Chrobak normal form revisited, with applications

Abstract

It is well known that any nondeterministic finite automata over a unary alphabet can be represented in a certain normal form called the Chrobak normal form [1]. We present a very simple conversion procedure working in O(n 3) time. Then we extend the algorithm to improve two trade-offs concerning conversions between different representations of unary regular languages. Given an n-state NFA, we are able to find a regular expression of size O(n2/log2n) describing the same language (which improves the previously known O(n2) size bound [8]) and a context-free grammar in Chomsky normal form with O(√n log n) nonterminals (which improves the previously known O(n2/3) bound [3]). © 2011 Springer-Verlag.