On Separating Constant from Polynomial Ambiguity of Finite Automata: Extended Abstract

The degree of nondeterminism of a finite automaton can be measured by means of its ambiguity function. In many instances, whenever automata are allowed to be (substantially) less ambiguous, it is known that the number of states needed to recognize at least some languages increases exponentially. However, when comparing constantly ambiguous automata with polynomially ambiguous ones, the question whether there are languages such that the inferior class of automata requires exponentially many states more than the superior class to recognize them is still an open problem. The purpose of this paper is to suggest a family of languages that seems apt for a proof of this (conjectured) gap. As a byproduct, we derive a new variant of the proof of the existence of a superpolynomial gap between polynomial and fixed-constant ambiguity. Although our candidate languages are defined over a huge alphabet, we show how to overcome this drawback.