Branching-time model checking of one-counter processes

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

Branching-time Model Checking of One-counter Processes

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 $CTL$\u27s 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. For the latter, 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 answer further open questions

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

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

Model-checking branching-time properties of probabilistic automata and probabilistic one-counter automata

This paper studies the problem of model-checking of probabilistic automaton and probabilistic one-counter automata against probabilistic branching-time temporal logics (PCTL and PCTL$^*$). We show that it is undecidable for these problems. We first show, by reducing to emptiness problem of probabilistic automata, that the model-checking of probabilistic finite automata against branching-time temporal logics are undecidable. And then, for each probabilistic automata, by constructing a probabilistic one-counter automaton with the same behavior as questioned probabilistic automata the undecidability of model-checking problems against branching-time temporal logics are derived, herein.Comment: Comments are welcom

Leader Election in Anonymous Rings: Franklin Goes Probabilistic

We present a probabilistic leader election algorithm for anonymous, bidirectional, asynchronous rings. It is based on an algorithm from Franklin, augmented with random identity selection, hop counters to detect identity clashes, and round numbers modulo 2. As a result, the algorithm is finite-state, so that various model checking techniques can be employed to verify its correctness, that is, eventually a unique leader is elected with probability one. We also sketch a formal correctness proof of the algorithm for rings with arbitrary size

Parameterized Model Checking of Token-Passing Systems

We revisit the parameterized model checking problem for token-passing systems and specifications in indexed $\textsf{CTL}^\ast \backslash \textsf{X}$. Emerson and Namjoshi (1995, 2003) have shown that parameterized model checking of indexed $\textsf{CTL}^\ast \backslash \textsf{X}$ in uni-directional token rings can be reduced to checking rings up to some \emph{cutoff} size. Clarke et al. (2004) have shown a similar result for general topologies and indexed $\textsf{LTL} \backslash \textsf{X}$, provided processes cannot choose the directions for sending or receiving the token. We unify and substantially extend these results by systematically exploring fragments of indexed $\textsf{CTL}^\ast \backslash \textsf{X}$ with respect to general topologies. For each fragment we establish whether a cutoff exists, and for some concrete topologies, such as rings, cliques and stars, we infer small cutoffs. Finally, we show that the problem becomes undecidable, and thus no cutoffs exist, if processes are allowed to choose the directions in which they send or from which they receive the token.Comment: We had to remove an appendix until the proofs and notations there is cleare

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)