3 research outputs found
Undirected Graphs of Entanglement Two
Entanglement is a complexity measure of directed graphs that origins in fixed
point theory. This measure has shown its use in designing efficient algorithms
to verify logical properties of transition systems. We are interested in the
problem of deciding whether a graph has entanglement at most k. As this measure
is defined by means of games, game theoretic ideas naturally lead to design
polynomial algorithms that, for fixed k, decide the problem. Known
characterizations of directed graphs of entanglement at most 1 lead, for k = 1,
to design even faster algorithms. In this paper we present an explicit
characterization of undirected graphs of entanglement at most 2. With such a
characterization at hand, we devise a linear time algorithm to decide whether
an undirected graph has this property
The Variable Hierarchy for the Games mu-Calculus
Parity games are combinatorial representations of closed Boolean mu-terms. By
adding to them draw positions, they have been organized by Arnold and one of
the authors into a mu-calculus. As done by Berwanger et al. for the
propositional modal mu-calculus, it is possible to classify parity games into
levels of a hierarchy according to the number of fixed-point variables. We ask
whether this hierarchy collapses w.r.t. the standard interpretation of the
games mu-calculus into the class of all complete lattices. We answer this
question negatively by providing, for each n >= 1, a parity game Gn with these
properties: it unravels to a mu-term built up with n fixed-point variables, it
is semantically equivalent to no game with strictly less than n-2 fixed-point
variables
The Variable Hierarchy for the Games mu-Calculus
To appear in the journal Annals of Pure and Applied LogicInternational audienceParity games are combinatorial representations of closed Boolean mu-terms. By adding to them draw positions, they have been organized by Arnold and one of the authors into a mu-calculus. As done by Berwanger et al. for the propositional modal mu-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games mu-calculus into the class of all complete lattices. We answer this question negatively by providing, for each n >= 1, a parity game Gn with these properties: it unravels to a mu-term built up with n fixed-point variables, it is semantically equivalent to no game with strictly less than n-2 fixed-point variables