[Show abstract][Hide abstract] ABSTRACT: Determining the complexity of the reachability problem for vector addition
systems with states (VASS) is a long-standing 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 two-dimensional VASS is shown
PSPACE-complete. 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 PSPACE-complete. In addition,
some complexity results are given for the reachability problem in
two-dimensional VASS and in integer VASS when numbers are encoded in unary.
[Show abstract][Hide abstract] ABSTRACT: A one-counter automaton is a pushdown automaton with a singleton stack alphabet, where stack emptiness can be tested; it is a real-time automaton if it contains no ε-transitions. We study the computational complexity of the problems of equivalence and regularity (i.e. semantic finiteness) on real-time one-counter automata. The first main result shows PSPACEPSPACE-completeness of bisimulation equivalence; this closes the complexity gap between decidability [23] and PSPACEPSPACE-hardness [25]. The second main result shows NLNL-completeness of language equivalence of deterministic real-time one-counter automata; this improves the known PSPACEPSPACE upper bound (indirectly shown by Valiant and Paterson [27]). Finally we prove PP-completeness of the problem if a given one-counter automaton is bisimulation equivalent to a finite system, and NLNL-completeness of the problem if the language accepted by a given deterministic real-time one-counter automaton is regular.
Journal of Computer and System Sciences 06/2014; 80(4):720–743. DOI:10.1016/j.jcss.2013.11.003 · 1.14 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We prove that language equivalence of deterministic one-counter automata is
NL-complete. This improves the superpolynomial time complexity upper bound
shown by Valiant and Paterson in 1975. Our main contribution is to prove that
two deterministic one-counter 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.
Proceedings of the Annual ACM Symposium on Theory of Computing 01/2013; DOI:10.1145/2488608.2488626
[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 PSPACE-complete 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 range-constraining formula.
[Show abstract][Hide abstract] ABSTRACT: One-counter 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 μ-calculus 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 PSPACE-hard, i.e., expression complexity is PSPACE-hard. Third, we show that there already exists a fixed CTL formula for which model checking of OCA is PSPACE-hard, i.e., data complexity is PSPACE-hard 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 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 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 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 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 2-clock timed automata is PSPACE-hard, generalizing a PSPACE-hardness 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 EXPTIME-hardness, which was the previously
best known lower bound for this problem. Our lower bound result holds for
normed pushdown systems as well.
[Show abstract][Hide abstract] 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.
Proceedings of the 23rd International Conference on Concurrency Theory (CONCUR'12); 09/2012
[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
non-elementary 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
non-elementary over linear orders, but its complexity drops to
PSPACE-completeness over N. We categorize the strict fragments arising from
different combinations of modal and hybrid operators into NP-complete and
tractable (i.e. complete for NC1or LOGSPACE). Interestingly, NP-completeness
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 model-theoretic properties of the
fragments in question.
[Show abstract][Hide abstract] ABSTRACT: We study the computational complexity of model checking EF logic and modal logic on parametric one-counter automata (POCA). A POCA is a one-counter automaton whose counter updates are either integer values encoded in binary or integer-valued parameters. Given a formula and a configuration of a POCA, the model-checking 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 PSPACE-completeness. 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 PSPACE-complete.
Proceedings of the 15th international conference on Foundations of Software Science and Computational Structures; 03/2012
[Show abstract][Hide abstract] 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 Hennessy-Milner (HM) logic and its extension with the reachability operator (EF-logic) from PSPACE to nonelementary. In particular, this nonelementary jump happens for EF-logic 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 Feferman-Vaught (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.
29th International Symposium on Theoretical Aspects of Computer Science, STACS 2012, February 29th - March 3rd, 2012, Paris, France; 02/2012
[Show abstract][Hide abstract] ABSTRACT: We show that bisimulation equivalence of order-two pushdown automata is undecidable. Moreover, we study the lower order problem of higher-order pushdown automata, which asks, given an order-k pushdown automaton and some k < k, to determine if there exists a reachable configuration that is bisimilar to some order-k pushdown automaton. We show that the lower order problem is undecidable for each k ≥ 2 even when the input k-PDA is deterministic and real-time.
[Show abstract][Hide abstract] ABSTRACT: We prove that the complexity of the uniform first-order 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
first-order 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 non-elementary first-order
theory.
[Show abstract][Hide abstract] ABSTRACT: We prove that deciding language equivalence of deterministic real-time one-counter automata is NL-complete, 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 NL-complete as well.
Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Warsaw, Poland, August 22-26, 2011. Proceedings; 01/2011
[Show abstract][Hide abstract] ABSTRACT: In his seminal paper, R. Mayr introduced the well-known Process Rewrite Systems (PRS) hierarchy, which contains many well-studied
classes of infinite systems including pushdown systems, Petri nets and PA-processes. 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.
CONCUR 2011 - Concurrency Theory - 22nd International Conference, CONCUR 2011, Aachen, Germany, September 6-9, 2011. Proceedings; 01/2011
[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 model-checking problem for the modal µ-calculus and (monadic) least fixpoint logic on hierarchically defined input graphs is investigated. In order to analyze the modal µ-calculus, 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 model-checking algorithms is presented.
Theory of Computing Systems 01/2011; 48(1):93-131. DOI:10.1007/s00224-009-9227-1 · 0.53 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A one-counter automaton is a pushdown automaton over a singleton stack alphabet. We prove that the bisimilarity of processes
generated by nondeterministic one-counter automata (with no ε-steps) is in PSPACE. This improves the previously known decidability result (Jančar 2000), and matches the known PSPACE lower
bound (Srba 2009). We add the PTIME-completeness result for deciding regularity (i.e. finiteness up to bisimilarity) of one-counter
processes.
[Show abstract][Hide abstract] ABSTRACT: We investigate the decidability and complexity of various model checking problems over one-counter automata. More specifically,
we consider succinct one-counter automata, in which additive updates are encoded in binary, as well as parametric one-counter 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 μ-calculus specifications.
[Show abstract][Hide abstract] ABSTRACT: 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. 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.
[Show abstract][Hide abstract] ABSTRACT: We study satisfiability and infinite-state 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 2EXPTIME-complete by an existing lower bound, and (ii) infinite-state model check- ing of basic process algebras and pushdown systems is also 2EXPTIME-complete. Both upper bounds are obtained by polynomial time computable reductions to ω-regular 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 two-way 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 infinite-state model checking using an encoding of alternating Turing machines.