Maximal Existential and Universal Width1

The tree width of an alternating finite automaton (AFA) measures the parallelism in all computations of the AFA on a given input. The maximal existential (respectively, universal) width of an AFA A on string w measures the maximal number of existential choices (respectively, of parallel universal branches) in one computation of A on w. We give polynomial time algorithms deciding finiteness of an AFA's tree width and maximal universal width. Also we give a polynomial time algorithm that for an AFA A with finite maximal universal width decides whether or not the maximal existential width of A is finite. Finiteness of maximal existential width is decidable in the general case but the algorithm uses exponential time. Additionally, we establish necessary and sufficient conditions for an AFA to have exponential tree width growth rate, as well as sufficient conditions for an AFA to have exponential maximal existential width or exponential maximal universal width.