709 research outputs found
Separating Regular Languages over Infinite Words with Respect to the Wagner Hierarchy
We investigate the separation problem for regular ?-languages with respect to the Wagner hierarchy where the input languages are given as deterministic Muller automata (DMA). We show that a minimal separating DMA can be computed in exponential time and that some languages require separators of exponential size. Further, we show that in this setting it can be decided in polynomial time whether a separator exists on a certain level of the Wagner hierarchy and that emptiness of the intersection of two languages given by DMAs can be decided in polynomial time. Finally, we show that separation can also be decided in polynomial time if the input languages are given as deterministic parity automata
Determinization of B\"uchi Automata: Unifying the Approaches of Safra and Muller-Schupp
Determinization of B\"uchi automata is a long-known difficult problem and
after the seminal result of Safra, who developed the first asymptotically
optimal construction from B\"uchi into Rabin automata, much work went into
improving, simplifying or avoiding Safra's construction. A different, less
known determinization construction was derived by Muller and Schupp and appears
to be unrelated to Safra's construction on the first sight. In this paper we
propose a new meta-construction from nondeterministic B\"uchi to deterministic
parity automata which strictly subsumes both the construction of Safra and the
construction of Muller and Schupp. It is based on a correspondence between
structures that are encoded in the macrostates of the determinization
procedures - Safra trees on one hand, and levels of the split-tree, which
underlies the Muller and Schupp construction, on the other. Our construction
allows for combining the mentioned constructions and opens up new directions
for the development of heuristics.Comment: Full version of ICALP 2019 pape
An expressive completeness theorem for coalgebraic modal mu-calculi
Generalizing standard monadic second-order logic for Kripke models, we
introduce monadic second-order logic interpreted over coalgebras for an
arbitrary set functor. We then consider invariance under behavioral equivalence
of MSO-formulas. More specifically, we investigate whether the coalgebraic
mu-calculus is the bisimulation-invariant fragment of the monadic second-order
language for a given functor. Using automatatheoretic techniques and building
on recent results by the third author, we show that in order to provide such a
characterization result it suffices to find what we call an adequate uniform
construction for the coalgebraic type functor. As direct applications of this
result we obtain a partly new proof of the Janin-Walukiewicz Theorem for the
modal mu-calculus, avoiding the use of syntactic normal forms, and bisimulation
invariance results for the bag functor (graded modal logic) and all exponential
polynomial functors (including the "game functor"). As a more involved
application, involving additional non-trivial ideas, we also derive a
characterization theorem for the monotone modal mu-calculus, with respect to a
natural monadic second-order language for monotone neighborhood models.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0721
Computing the Width of Non-deterministic Automata
International audienceWe introduce a measure called width, quantifying the amount of nondetermin-ism in automata. Width generalises the notion of good-for-games (GFG) automata, that correspond to NFAs of width 1, and where an accepting run can be built on-the-fly on any accepted input. We describe an incremental determinisation construction on NFAs, which can be more efficient than the full powerset determinisation, depending on the width of the input NFA. This construction can be generalised to infinite words, and is particularly well-suited to coBüchi automata. For coBüchi automata, this procedure can be used to compute either a deterministic automaton or a GFG one, and it is algorithmically more efficient in the last case. We show this fact by proving that checking whether a coBüchi automaton is determinisable by pruning is NP-complete. On finite or infinite words, we show that computing the width of an automaton is EXPTIME-complete. This implies EXPTIME-completeness for multipebble simulation games on NFAs
Good for Games Automata: From Nondeterminism to Alternation
A word automaton recognizing a language is good for games (GFG) if its
composition with any game with winning condition preserves the game's
winner. While all deterministic automata are GFG, some nondeterministic
automata are not. There are various other properties that are used in the
literature for defining that a nondeterministic automaton is GFG, including
"history-deterministic", "compliant with some letter game", "good for trees",
and "good for composition with other automata". The equivalence of these
properties has not been formally shown.
We generalize all of these definitions to alternating automata and show their
equivalence. We further show that alternating GFG automata are as expressive as
deterministic automata with the same acceptance conditions and indices. We then
show that alternating GFG automata over finite words, and weak automata over
infinite words, are not more succinct than deterministic automata, and that
determinizing B\"uchi and co-B\"uchi alternating GFG automata involves a
state blow-up. We leave open the question of whether
alternating GFG automata of stronger acceptance conditions allow for
doubly-exponential succinctness compared to deterministic automata.Comment: Full version of a paper of the same name accepted fr publication at
the 30th International Conference on Concurrency Theor
Index appearance record with preorders
Transforming ω-automata into parity automata is traditionally done using appearance records. We present an efficient variant of this idea, tailored to Rabin automata, and several optimizations applicable to all appearance records. We compare the methods experimentally and show that our method produces significantly smaller automata than previous approaches
Heuristics for the refinement of assumptions in generalized reactivity formulae
Reactive synthesis is concerned with automatically generating implementations from formal specifications. These specifications are typically written in the language of generalized reactivity (GR(1)), a subset of linear temporal logic capable of expressing the most common industrial specification patterns, and describe the requirements about the behavior of a system under assumptions about the environment where the system is to be deployed. Oftentimes no implementation exists which guarantees the required behavior under all possible environments, typically due to missing assumptions (this is usually referred to as unrealizability). To address this issue, new assumptions need to be added to complete the specification, a problem known as assumptions refinement. Since the space of candidate assumptions is intractably large, searching for the best solutions is inherently hard. In particular, new methods are needed to (i) increase the effectiveness of the search procedures, measured as the ratio between the number of solutions found and of refinements explored; and (ii) improve the results' quality, defined as the weakness of the solutions. In this thesis we propose a set of heuristics to meet these goals, and a methodology to assess and compare assumptions refinement methods based on quantitative metrics. The heuristics are in the form of algorithms to generate candidate refinements during the search, and quantitative measures to assess the quality of the candidates.
We first discuss a heuristic method to generate assumptions that target the cause of unrealizability. This is done by selecting candidate refinement formulas based on Craig's interpolation. We provide a formal underpinning of the technique and evaluate it in terms of our new metric of effectiveness, as defined above, whose value is improved with respect to the state of the art. We demonstrate this on a set of popular benchmarks of embedded software.
We then provide a formal, quantitative characterization of the permissiveness of environment assumptions in the form of a weakness measure. We prove that the partial order induced by this measure is consistent with the one induced by implication. The key advantage of this measure is that it allows for prioritizing candidate solutions, as we show experimentally.
Lastly, we propose a notion of minimal refinements with respect to the observed counterstrategies. We demonstrate that exploring minimal refinements produces weaker solutions, and reduces the amount of computations needed to explore each refinement. However, this may come at the cost of reducing the effectiveness of the search. To counteract this effect, we propose a hybrid search approach in which both minimal and non-minimal refinements are explored.Open Acces
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