NFAs With and Without ε-Transitions

Abstract

The construction of an ε-free nondeterministic finite automaton (NFA) from a given NFA is a basic step in the development of compilers and computer systems. The standard conversion may increase the number of transitions quadratically and its optimality with respect to the number of transitions is a long standing open problem. We show that there exist regular languages L _n that can be recognized by NFAs with O(n log₂ n) transitions, but ε-free NFAs need Ω(n ²) transitions to accept L _n . Hence the standard conversion cannot be improved significantly. However L _n requires an alphabet of size n, but we also construct regular languages K _n over {0,1} with NFAs of size O(n log₂ n), whereas ε-free NFAs require size $n \cdot 2^{c \cdot\sqrt{{\rm log}_{2}n}}$ for every c < 1/2.

Supported in part by SNF grant 200021-107327/1 and DFG grant SCHN 503/2-2.