Computation Widths for Alternating Finite Automata

Abstract

Alternating finite automata (AFA) were first introduced in 1981, and are a generalization of nondeterministic finite automata (NFA), which utilizes both nondeterminism and parallelism for describing regular languages. Though AFAs describe the same set of languages as NFAs and DFAs (deterministic finite automata), they are capable of doing so in a very succinct way; in the extreme case exhibiting exponential savings in state complexity compared to an equivalent NFA, and double exponential savings when compared to an equivalent DFA. This makes AFAs very useful for representing regular languages; however understanding which regular languages benefit the most from this representation is still not very well understood. To this end, one can study the measures known as existential and universal width, which respectively limit the number of existential (i.e., nondeterministic) choices and the amount of universal parallelism in computations of an AFA. In this thesis, we will be tackling the problems of computing existential and universal width for AFAs, as well as related problems such as deciding whether the width of an AFA (be it existential or universal) is less than a given integer k. These problems have been studied before using the best case measures known as optimal existential width and optimal universal width, and all found to be at least PSPACE-hard. Therefore, we instead choose to consider the variants known as maximal existential and universal width, and demonstrate that the same problems in fact behave rather differently, and yield much more efficient algorithms. In particular, we provide a polynomial time algorithm for computing the maximal existential width of an NFA, as well as the maximal universal width of an AFA under the assumption that its maximal universal width is bounded by a constant l. Additionally, in the unary case we show that both the maximal universal width and maximal existential width of an AFA can be computed in cubic time. We also show that the problem of deciding whether the maximal universal width of a general AFA is less than an integer k is at least NP-hard, which stands as the only hardness result currently known for the maximal widths.