Descriptional Complexity of (Un)ambiguous Finite State Machines and Pushdown Automata

Unambiguity and its generalization to quantified ambiguity are important concepts in, e.g., automata and complexity theory. Basically, an unambiguous machine has at most one accepting computation path for each accepted word. While unambiguous pushdown automata induce a language family strictly in between the deterministic and general context-free languages, unambiguous finite automata capture the regular languages, that is, they are equally powerful as deterministic and nondeterministic finite automata. However, their descriptional capacity is significantly different. In the present paper, we summarize and discuss developments relevant to (un)ambiguous finite automata and pushdown automata problems from the descriptional complexity point of view. We do not prove these results but we merely draw attention to the big picture and some of the main ideas involved.