Ambiguity and Communication

TL;DR

This paper investigates the ambiguity of nondeterministic finite automata (NFA), constructing language hierarchies based on ambiguity levels and demonstrating exponential size requirements for certain ambiguity constraints, thus resolving a long-standing open problem.

Contribution

It introduces a hierarchy for polynomial ambiguity in NFAs and proves exponential size lower bounds, solving a longstanding open problem in automata theory.

Findings

Established a hierarchy for polynomial ambiguity in NFAs.

Proved exponential size lower bounds for NFAs with restricted ambiguity.

Solved a long-standing open problem from Ravikumar and Ibarra (1989).

Abstract

The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for an input of size n. For all k, r 2 N we construct languages Lr,k which can be recognized by NFA's with size k poly(r) and ambiguity O(nk), but Lr,k has only NFA's with exponential size, if ambiguity o(nk) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, 1989, Leung, 1998).