102 research outputs found
State of B\"uchi Complementation
Complementation of B\"uchi automata has been studied for over five decades
since the formalism was introduced in 1960. Known complementation constructions
can be classified into Ramsey-based, determinization-based, rank-based, and
slice-based approaches. Regarding the performance of these approaches, there
have been several complexity analyses but very few experimental results. What
especially lacks is a comparative experiment on all of the four approaches to
see how they perform in practice. In this paper, we review the four approaches,
propose several optimization heuristics, and perform comparative
experimentation on four representative constructions that are considered the
most efficient in each approach. The experimental results show that (1) the
determinization-based Safra-Piterman construction outperforms the other three
in producing smaller complements and finishing more tasks in the allocated time
and (2) the proposed heuristics substantially improve the Safra-Piterman and
the slice-based constructions.Comment: 28 pages, 4 figures, a preliminary version of this paper appeared in
the Proceedings of the 15th International Conference on Implementation and
Application of Automata (CIAA
Tight Upper Bounds for Streett and Parity Complementation
Complementation of finite automata on infinite words is not only a
fundamental problem in automata theory, but also serves as a cornerstone for
solving numerous decision problems in mathematical logic, model-checking,
program analysis and verification. For Streett complementation, a significant
gap exists between the current lower bound and upper
bound , where is the state size, is the number of
Streett pairs, and can be as large as . Determining the complexity
of Streett complementation has been an open question since the late '80s. In
this paper show a complementation construction with upper bound for and for ,
which matches well the lower bound obtained in \cite{CZ11a}. We also obtain a
tight upper bound for parity complementation.Comment: Corrected typos. 23 pages, 3 figures. To appear in the 20th
Conference on Computer Science Logic (CSL 2011
Benchmarks for Parity Games (extended version)
We propose a benchmark suite for parity games that includes all benchmarks
that have been used in the literature, and make it available online. We give an
overview of the parity games, including a description of how they have been
generated. We also describe structural properties of parity games, and using
these properties we show that our benchmarks are representative. With this work
we provide a starting point for further experimentation with parity games.Comment: The corresponding tool and benchmarks are available from
https://github.com/jkeiren/paritygame-generator. This is an extended version
of the paper that has been accepted for FSEN 201
A Tight Lower Bound for Streett Complementation
Finite automata on infinite words (-automata) proved to be a powerful
weapon for modeling and reasoning infinite behaviors of reactive systems.
Complementation of -automata is crucial in many of these applications.
But the problem is non-trivial; even after extensive study during the past four
decades, we still have an important type of -automata, namely Streett
automata, for which the gap between the current best lower bound and upper bound is substantial, for the
Streett index size can be exponential in the number of states . In
arXiv:1102.2960 we showed a construction for complementing Streett automata
with the upper bound for and for . In this paper we establish a matching lower bound
for and for
, and therefore showing that the construction is asymptotically
optimal with respect to the notation.Comment: Typo correction and section reorganization. To appear in the
proceeding of the 31st Foundations of Software Technology and Theoretical
Computer Science conference (FSTTCS 2011
Succinct progress measures for solving parity games
The recent breakthrough paper by Calude et al. has given the first algorithm
for solving parity games in quasi-polynomial time, where previously the best
algorithms were mildly subexponential. We devise an alternative
quasi-polynomial time algorithm based on progress measures, which allows us to
reduce the space required from quasi-polynomial to nearly linear. Our key
technical tools are a novel concept of ordered tree coding, and a succinct tree
coding result that we prove using bounded adaptive multi-counters, both of
which are interesting in their own right
Model counting for reactive systems
Model counting is the problem of computing the number of solutions for a logical formula. In the last few years, it has been primarily studied for propositional logic, and has been shown to be useful in many applications. In planning, for example, propositional model counting has been used to compute the robustness of a plan in an incomplete domain. In information-flow control, model counting has been applied to measure the amount of information leaked by a security-critical system. In this thesis, we introduce the model counting problem for linear-time properties, and show its applications in formal verification. In the same way propositional model counting generalizes the satisfiability problem for propositional logic, counting models for linear-time properties generalizes the emptiness problem for languages over infinite words to one that asks for the number of words in a language. The model counting problem, thus, provides a foundation for quantitative extensions of model checking, where not only the existence of computations that violate the specification is determined, but also the number of such violations. We solve the model counting problem for the prominent class of omega-regular properties. We present algorithms for solving the problem for different classes of properties, and show the advantages of our algorithms in comparison to indirect approaches based on encodings into propositional logic. We further show how model counting can be used for solving a variety of quantitative problems in formal verification, including probabilistic model checking, quantitative information-flow in security-critical systems, and the synthesis of approximate implementations for reactive systems.Das ModellzĂ€hlproblem fragt nach der Anzahl der Lösungen einer logischen Formel, und wurde in den letzten Jahren hauptsĂ€chlich fĂŒr Aussagenlogik untersucht. Das ZĂ€hlen von Modellen aussagenlogischer Formeln hat sich in vielen Anwendungen als nĂŒtzlich erwiesen. Im Bereich der kĂŒnstlichen Intelligenz wurde das ZĂ€hlen von Modellen beispielsweise verwendet, um die Robustheit eines Plans in einem unvollstĂ€ndigen Weltmodell zu bewerten. Das ZĂ€hlen von Modellen kann auch verwendet werden, um in sicherheitskritischen Systemen die Menge an enthĂŒllten vertraulichen Daten zu messen. Diese Dissertation stellt das ModellzĂ€hlproblem fĂŒr Linearzeiteigenschaften vor, und untersucht dessen Rolle in der Welt der formalen Verifikation. Das ZĂ€hlen von Modellen fĂŒr Linearzeiteigenschaften fĂŒhrt zu neuen quantitativen Erweiterungen klassischer Verifikationsprobleme, bei denen nicht nur die Existenz eines Fehlers in einem System zu ĂŒberprĂŒfen ist, sondern auch die Anzahl solcher Fehler. Wir prĂ€sentieren Algorithmen zur Lösung des ModellzĂ€hlproblems fĂŒr verschiedene Klassen von Linearzeiteigenschaften und zeigen die Vorteile unserer Algorithmen im Vergleich zu indirekten AnsĂ€tzen, die auf Kodierungen der untersuchten Probleme in Aussagenlogik basieren. DarĂŒberhinaus zeigen wir wie das ZĂ€hlen von Modellen zur Lösung einer Vielzahl quantitativer Probleme in der formalen Verifikation verwendet werden kann. Dies beinhaltet unter anderem die Analyse probabilistischer Modelle, die Kontrolle quantitativen Informationsflusses in sicherheitskritischen Systemen, und die Synthese von approximativen Implementierungen fĂŒr reaktive Systeme
A survey of stochastic Ï regular games
We summarize classical and recent results about two-player games played on graphs with Ï-regular objectives. These games have applications in the verification and synthesis of reactive systems. Important distinctions are whether a graph game is turn-based or concurrent; deterministic or stochastic; zero-sum or not. We cluster known results and open problems according to these classifications
B\"uchi Complementation and Size-Change Termination
We compare tools for complementing nondeterministic B\"uchi automata with a
recent termination-analysis algorithm. Complementation of B\"uchi automata is a
key step in program verification. Early constructions using a Ramsey-based
argument have been supplanted by rank-based constructions with exponentially
better bounds. In 2001 Lee et al. presented the size-change termination (SCT)
problem, along with both a reduction to B\"uchi automata and a Ramsey-based
algorithm. The Ramsey-based algorithm was presented as a more practical
alternative to the automata-theoretic approach, but strongly resembles the
initial complementation constructions for B\"uchi automata. We prove that the
SCT algorithm is a specialized realization of the Ramsey-based complementation
construction. To do so, we extend the Ramsey-based complementation construction
to provide a containment-testing algorithm. Surprisingly, empirical analysis
suggests that despite the massive gap in worst-case complexity, Ramsey-based
approaches are superior over the domain of SCT problems. Upon further analysis
we discover an interesting property of the problem space that both explains
this result and provides a chance to improve rank-based tools. With these
improvements, we show that theoretical gains in efficiency of the rank-based
approach are mirrored in empirical performance
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