The Boundary of Prefix-Free Languages

Abstract

AbstractWe investigate the boundary operation on the class of prefix-free regular languages. We show that if a prefix-free language is recognized by a deterministic finite automaton of n states, then its boundary is recognized by a deterministic automaton of at most \((n-1)\cdot 2^{n-4}+n+1\) states. We prove that this bound is tight, and to describe worst-case examples, we use a three-letter alphabet. Next we show that the tight bound for boundary on binary prefix-free languages is \(2n-2\), and that in the unary case, the tight bound is \(n-2\).