36 research outputs found
Complementing deterministic BĂźchi automata in polynomial time
AbstractFor any Buchi automaton Î with n states which accepts the (Ď-regular) language L(Î), an explicit construction is given for a BĂźchi automaton Î with 2n states which, when Î is deterministic, accepts exactly the complementary language L(Î)â˛. It follows that the nonemptiness of complement problem for deterministic Buchi automata (i.e., whether L(Î)Ⲡ= â) is solvable in polynomial time. The best previously known construction for complementing a deterministic BĂźchi automaton with n states has O(24n2) states; for nondeterministic Î, determining whether L(Î)Ⲡ= â, is known to be PSPACE-complete. Interest in deterministic BĂźchi automata arises from the suitability of deterministic automata in general to describe properties of physical systems; such properties have been found to be more naturally expressible by deterministic automata than by nondeterministic automata. However, if Î is nondeterministic, then Î provides a âpoor man'sâ approximate inverse to Î in the following sense: L(Î)Ⲡâ L(Î), and as nondeterministic branches of T are removed, the two languages become closer. Hence, for example, given two nondeterministic Buchi automata Î and Î, one may test for containment of their associated languages through use of the corollary that L (Î â Î = â â L (Î) â L(Î) (where Î â Î is one of the standard constructions satisfying L (Î â Î) = L (Î) ⊠L(Î)). The âerror termâ L = L(Î) ⧚ L(Î)Ⲡmay be deter exactly, and whether L = â may be determined in time O(e2), where e is the number of edges of Î
Looking at Mean-Payoff through Foggy Windows
Mean-payoff games (MPGs) are infinite duration two-player zero-sum games
played on weighted graphs. Under the hypothesis of perfect information, they
admit memoryless optimal strategies for both players and can be solved in
NP-intersect-coNP. MPGs are suitable quantitative models for open reactive
systems. However, in this context the assumption of perfect information is not
always realistic. For the partial-observation case, the problem that asks if
the first player has an observation-based winning strategy that enforces a
given threshold on the mean-payoff, is undecidable. In this paper, we study the
window mean-payoff objectives that were introduced recently as an alternative
to the classical mean-payoff objectives. We show that, in sharp contrast to the
classical mean-payoff objectives, some of the window mean-payoff objectives are
decidable in games with partial-observation
The Infimum Problem as a Generalization of the Inclusion Problem for Automata
This thesis is concerned with automata over infinite trees. They are given a labeled infinite tree and accept or reject this tree based on its labels. A generalization of these automata with binary decisions are weighted automata. They do not just decide 'yes' or 'no', but rather compute an arbitrary value from a given algebraic structure, e.g., a semiring or a lattice. When passing from unweighted to weighted formalisms, many problems can be translated accordingly. The purpose of this work is to determine the feasibility of solving the inclusion problem for automata on infinite trees and its generalization to weighted automata, the infimum aggregation problem
An Efficient Normalisation Procedure for Linear Temporal Logic and Very Weak Alternating Automata
In the mid 80s, Lichtenstein, Pnueli, and Zuck proved a classical theorem
stating that every formula of Past LTL (the extension of LTL with past
operators) is equivalent to a formula of the form , where
and contain only past operators. Some years later, Chang,
Manna, and Pnueli built on this result to derive a similar normal form for LTL.
Both normalisation procedures have a non-elementary worst-case blow-up, and
follow an involved path from formulas to counter-free automata to star-free
regular expressions and back to formulas. We improve on both points. We present
a direct and purely syntactic normalisation procedure for LTL yielding a normal
form, comparable to the one by Chang, Manna, and Pnueli, that has only a single
exponential blow-up. As an application, we derive a simple algorithm to
translate LTL into deterministic Rabin automata. The algorithm normalises the
formula, translates it into a special very weak alternating automaton, and
applies a simple determinisation procedure, valid only for these special
automata.Comment: This is the extended version of the referenced conference paper and
contains an appendix with additional materia