380 research outputs found
Unambiguous Separators for Tropical Tree Automata
In this paper we show that given a max-plus automaton (over trees, and with real weights) computing a function f and a min-plus automaton (similar) computing a function g such that f ? g, there exists effectively an unambiguous tropical automaton computing h such that f ? h ? g.
This generalizes a result of Lombardy and Mairesse of 2006 stating that series which are both max-plus and min-plus rational are unambiguous. This generalization goes in two directions: trees are considered instead of words, and separation is established instead of characterization (separation implies characterization). The techniques in the two proofs are very different
Finitary languages
The class of omega-regular languages provides a robust specification language
in verification. Every omega-regular condition can be decomposed into a safety
part and a liveness part. The liveness part ensures that something good happens
"eventually". Finitary liveness was proposed by Alur and Henzinger as a
stronger formulation of liveness. It requires that there exists an unknown,
fixed bound b such that something good happens within b transitions. In this
work we consider automata with finitary acceptance conditions defined by
finitary Buchi, parity and Streett languages. We study languages expressible by
such automata: we give their topological complexity and present a
regular-expression characterization. We compare the expressive power of
finitary automata and give optimal algorithms for classical decisions
questions. We show that the finitary languages are Sigma 2-complete; we present
a complete picture of the expressive power of various classes of automata with
finitary and infinitary acceptance conditions; we show that the languages
defined by finitary parity automata exactly characterize the star-free fragment
of omega B-regular languages; and we show that emptiness is NLOGSPACE-complete
and universality as well as language inclusion are PSPACE-complete for finitary
parity and Streett automata
Near-Optimal Scheduling for LTL with Future Discounting
We study the search problem for optimal schedulers for the linear temporal
logic (LTL) with future discounting. The logic, introduced by Almagor, Boker
and Kupferman, is a quantitative variant of LTL in which an event in the far
future has only discounted contribution to a truth value (that is a real number
in the unit interval [0, 1]). The precise problem we study---it naturally
arises e.g. in search for a scheduler that recovers from an internal error
state as soon as possible---is the following: given a Kripke frame, a formula
and a number in [0, 1] called a margin, find a path of the Kripke frame that is
optimal with respect to the formula up to the prescribed margin (a truly
optimal path may not exist). We present an algorithm for the problem; it works
even in the extended setting with propositional quality operators, a setting
where (threshold) model-checking is known to be undecidable
Index problems for game automata
For a given regular language of infinite trees, one can ask about the minimal
number of priorities needed to recognize this language with a
non-deterministic, alternating, or weak alternating parity automaton. These
questions are known as, respectively, the non-deterministic, alternating, and
weak Rabin-Mostowski index problems. Whether they can be answered effectively
is a long-standing open problem, solved so far only for languages recognizable
by deterministic automata (the alternating variant trivializes).
We investigate a wider class of regular languages, recognizable by so-called
game automata, which can be seen as the closure of deterministic ones under
complementation and composition. Game automata are known to recognize languages
arbitrarily high in the alternating Rabin-Mostowski index hierarchy; that is,
the alternating index problem does not trivialize any more.
Our main contribution is that all three index problems are decidable for
languages recognizable by game automata. Additionally, we show that it is
decidable whether a given regular language can be recognized by a game
automaton
Ambiguity, Weakness, and Regularity in Probabilistic B\"uchi Automata
Probabilistic B\"uchi automata are a natural generalization of PFA to
infinite words, but have been studied in-depth only rather recently and many
interesting questions are still open. PBA are known to accept, in general, a
class of languages that goes beyond the regular languages. In this work we
extend the known classes of restricted PBA which are still regular, strongly
relying on notions concerning ambiguity in classical omega-automata.
Furthermore, we investigate the expressivity of the not yet considered but
natural class of weak PBA, and we also show that the regularity problem for
weak PBA is undecidable
Permutation Games for the Weakly Aconjunctive -Calculus
We introduce a natural notion of limit-deterministic parity automata and
present a method that uses such automata to construct satisfiability games for
the weakly aconjunctive fragment of the -calculus. To this end we devise a
method that determinizes limit-deterministic parity automata of size with
priorities through limit-deterministic B\"uchi automata to deterministic
parity automata of size and with
priorities. The construction relies on limit-determinism to avoid the full
complexity of the Safra/Piterman-construction by using partial permutations of
states in place of Safra-Trees. By showing that limit-deterministic parity
automata can be used to recognize unsuccessful branches in pre-tableaux for the
weakly aconjunctive -calculus, we obtain satisfiability games of size
with priorities for weakly aconjunctive
input formulas of size and alternation-depth . A prototypical
implementation that employs a tableau-based global caching algorithm to solve
these games on-the-fly shows promising initial results
Coalgebraic Trace Semantics for Buechi and Parity Automata
Despite its success in producing numerous general results on state-based dynamics, the theory of coalgebra has struggled to accommodate the Buechi acceptance condition---a basic notion in the
theory of automata for infinite words or trees. In this paper we present a clean answer to the question that builds on the "maximality" characterization of infinite traces (by Jacobs and Cirstea): the accepted language of a Buechi automaton is characterized by two commuting diagrams, one for a least homomorphism and the other for a greatest, much like in a system of (least and greatest) fixed-point equations. This characterization works uniformly for the nondeterministic branching and the probabilistic one; and for words and trees alike. We present our results in terms of the parity acceptance condition that generalizes Buechi\u27s
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