Weak alternating automata and tree automata emptiness

Automata on infinite words and trees are used for specification and verification of nonterminating programs. The verification and the satisfiability problems of specifications can be reduced to the nonemptiness problem of such automata. In a weak automaton, the state space is partitioned into partially ordered sets, and the automaton can proceed from a certain set only to smaller sets. Reasoning about weak automata is easier than reasoning about automata with no restricted structure. In particular, the nonemptiness problem for weak alternating automata over a singleton alphabet can be solved in linear time. Known translations of alternating automata to weak alternating automata involve determinization, and therefore involve a double exponential blow-up. In this paper we describe simple and efficient translations, which circumvent the need for determinization, of parity and Rabin alternating word automata to weak alternating word automata. Beyond the independent interest of such translations, they give rise to a simple algorithm for deciding the nonemptiness of nondeterministic parity and Rabin tree automata. In particular, our algorithm for Rabin automata runs in time O(n^2k+1·k!), where n is the number of states in the automaton and k is the number of pairs in the acceptance condition. This improves the known O((nk)^3k) bound for the problem.