856 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
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Buffered Simulation Games for B\"uchi Automata
Simulation relations are an important tool in automata theory because they
provide efficiently computable approximations to language inclusion. In recent
years, extensions of ordinary simulations have been studied, for instance
multi-pebble and multi-letter simulations which yield better approximations and
are still polynomial-time computable.
In this paper we study the limitations of approximating language inclusion in
this way: we introduce a natural extension of multi-letter simulations called
buffered simulations. They are based on a simulation game in which the two
players share a FIFO buffer of unbounded size. We consider two variants of
these buffered games called continuous and look-ahead simulation which differ
in how elements can be removed from the FIFO buffer. We show that look-ahead
simulation, the simpler one, is already PSPACE-hard, i.e. computationally as
hard as language inclusion itself. Continuous simulation is even EXPTIME-hard.
We also provide matching upper bounds for solving these games with infinite
state spaces.Comment: In Proceedings AFL 2014, arXiv:1405.527
Satisfiability Games for Branching-Time Logics
The satisfiability problem for branching-time temporal logics like CTL*, CTL
and CTL+ has important applications in program specification and verification.
Their computational complexities are known: CTL* and CTL+ are complete for
doubly exponential time, CTL is complete for single exponential time. Some
decision procedures for these logics are known; they use tree automata,
tableaux or axiom systems. In this paper we present a uniform game-theoretic
framework for the satisfiability problem of these branching-time temporal
logics. We define satisfiability games for the full branching-time temporal
logic CTL* using a high-level definition of winning condition that captures the
essence of well-foundedness of least fixpoint unfoldings. These winning
conditions form formal languages of \omega-words. We analyse which kinds of
deterministic {\omega}-automata are needed in which case in order to recognise
these languages. We then obtain a reduction to the problem of solving parity or
B\"uchi games. The worst-case complexity of the obtained algorithms matches the
known lower bounds for these logics. This approach provides a uniform, yet
complexity-theoretically optimal treatment of satisfiability for branching-time
temporal logics. It separates the use of temporal logic machinery from the use
of automata thus preserving a syntactical relationship between the input
formula and the object that represents satisfiability, i.e. a winning strategy
in a parity or B\"uchi game. The games presented here work on a Fischer-Ladner
closure of the input formula only. Last but not least, the games presented here
come with an attempt at providing tool support for the satisfiability problem
of complex branching-time logics like CTL* and CTL+
A Hierarchy of Scheduler Classes for Stochastic Automata
Stochastic automata are a formal compositional model for concurrent
stochastic timed systems, with general distributions and non-deterministic
choices. Measures of interest are defined over schedulers that resolve the
nondeterminism. In this paper we investigate the power of various theoretically
and practically motivated classes of schedulers, considering the classic
complete-information view and a restriction to non-prophetic schedulers. We
prove a hierarchy of scheduler classes w.r.t. unbounded probabilistic
reachability. We find that, unlike Markovian formalisms, stochastic automata
distinguish most classes even in this basic setting. Verification and strategy
synthesis methods thus face a tradeoff between powerful and efficient classes.
Using lightweight scheduler sampling, we explore this tradeoff and demonstrate
the concept of a useful approximative verification technique for stochastic
automata
Approximating the Termination Value of One-Counter MDPs and Stochastic Games
One-counter MDPs (OC-MDPs) and one-counter simple stochastic games (OC-SSGs)
are 1-player, and 2-player turn-based zero-sum, stochastic games played on the
transition graph of classic one-counter automata (equivalently, pushdown
automata with a 1-letter stack alphabet). A key objective for the analysis and
verification of these games is the termination objective, where the players aim
to maximize (minimize, respectively) the probability of hitting counter value
0, starting at a given control state and given counter value. Recently, we
studied qualitative decision problems ("is the optimal termination value = 1?")
for OC-MDPs (and OC-SSGs) and showed them to be decidable in P-time (in NP and
coNP, respectively). However, quantitative decision and approximation problems
("is the optimal termination value ? p", or "approximate the termination value
within epsilon") are far more challenging. This is so in part because optimal
strategies may not exist, and because even when they do exist they can have a
highly non-trivial structure. It thus remained open even whether any of these
quantitative termination problems are computable. In this paper we show that
all quantitative approximation problems for the termination value for OC-MDPs
and OC-SSGs are computable. Specifically, given a OC-SSG, and given epsilon >
0, we can compute a value v that approximates the value of the OC-SSG
termination game within additive error epsilon, and furthermore we can compute
epsilon-optimal strategies for both players in the game. A key ingredient in
our proofs is a subtle martingale, derived from solving certain LPs that we can
associate with a maximizing OC-MDP. An application of Azuma's inequality on
these martingales yields a computable bound for the "wealth" at which a "rich
person's strategy" becomes epsilon-optimal for OC-MDPs.Comment: 35 pages, 1 figure, full version of a paper presented at ICALP 2011,
invited for submission to Information and Computatio
Enriched MU-Calculi Module Checking
The model checking problem for open systems has been intensively studied in
the literature, for both finite-state (module checking) and infinite-state
(pushdown module checking) systems, with respect to Ctl and Ctl*. In this
paper, we further investigate this problem with respect to the \mu-calculus
enriched with nominals and graded modalities (hybrid graded Mu-calculus), in
both the finite-state and infinite-state settings. Using an automata-theoretic
approach, we show that hybrid graded \mu-calculus module checking is solvable
in exponential time, while hybrid graded \mu-calculus pushdown module checking
is solvable in double-exponential time. These results are also tight since they
match the known lower bounds for Ctl. We also investigate the module checking
problem with respect to the hybrid graded \mu-calculus enriched with inverse
programs (Fully enriched \mu-calculus): by showing a reduction from the domino
problem, we show its undecidability. We conclude with a short overview of the
model checking problem for the Fully enriched Mu-calculus and the fragments
obtained by dropping at least one of the additional constructs
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