On Emptiness and Counting for Alternating Finite Automata

We investigate problems for regular languages coded by alternating nite automata. In automata and formal language theory, it is well-known that there is a language accepted by an n-state nondeterministic nite automaton but any deterministic nite automaton accepting that language must have at least 2 n states. Diierent representations of regular languages are used to deene problems of varying levels of diiculty, i.e., the complexity of problems increases when their instances are described in a more compact way. Examples of this phenomenon can be found, e.g., in Jiang and Ravikumar 2] and Jones 3]. The complexity of problems for deterministic and nondeterministic nite automata is in most cases well determined, but only few results are known for alternating nite automata. One classical problem of complexity theory is emptiness. For both deter-ministic and nondeterministic nite automata the exact complexity, i.e., NL-completeness, was determined by Jones 3]. In fact, Jones proved that even for automata over a unary input alphabet the problem remains NL-complete. In case of alternating nite automata, a partial solution to emptiness was given by Jiang and Ravikumar 2]. Here, partial solution means that the unary input alphabet case was left open. We show that emptiness for alternating nite automata over a unary input alphabet is as hard as emptiness of extended 0L Lindenmayer languages. The complexity of the latter problem was a long open standing problem (Lange and Schudy 5]), but was recently solved by Monti and Roncato 6] in an "`unex-pected"' way. Using this result we obtain: Theorem1. The emptiness problem for alternating nite automata is PSpace-complete. PSpace-completeness holds even for alternating nite automata with unary input alphabet. We mention that the complexity of emptiness for alternating nite automata remains unchanged if one generalizes the problem to, e.g., intersection emptiness, or restrict it to, e.g., alternating nite automata where every state is determin-istic except for one state which is universal. A problem closely connected with emptiness is the so-called short word problem which was investigated by Lange and Rossmanith 4]. For both deterministic and nondeterministic nite automata this problem can be used to characterize classes of bounded nondeterminism. The main idea underlying short words is that in general the shortest word accepted of an alternating nite automaton ?