Publications (27)5.56 Total impact
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ABSTRACT: Determining the complexity of the reachability problem for vector addition systems with states (VASS) is a longstanding open problem in computer science. Long known to be decidable, the problem to this day lacks any complexity upper bound whatsoever. In this paper, reachability for twodimensional VASS is shown PSPACEcomplete. This improves on a previously known doubly exponential time bound established by Howell, Rosier, Huynh and Yen in 1986. The coverability and boundedness problems are also noted to be PSPACEcomplete. In addition, some complexity results are given for the reachability problem in twodimensional VASS and in integer VASS when numbers are encoded in unary.  [Show abstract] [Hide abstract]
ABSTRACT: A onecounter automaton is a pushdown automaton with a singleton stack alphabet, where stack emptiness can be tested; it is a realtime automaton if it contains no εtransitions. We study the computational complexity of the problems of equivalence and regularity (i.e. semantic finiteness) on realtime onecounter automata. The first main result shows PSPACEPSPACEcompleteness of bisimulation equivalence; this closes the complexity gap between decidability [23] and PSPACEPSPACEhardness [25]. The second main result shows NLNLcompleteness of language equivalence of deterministic realtime onecounter automata; this improves the known PSPACEPSPACE upper bound (indirectly shown by Valiant and Paterson [27]). Finally we prove PPcompleteness of the problem if a given onecounter automaton is bisimulation equivalent to a finite system, and NLNLcompleteness of the problem if the language accepted by a given deterministic realtime onecounter automaton is regular.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that language equivalence of deterministic onecounter automata is NLcomplete. This improves the superpolynomial time complexity upper bound shown by Valiant and Paterson in 1975. Our main contribution is to prove that two deterministic onecounter automata are inequivalent if and only if they can be distinguished by a word of length polynomial in the size of the two input automata.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
ABSTRACT: Onecounter automata (OCA) are pushdown automata which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) on transition systems induced by OCA. A PSPACE upper bound is inherited from the modal μcalculus for this problem proved by Serre. First, we analyze the periodic behavior of CTL over OCA 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 OCA against CTL formulas with a fixed leftward until depth is in P. This generalizes a corresponding recent result of Göller, Mayr, and To for the expression complexity of CTL’s fragment EF. Second, we prove that already over some fixed OCA, 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 OCA 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 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 OCA is hard for P NP , thus establishing a matching lower bound. Moreover, we 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 recently proved lower bound for every level of the boolean hierarchy shown by Brázdil et al. Finally, we prove that there is a fixed CTL formula for which model checking 2clock timed automata is PSPACEhard, generalizing a PSPACEhardness result for the combined complexity by Laroussinie, Markey, and Schnoebelen.  [Show abstract] [Hide abstract]
ABSTRACT: Given two pushdown systems, the bisimilarity problem asks whether they are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIMEhardness, which was the previously best known lower bound for this problem. Our lower bound result holds for normed pushdown systems as well. 
Conference Paper: A~Comparison of Succinctly Represented FiniteState Systems
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ABSTRACT: We study the succinctness of different classes of succinctly presented finite transition systems with respect to bisimulation equivalence. Our results show that synchronized product of finite automata, hierarchical graphs, and timed automata are pairwise incomparable in this sense. We moreover study the computational complexity of deciding simulation preorder and bisimulation equivalence on these classes.  [Show abstract] [Hide abstract]
ABSTRACT: Hybrid logic with binders is an expressive specification language. Its satisfiability problem is undecidable in general. If frames are restricted to N or general linear orders, then satisfiability is known to be decidable, but of nonelementary complexity. In this paper, we consider monotone hybrid logics (i.e., the Boolean connectives are conjunction and disjunction only) over N and general linear orders. We show that the satisfiability problem remains nonelementary over linear orders, but its complexity drops to PSPACEcompleteness over N. We categorize the strict fragments arising from different combinations of modal and hybrid operators into NPcomplete and tractable (i.e. complete for NC1or LOGSPACE). Interestingly, NPcompleteness depends only on the fragment and not on the frame. For the cases above NP, satisfiability over linear orders is harder than over N, while below NP it is at most as hard. In addition we examine modeltheoretic properties of the fragments in question. 
Conference Paper: BranchingTime Model Checking of Parametric OneCounter Automata
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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. 
Conference Paper: Concurrency Makes Simple Theories Hard
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ABSTRACT: A standard way of building concurrent systems is by composing several individual processes by product operators. We show that even the simplest notion of product operators (i.e. asynchronous products) suffices to increase the complexity of model checking simple logics like HennessyMilner (HM) logic and its extension with the reachability operator (EFlogic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EFlogic when we consider individual processes represented by pushdown systems (indeed, even with only one control state). Using this result, we prove nonelementary lower bounds on the size of formula decompositions provided by FefermanVaught (de)compositional methods for HM and EF logics, which reduce theories of asynchronous products to theories of the components. Finally, we show that the same nonelementary lower bounds also hold when we consider the relativization of such compositional methods to finite systems.  [Show abstract] [Hide abstract]
ABSTRACT: We show that bisimulation equivalence of ordertwo pushdown automata is undecidable. Moreover, we study the lower order problem of higherorder pushdown automata, which asks, given an orderk pushdown automaton and some k < k, to determine if there exists a reachable configuration that is bisimilar to some orderk pushdown automaton. We show that the lower order problem is undecidable for each k ≥ 2 even when the input kPDA is deterministic and realtime.  [Show abstract] [Hide abstract]
ABSTRACT: In his seminal paper, R. Mayr introduced the wellknown Process Rewrite Systems (PRS) hierarchy, which contains many wellstudied classes of infinite systems including pushdown systems, Petri nets and PAprocesses. A seperate development in the term rewriting community introduced the notion of Ground Tree Rewrite Systems (GTRS), which is a model that strictly extends pushdown systems while still enjoying desirable decidable properties. There have been striking similarities between the verification problems that have been shown decidable (and undecidable) over GTRS and over models in the PRS hierarchy such as PA and PAD processes. It is open to what extent PRS and GTRS are connected in terms of their expressive power. In this paper we pinpoint the exact connection between GTRS and models in the PRS hierarchy in terms of their expressive power with respect to strong, weak, and branching bisimulation. Among others, this connection allows us to give new insights into the decidability results for subclasses of PRS, e.g., simpler proofs of known decidability results of verifications problems on PAD. 
Conference Paper: Language Equivalence of Deterministic RealTime OneCounter Automata Is NLComplete
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ABSTRACT: We prove that deciding language equivalence of deterministic realtime onecounter automata is NLcomplete, in stark contrast to the inclusion problem which is known to be undecidable. This yields a subclass of deterministic pushdown automata for which the precise complexity of the equivalence problem can be determined. Moreover, we show that deciding regularity is NLcomplete as well.  [Show abstract] [Hide abstract]
ABSTRACT: We prove that the complexity of the uniform firstorder theory of ground tree rewrite graphs is in ATIME(2^{2^{poly(n)}},O(n)). Providing a matching lower bound, we show that there is some fixed ground tree rewrite graph whose firstorder theory is hard for ATIME(2^{2^{poly(n)}},poly(n)) with respect to logspace reductions. Finally, we prove that there exists a fixed ground tree rewrite graph together with a single unary predicate in form of a regular tree language such that the resulting structure has a nonelementary firstorder theory.  [Show abstract] [Hide abstract]
ABSTRACT: Hierarchical graph definitions allow a modular description of graphs using mod ules for the specification of repeated substructures. Beside this modularity, hierarchi cal graph definitions also allow to specify graphs of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational com plexity of decision problems. In this paper, the modelchecking problem for the modal µcalculus and (monadic) least fixpoint logic on hierarchically defined input graphs is investigated. In order to analyze the modal µcalculus, parity games on hierar chically defined input graphs are investigated. Precise upper and lower complexity bounds are derived. A restriction on hierarchical graph definitions that leads to more efficient modelchecking algorithms is presented.  [Show abstract] [Hide abstract]
ABSTRACT: A onecounter automaton is a pushdown automaton over a singleton stack alphabet. We prove that the bisimilarity of processes generated by nondeterministic onecounter automata (with no εsteps) is in PSPACE. This improves the previously known decidability result (Jančar 2000), and matches the known PSPACE lower bound (Srba 2009). We add the PTIMEcompleteness result for deciding regularity (i.e. finiteness up to bisimilarity) of onecounter processes.  [Show abstract] [Hide abstract]
ABSTRACT: We investigate the decidability and complexity of various model checking problems over onecounter automata. More specifically, we consider succinct onecounter automata, in which additive updates are encoded in binary, as well as parametric onecounter automata, in which additive updates may be given as unspecified parameters. We fully determine the complexity of model checking these automata against CTL, LTL, and modal μcalculus specifications. 
Conference Paper: Bisimilarity of OneCounter Processes Is PSPACEComplete.
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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 mucalculus 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 PSPACEhard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACEhard. For the latter, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspaceuniform NC^1 and (ii) PSPACE is AC^0serializable. We demonstrate that our approach can be used to answer further open questions.  [Show abstract] [Hide abstract]
ABSTRACT: We study satisfiability and infinitestate model checking in ICPDL, which extends Propositional Dynamic Logic (PDL) with intersection and converse operators on programs. The two main results of this paper are that (i) satisfiability is in 2EXPTIME, thus 2EXPTIMEcomplete by an existing lower bound, and (ii) infinitestate model check ing of basic process algebras and pushdown systems is also 2EXPTIMEcomplete. Both upper bounds are obtained by polynomial time computable reductions to ωregular tree satisfiability in ICPDL, a reasoning problem that we introduce specifically for this pur pose. This problem is then reduced to the emptiness problem for alternating twoway automata on infinite trees. Our approach to (i) also provides a shorter and more elegant proof of Danecki's difficult result that satisfiability in IPDL is in 2EXPTIME. We prove the lower bound(s) for infinitestate model checking using an encoding of alternating Turing machines.
Publication Stats
102  Citations  
5.56  Total Impact Points  
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Institutions

20102014

Universität Bremen
Bremen, Bremen, Germany


2012

Paris Diderot University
Lutetia Parisorum, ÎledeFrance, France


20082009

University of Leipzig
 Institute of Computer Science
Leipzig, Saxony, Germany


20052007

Universität Stuttgart
Stuttgart, BadenWürttemberg, Germany
