Combining Limited Parallelism and Nondeterminism in Alternating Finite Automata

Abstract

We introduce the existential width measure (respectively, the maximal existential width), which, roughly speaking, for an alternating finite automaton (AFA), counts the number of branches which do not need to be traversed in an accepting computation (respectively, the maximum number of branches which can be ignored in any computation tree of the AFA). We also define the combined width (respectively, the maximal combined width), by combining this new measure with an existing measure, the universal width (respectively, the maximal universal width), which counts the minimum number of branches of a computation tree which must be traversed for an AFA to accept a computation (respectively, the maximum number of branches which can be traversed in any computation tree of the AFA). We give a polynomial algorithm to decide whether the (maximal) combined width is bounded, and a construction showing that an AFA with finite combined width can be simulated by an NFA with only a polynomial blow-up in the number of states. We also improve the upper bound for deciding finiteness of an m-state NFA’s tree width from $$O(m^3)$$ to $$O(m^2)$$ .