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Branchingtime Model Checking of Onecounter Processes
 In Proc. of STACS, volume 5 of LIPIcs
, 2010
"... Abstract. Onecounter 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 µcalculus for this problem [20]. First, we ana ..."
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Abstract. Onecounter 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 µcalculus for this problem [20]. 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. In particular, model checking fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This generalizes a corresponding result from [12] for the expression complexity of CTL’s fragment EF. Second, we prove that already over some fixed OCP, CTL model checking is PSPACEhard, i.e., expression complexity is PSPACEhard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACEhard, i.e., data complexity is PSPACEhard as well. 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 logspaceuniform NC 1 [8] and (ii) PSPACE is AC 0serializable [14]. We demonstrate that our approach can be used to obtain further results. We show that modelchecking CTL’s fragment EF over OCPs is hard for P NP, thus establishing a matching lower bound and answering an open question from [12]. We moreover show that the following problem is hard for PSPACE: Given a onecounter 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 shown in [5]. 1
BranchingTime Model Checking of Parametric OneCounter Automata
"... Abstract. We study the computational complexity of model checking EF logic and modal logic on parametric onecounter automata (POCA). A POCA is a onecounter automaton whose counter updates are either integer values encoded in binary or integervalued parameters. Given a formula and a configuration o ..."
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Abstract. We study the computational complexity of model checking EF logic and modal logic on parametric onecounter automata (POCA). A POCA is a onecounter automaton whose counter updates are either integer values encoded in binary or integervalued parameters. Given a formula and a configuration of a POCA, the modelchecking problem asks whether the formula is true in this configuration for all possible valuations of the parameters. We show that this problem is undecidable for EF logic via reduction from Hilbert’s tenth problem, however for modal logic we prove PSPACEcompleteness. Obtaining the PSPACE upper bound involves analysing systems of linear Diophantine inequalities of exponential size that admit solutions of polynomial size. Finally, we show that model checking EF logic on POCA without parameters is PSPACEcomplete. 1
When ModelChecking Freeze LTL over Counter Machines Becomes Decidable
, 2010
"... We study the decidability status of modelchecking freeze LTL over various subclasses of counter machines for which the reachability problem is known to be decidable (reversalbounded counter machines, vector additions systems with states, flat counter machines, onecounter machines). In freeze LTL, ..."
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We study the decidability status of modelchecking freeze LTL over various subclasses of counter machines for which the reachability problem is known to be decidable (reversalbounded counter machines, vector additions systems with states, flat counter machines, onecounter machines). In freeze LTL, a register can store a counter value and at some future position an equality test can be done between a register and a counter value. Herein, we complete an earlier work started on onecounter machines by considering other subclasses of counter machines, and especially the class of reversalbounded counter machines. This gives us the opportuniy to provide a systematic classification that distinguishes determinism vs. nondeterminism and we consider subclasses of formulae by restricting the set of atomic formulae or/and the polarity of the occurrences of the freeze operators, leading to the flat fragment.
Taming past LTL and flat counter systems
, 2012
"... Abstract. Reachability and LTL modelchecking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in NP, the best known complexity upper bound for the latter problem is made of a tower of several exponentials. Herein, we show that the problem ..."
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Abstract. Reachability and LTL modelchecking problems for flat counter systems are known to be decidable but whereas the reachability problem can be shown in NP, the best known complexity upper bound for the latter problem is made of a tower of several exponentials. Herein, we show that the problem is only NPcomplete even if LTL admits pasttime operators and arithmetical constraints on counters. Actually, the NP upper bound is shown by adequately combining a new stuttering theorem for Past LTL and the property of small integer solutions for quantifierfree Presburger formulae. Other complexity results are proved, for instance for restricted classes of flat counter systems. 1
Reachability in Register Machines with Polynomial Updates
"... Abstract. This paper introduces a class of register machines whose registers can be updated by polynomial functions when a transition is taken, and the domain of the registers can be constrained by linear constraints. This model strictly generalises a variety of known formalisms such as various clas ..."
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Abstract. This paper introduces a class of register machines whose registers can be updated by polynomial functions when a transition is taken, and the domain of the registers can be constrained by linear constraints. This model strictly generalises a variety of known formalisms such as various classes of Vector Addition Systems with States. Our main result is that reachability in our class is PSPACEcomplete when restricted to one register. We moreover give a classification of the complexity of reachability according to the type of polynomials allowed and the geometry induced by the rangeconstraining formula. 1