Weak Parity Games and Language Containment of Weak Alternating Parity Automata

Optimisations of the state space of weak alternating parity automata (WAPA) are needed for example to speed up μTL model-checking algorithms which involve WAPA. It is assumed that deciding language containment for WAPA is helpful to perform such optimisations. In this paper the problem of language containment is reduced to the problem of computing winning sets in weak parity games. For the latter a linear time algorithm is presented. This section gives some notes on previous work on language containment and optimisations of finite automata on finite and infinite words as well as an overview of the text. In the second section one finds the definition of parity games and weak parity games as well as an algorithm that computes the winning sets in weak parity games in linear time. This matches the best known algorithms for deciding winning sets in this games. An algorithm that computes winning set in arbitrary parity games can be found in [Jur00]. As a special case the algorithm of Jurdzinski ([Jur00]) solves weak parity games in linear time too but it is not as intuitive and easy too implement as the algorithm which is presented here. In the third section the definition of WAPA is repeated and some notes on the optimisation of the state space with the help of language containment in this automata are made. The last section describes two different reductions of language containment of WAPA into deciding winning sets in weak parity games which delivers a polynomial time algorithm for language containment that is sound but not compete and a PSPACE algorithm for language containment that is sound and complete. The algorithms have been implemented in the μSabre project of Martin Lange and Jan Johannsen at the University of Munich. It is well known that the problem to decide weather L(A1) is a subset of L(A2) or not for given deterministic finite automaton (DFA) A1 and A2 is quite easy. One only has to construct a DFA for the language L(A1)∩L(A2) and test if there is a final state reachable from the initial state. In fact this can be done in time O(n2). An algorithm is for example given in [Hop02]. A generalisation to nondeterministic finite automaton (NFA) makes the language containment problem quite difficult: language containment for NFA is