Igor Walukiewicz — Page 2

arXiv2015

Ordered Tree-Pushdown Systems

We define a new class of pushdown systems where the pushdown is a tree instead of a word. We allow a limited form of lookahead on the pushdown conforming to a certain ordering restriction, and we show that the resulting class enjoys a decidable reachability problem. This follows from a preservation of recognizability result for the backward reachability relation of such systems. As an application, we show that our simple model can encode several formalisms generalizing pushdown systems, such as ordered multi-pushdown systems, annotated higher-order pushdown systems, the Krivine machine, and ordered annotated multi-pushdown systems. In each case, our procedure yields tight complexity.

cs.FL↗

arXiv2015

A Note on Monitors and Büchi automata

When a property needs to be checked against an unknown or very complex system, classical exploration techniques like model-checking are not applicable anymore. Sometimes a~monitor can be used, that checks a given property on the underlying system at runtime. A monitor for a property $L$ is a deterministic finite automaton $M_L$ that after each finite execution tells whether (1) every possible extension of the execution is in $L$, or (2) every possible extension is in the complement of $L$, or neither (1) nor (2) holds. Moreover, $L$ being monitorable means that it is always possible that in some future the monitor reaches (1) or (2). Classical examples for monitorable properties are safety and cosafety properties. On the other hand, deterministic liveness properties like "infinitely many $a$'s" are not monitorable. We discuss various monitor constructions with a focus on deterministic omega-regular languages. We locate a proper subclass of of deterministic omega-regular languages but also strictly large than the subclass of languages which are deterministic and codeterministic, and for this subclass there exists a canonical monitor which also accepts the language itself. We also address the problem to decide monitorability in comparison with deciding liveness. The state of the art is as follows. Given a Büchi automaton, it is PSPACE-complete to decide liveness or monitorability. Given an LTL formula, deciding liveness becomes EXPSPACE-complete, but the complexity to decide monitorability remains open.

cs.FL↗

arXiv2015

Using models to model-check recursive schemes

We propose a model-based approach to the model checking problem for recursive schemes. Since simply typed lambda calculus with the fixpoint operator, lambda-Y-calculus, is equivalent to schemes, we propose the use of a model of lambda-Y-calculus to discriminate the terms that satisfy a given property. If a model is finite in every type, this gives a decision procedure. We provide a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing. Such properties pose already interesting challenges for model construction. Moreover, we argue that having models capturing some class of properties has several other virtues in addition to providing decidability of the model-checking problem. As an illustration, we show a very simple construction transforming a scheme to a scheme reflecting a property captured by a given model.

cs.LO↗

arXiv2014

Fast detection of cycles in timed automata

We propose a new efficient algorithm for detecting if a cycle in a timed automaton can be iterated infinitely often. Existing methods for this problem have a complexity which is exponential in the number of clocks. Our method is polynomial: it essentially does a logarithmic number of zone canonicalizations. This method can be incorporated in algorithms for verifying Büchi properties on timed automata. We report on some experiments that show a significant reduction in search space when our iteratability test is used.

cs.LO↗

arXiv2014

Distributed synthesis for acyclic architectures

The distributed synthesis problem is about constructing cor- rect distributed systems, i.e., systems that satisfy a given specification. We consider a slightly more general problem of distributed control, where the goal is to restrict the behavior of a given distributed system in order to satisfy the specification. Our systems are finite state machines that communicate via rendez-vous (Zielonka automata). We show decidability of the synthesis problem for all omega-regular local specifications, under the restriction that the communication graph of the system is acyclic. This result extends a previous decidability result for a restricted form of local reachability specifications.

cs.LO↗

arXiv2013

Asynchronous Games over Tree Architectures

We consider the task of controlling in a distributed way a Zielonka asynchronous automaton. Every process of a controller has access to its causal past to determine the next set of actions it proposes to play. An action can be played only if every process controlling this action proposes to play it. We consider reachability objectives: every process should reach its set of final states. We show that this control problem is decidable for tree architectures, where every process can communicate with its parent, its children, and with the environment. The complexity of our algorithm is l-fold exponential with l being the height of the tree representing the architecture. We show that this is unavoidable by showing that even for three processes the problem is EXPTIME-complete, and that it is non-elementary in general.

cs.FL↗

arXiv2013

Lazy abstractions for timed automata

We consider the reachability problem for timed automata. A standard solution to this problem involves computing a search tree whose nodes are abstractions of zones. For efficiency reasons, they are parametrized by the maximal lower and upper bounds (LU-bounds) occurring in the guards of the automaton. We propose an algorithm that is updating LU-bounds during exploration of the search tree. In order to keep them as small as possible, the bounds are refined only when they enable a transition that is impossible in the unabstracted system. So our algorithm can be seen as a kind of lazy CEGAR algorithm for timed automata. We show that on several standard benchmarks, the algorithm is capable of keeping very small LU-bounds, and in consequence reduce the search space substantially.

cs.LO↗

arXiv2012

Weak Alternating Timed Automata

Alternating timed automata on infinite words are considered. The main result is a characterization of acceptance conditions for which the emptiness problem for these automata is decidable. This result implies new decidability results for fragments of timed temporal logics. It is also shown that, unlike for MITL, the characterisation remains the same even if no punctual constraints are allowed.

cs.LO↗

arXiv2012

Wreath Products of Forest Algebras, with Applications to Tree Logics

We use the recently developed theory of forest algebras to find algebraic characterizations of the languages of unranked trees and forests definable in various logics. These include the temporal logics CTL and EF, and first-order logic over the ancestor relation. While the characterizations are in general non-effective, we are able to use them to formulate necessary conditions for definability and provide new proofs that a number of languages are not definable in these logics.

cs.LO↗

arXiv2012

Postinal Determinacy of Games with Infinitely Many Priorities

We study two-player games of infinite duration that are played on finite or infinite game graphs. A winning strategy for such a game is positional if it only depends on the current position, and not on the history of the play. A game is positionally determined if, from each position, one of the two players has a positional winning strategy. The theory of such games is well studied for winning conditions that are defined in terms of a mapping that assigns to each position a priority from a finite set. Specifically, in Muller games the winner of a play is determined by the set of those priorities that have been seen infinitely often; an important special case are parity games where the least (or greatest) priority occurring infinitely often determines the winner. It is well-known that parity games are positionally determined whereas Muller games are determined via finite-memory strategies. In this paper, we extend this theory to the case of games with infinitely many priorities. Such games arise in several application areas, for instance in pushdown games with winning conditions depending on stack contents. For parity games there are several generalisations to the case of infinitely many priorities. While max-parity games over omega or min-parity games over larger ordinals than omega require strategies with infinite memory, we can prove that min-parity games with priorities in omega are positionally determined. Indeed, it turns out that the min-parity condition over omega is the only infinitary Muller condition that guarantees positional determinacy on all game graphs.

cs.LO↗

arXiv2011

Using non-convex approximations for efficient analysis of timed automata

The reachability problem for timed automata asks if there exists a path from an initial state to a target state. The standard solution to this problem involves computing the zone graph of the automaton, which in principle could be infinite. In order to make the graph finite, zones are approximated using an extrapolation operator. For reasons of efficiency in current algorithms extrapolation of a zone is always a zone and in particular it is convex. In this paper, we propose to solve the reachability problem without such extrapolation operators. To ensure termination, we provide an efficient algorithm to check if a zone is included in the so called region closure of another. Although theoretically better, closure cannot be used in the standard algorithm since a closure of a zone may not be convex. An additional benefit of the proposed approach is that it permits to calculate approximating parameters on-the-fly during exploration of the zone graph, as opposed to the current methods which do it by a static analysis of the automaton prior to the exploration. This allows for further improvements in the algorithm. Promising experimental results are presented.

cs.LO↗

arXiv2008

A lower bound on web services composition

A web service is modeled here as a finite state machine. A composition problem for web services is to decide if a given web service can be constructed from a given set of web services; where the construction is understood as a simulation of the specification by a fully asynchronous product of the given services. We show an EXPTIME-lower bound for this problem, thus matching the known upper bound. Our result also applies to richer models of web services, such as the Roman model.

cs.LO↗

arXiv2007

The Complexity of Games on Higher Order Pushdown Automata

We prove an n-EXPTIME lower bound for the problem of deciding the winner in a reachability game on Higher Order Pushdown Automata (HPDA) of level n. This bound matches the known upper bound for parity games on HPDA. As a consequence the mu-calculus model checking over graphs given by n-HPDA is n-EXPTIME complete.

cs.GT↗

arXiv2006

Positional Determinacy of Games with Infinitely Many Priorities

We study two-player games of infinite duration that are played on finite or infinite game graphs. A winning strategy for such a game is positional if it only depends on the current position, and not on the history of the play. A game is positionally determined if, from each position, one of the two players has a positional winning strategy. The theory of such games is well studied for winning conditions that are defined in terms of a mapping that assigns to each position a priority from a finite set. Specifically, in Muller games the winner of a play is determined by the set of those priorities that have been seen infinitely often; an important special case are parity games where the least (or greatest) priority occurring infinitely often determines the winner. It is well-known that parity games are positionally determined whereas Muller games are determined via finite-memory strategies. In this paper, we extend this theory to the case of games with infinitely many priorities. Such games arise in several application areas, for instance in pushdown games with winning conditions depending on stack contents. For parity games there are several generalisations to the case of infinitely many priorities. While max-parity games over omega or min-parity games over larger ordinals than omega require strategies with infinite memory, we can prove that min-parity games with priorities in omega are positionally determined. Indeed, it turns out that the min-parity condition over omega is the only infinitary Muller condition that guarantees positional determinacy on all game graphs.

cs.LO↗

arXiv2006

Alternating Timed Automata

A notion of alternating timed automata is proposed. It is shown that such automata with only one clock have decidable emptiness problem over finite words. This gives a new class of timed languages which is closed under boolean operations and which has an effective presentation. We prove that the complexity of the emptiness problem for alternating timed automata with one clock is non-primitive recursive. The proof gives also the same lower bound for the universality problem for nondeterministic timed automata with one clock. We investigate extension of the model with epsilon-transitions and prove that emptiness is undecidable. Over infinite words, we show undecidability of the universality problem.

cs.LO↗