications given as CTL and CTL* formulas. Recently, it has been shown that CTL (resp., CTL*) module checking offers a distinctly different perspective from the better-known problem of ATL (resp., ATL*) model checking. In particular, ATL (resp., ATL*) module checking strictly enhances the expressiveness of both CTL (resp., CTL*) module checking and ATL (resp. ATL*) model checking. In this paper, we provide asymptotically optimal bounds on the computational cost of module checking against ATL and ATL*, whose upper bounds are based on an automata-theoretic approach. We show that module-checking for ATL is EXPTIME-complete, which is the same complexity of module checking against CTL. On the other hand, ATL* module checking turns out to be 3EXPTIME-complete, hence exponentially harder than CTL* module checking.Comment: In Proceedings GandALF 2017, arXiv:1709.0176

    Branching-time model checking of one-counter processes

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    One-counter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from the modal mu-calculus for this problem. First, we analyze the periodic behaviour of CTL over OCPs and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. Thus, model checking fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This generalizes a result of the first author, Mayr, and To for the expression complexity of CTL's fragment EF. Second, we prove that already over some fixed OCP, CTL model checking is PSPACE-hard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACE-hard. To obtain the latter result, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspace-uniform NC^1 and (ii) PSPACE is AC^0-serializable. We demonstrate that our approach can be used to obtain further results. We show that model-checking CTL's fragment EF over OCPs is hard for P^NP, thus establishing a matching lower bound and answering an open question of the first author, Mayr, and To. We moreover show that the following problem is hard for PSPACE: Given a one-counter Markov decision process, a set of target states with counter value zero each, and an initial state, to decide whether the probability that the initial state will eventually reach one of the target states is arbitrarily close to 1. This improves a previously known lower bound for every level of the Boolean hierarchy by Brazdil et al

    Fixed-parameter tractability, definability, and model checking

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    In this article, we study parameterized complexity theory from the perspective of logic, or more specifically, descriptive complexity theory. We propose to consider parameterized model-checking problems for various fragments of first-order logic as generic parameterized problems and show how this approach can be useful in studying both fixed-parameter tractability and intractability. For example, we establish the equivalence between the model-checking for existential first-order logic, the homomorphism problem for relational structures, and the substructure isomorphism problem. Our main tractability result shows that model-checking for first-order formulas is fixed-parameter tractable when restricted to a class of input structures with an excluded minor. On the intractability side, for every t >= 0 we prove an equivalence between model-checking for first-order formulas with t quantifier alternations and the parameterized halting problem for alternating Turing machines with t alternations. We discuss the close connection between this alternation hierarchy and Downey and Fellows' W-hierarchy. On a more abstract level, we consider two forms of definability, called Fagin definability and slicewise definability, that are appropriate for describing parameterized problems. We give a characterization of the class FPT of all fixed-parameter tractable problems in terms of slicewise definability in finite variable least fixed-point logic, which is reminiscent of the Immerman-Vardi Theorem characterizing the class PTIME in terms of definability in least fixed-point logic.Comment: To appear in SIAM Journal on Computin
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