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 T. Lin Computer Science , 2015, Abstract: 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.
 Computer Science , 2013, Abstract: A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multi-type branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of {\em termination probabilities}, and computing these probabilities in turn boils down to computing the {\em least fixed point} (LFP) solution of a corresponding {\em monotone polynomial system} (MPS) of equations, denoted x=P(x). It was shown by Etessami & Yannakakis that a decomposed variant of Newton's method converges monotonically to the LFP solution for any MPS that has a non-negative solution. Subsequently, Esparza, Kiefer, & Luttenberger obtained upper bounds on the convergence rate of Newton's method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs. However, prior to this paper, for arbitrary (not necessarily strongly-connected) MPSs, no upper bounds at all were known on the convergence rate of Newton's method as a function of the encoding size |P| of the input MPS, x=P(x). In this paper we provide worst-case upper bounds, as a function of both the input encoding size |P|, and epsilon > 0, on the number of iterations required for decomposed Newton's method (even with rounding) to converge within additive error epsilon > 0 of q^*, for any MPS with LFP solution q^*. Our upper bounds are essentially optimal in terms of several important parameters. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) omega-regular or LTL property.
 Computer Science , 2005, DOI: 10.2168/LMCS-2(1:2)2006 Abstract: We consider the model checking problem for probabilistic pushdown automata (pPDA) and properties expressible in various probabilistic logics. We start with properties that can be formulated as instances of a generalized random walk problem. We prove that both qualitative and quantitative model checking for this class of properties and pPDA is decidable. Then we show that model checking for the qualitative fragment of the logic PCTL and pPDA is also decidable. Moreover, we develop an error-tolerant model checking algorithm for PCTL and the subclass of stateless pPDA. Finally, we consider the class of omega-regular properties and show that both qualitative and quantitative model checking for pPDA is decidable.
 Advances in Software Engineering , 2011, DOI: 10.1155/2011/869182 Abstract: The Property Specification (Prospec) tool uses patterns and scopes defined by Dwyer et al., to generate formal specifications in Linear Temporal Logic (LTL) and other languages. The work presented in this paper provides improved LTL specifications for patterns and scopes over those originally provided by Prospec. This improvement comes in the efficiency of the LTL formulas as measured in terms of the number of states in the Büchi automaton generated for the formula. Minimizing the size of the Büchi automata for an LTL specification provides a significant improvement for model checking software systems using such tools as the highly acclaimed Spin model checker. 1. Introduction The process of model checking a system consists of developing a model of the system to be verified and writing specifications in a temporal logic such as Linear Temporal Logic (LTL) [1] or Computational Tree Logic (CTL) [2]. In automata-based model checking, both the model and the complement of the temporal specification are represented by a special type of state machine called a Büchi Automaton (BA) [3]. To check the consistency of with , the model checker calculates the intersection of and where is the complement of . If the intersection is empty, then is consistent with . In other words, if and each represent a set of specifications and if , then the system satisfies the specification; otherwise, the system is inconsistent with the specification and a counter-example is returned. The process of writing formal specifications is not easy because of the required mathematical sophistication and depth of knowledge in the specification language. For this reason, tools that simplify the creation of formal specifications in logics such as LTL are of interest to the model checking community and others. In the case of automata-based model checkers such as Spin [4], it is important that these tools generate efficient formulas, since the model checker complements the formulas, translates the result into a BA, and intersects the BA with the automaton of the system. The size of the automaton that results from the intersection of two automata has as its upper bound the product of the number of states in each of the two. One way to avoid the classical problem of state space explosion is to minimize the number of states generated by the negation of the specification. This will reduce the number of states generated by the automaton of the intersection, and as a result, it will reduce the time required to model check a software system. The Property Specification (Prospec) [5–7] builds on the
 Computer Science , 2008, DOI: 10.2168/LMCS-4(2:9)2008 Abstract: We consider the model of priced (a.k.a. weighted) timed automata, an extension of timed automata with cost information on both locations and transitions, and we study various model-checking problems for that model based on extensions of classical temporal logics with cost constraints on modalities. We prove that, under the assumption that the model has only one clock, model-checking this class of models against the logic WCTL, CTL with cost-constrained modalities, is PSPACE-complete (while it has been shown undecidable as soon as the model has three clocks). We also prove that model-checking WMTL, LTL with cost-constrained modalities, is decidable only if there is a single clock in the model and a single stopwatch cost variable (i.e., whose slopes lie in {0,1}).
 Steen Vester Computer Science , 2014, Abstract: 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.
 Electronic Proceedings in Theoretical Computer Science , 2012, DOI: 10.4204/eptcs.78.4 Abstract: Model checking verifies that a model of a system satisfies a given property, and otherwise produces a counter-example explaining the violation. The verified properties are formally expressed in temporal logics. Some temporal logics, such as CTL, are branching: they allow to express facts about the whole computation tree of the model, rather than on each single linear computation. This branching aspect is even more critical when dealing with multi-modal logics, i.e. logics expressing facts about systems with several transition relations. A prominent example is CTLK, a logic that reasons about temporal and epistemic properties of multi-agent systems. In general, model checkers produce linear counter-examples for failed properties, composed of a single computation path of the model. But some branching properties are only poorly and partially explained by a linear counter-example. This paper proposes richer counter-example structures called tree-like annotated counter-examples (TLACEs), for properties in Action-Restricted CTL (ARCTL), an extension of CTL quantifying paths restricted in terms of actions labeling transitions of the model. These counter-examples have a branching structure that supports more complete description of property violations. Elements of these counter-examples are annotated with parts of the property to give a better understanding of their structure. Visualization and browsing of these richer counter-examples become a critical issue, as the number of branches and states can grow exponentially for deeply-nested properties. This paper formally defines the structure of TLACEs, characterizes adequate counter-examples w.r.t. models and failed properties, and gives a generation algorithm for ARCTL properties. It also illustrates the approach with examples in CTLK, using a reduction of CTLK to ARCTL. The proposed approach has been implemented, first by extending the NuSMV model checker to generate and export branching counter-examples, secondly by providing an interactive graphical interface to visualize and browse them.
 Computer Science , 2013, Abstract: Timed automata (TAs) are a common formalism for modeling timed systems. Bounded model checking (BMC) is a verification method that searches for runs violating a property using a SAT or SMT solver. MITL is a real-time extension of the linear time logic LTL. Originally, MITL was defined for traces of non-overlapping time intervals rather than the "super-dense" time traces allowing for intervals overlapping in single points that are employed by the nowadays common semantics of timed automata. In this paper we extend the semantics of a fragment of MITL to super-dense time traces and devise a bounded model checking encoding for the fragment. We prove correctness and completeness in the sense that using a sufficiently large bound a counter-example to any given non-holding property can be found. We have implemented the proposed bounded model checking approach and experimentally studied the efficiency and scalability of the implementation.
 Mathematics , 2006, DOI: 10.1142/S0218196708004901 Abstract: We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller-Schupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognised by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata.
 Computer Science , 2003, Abstract: Suitable extensions of the monadic second-order theory of k successors have been proposed in the literature to capture the notion of time granularity. In this paper, we provide the monadic second-order theories of downward unbounded layered structures, which are infinitely refinable structures consisting of a coarsest domain and an infinite number of finer and finer domains, and of upward unbounded layered structures, which consist of a finest domain and an infinite number of coarser and coarser domains, with expressively complete and elementarily decidable temporal logic counterparts. We obtain such a result in two steps. First, we define a new class of combined automata, called temporalized automata, which can be proved to be the automata-theoretic counterpart of temporalized logics, and show that relevant properties, such as closure under Boolean operations, decidability, and expressive equivalence with respect to temporal logics, transfer from component automata to temporalized ones. Then, we exploit the correspondence between temporalized logics and automata to reduce the task of finding the temporal logic counterparts of the given theories of time granularity to the easier one of finding temporalized automata counterparts of them.
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