Rabin Index Research Articles

The Rabin index of parity games: Its complexity and approximation

We study the descriptive complexity of parity games by taking into account the coloring of their game graphs whilst ignoring their ownership structure. Colorings of game graphs are identified if they determine the same winning regions and strategies, for all ownership structures of nodes. The Rabin index of a parity game is the minimum of the maximal color taken over all equivalent coloring functions. We show that deciding whether the Rabin index is at least k is in P for k=1 but NP-hard for all fixedk≥2. We present an EXPTIME algorithm that computes the Rabin index by simplifying its input coloring function. When replacing simple cycle with cycle detection in that algorithm, its output over-approximates the Rabin index in polynomial time. We evaluate this efficient algorithm as a preprocessor of solvers in detailed experiments: for Zielonka's solver on random and structured parity games and for our partial solver psolB on random games.

The Rabin index of parity games

We study the descriptive complexity of parity games by taking into account the coloring of their game graphs whilst ignoring their ownership structure. Colored game graphs are identified if they determine the same winning regions and strategies, for all ownership structures of nodes. The Rabin index of a parity game is the minimum of the maximal color taken over all equivalent coloring functions. We show that deciding whether the Rabin index is at least k is in PTIME for k=1 but NP-hard for all fixed k > 1. We present an EXPTIME algorithm that computes the Rabin index by simplifying its input coloring function. When replacing simple cycle with cycle detection in that algorithm, its output over-approximates the Rabin index in polynomial time. Experimental results show that this approximation yields good values in practice.

Computing the Rabin Index of a Parity Automaton

The Rabin index of a rational language of infinite words given by a parity automaton with n states is computable in time O(n2c) where c is the cardinality of the alphabet. The number of values used by a parity acceptance condition is always greater than the Rabin index and conversely, the acceptance condition of a parity automaton can always be replaced by an equivalent acceptance condition whose number of used values is exactly the Rabin index. This new acceptance condition can also be computed in time O(n2c).

Computing the Rabin Index of a Regular Language of Infinite Words

The Rabin index of a regular language of infinite words is the minimum number of accepting pairs used in any deterministic Rabin automaton recognizing this language. We show that the Rabin index of a language given by a Muller automaton withnstates andmaccepting sets is computable in timeO(m2nc) wherecis the cardinality of the alphabet.

Chain automata

We introduce a new version of Rabin automata for which the construction of an automaton for the union, the intersection and the complement is very simple. Furthermore, these automata admit a natural reduction of their acceptance condition. This leads to a new characterization of the Rabin index of an ω-language.

Hierarchies of weak automata and weak monadic formulas

The paper deals with alternating finite automata (a.f.a.) on trees. Rabin indices and Muller indices for automata, which are well known in relevant literature, are a natural measure of complexity of the automata. Muller index (or Rabin index) of a set of trees is a minimal index of a Muller (or Rabin) automaton which represents this set. In this paper it is proved that both indices, Rabin and Muller, are equal for weak as well as for strong conditions