Prefix-free languages: Left and right quotient and reversal

Abstract

We investigate the left and right quotient, and the reversal operation on the class of prefix-free regular languages. We get the tight upper bounds 2n−1, n−1, and 2n−2+1 on the state complexity of these three operations, respectively. To prove the tightness of these bounds, we use an (n−1)-letter alphabet for left quotient, a binary alphabet for right quotient, and a ternary alphabet for reversal. We also prove that these bounds cannot be met using languages defined over any smaller alphabet. For left quotient, we prove that the tight bound for an (n−2)-letter alphabet is 2n−1−1, and we provide exponential lower bounds for every smaller alphabet, except for the unary case. For the reversal operation on binary prefix-free languages, we get 2n−2−7 lower bound in the case of nmod3≠2, and 2n−2−15 lower bound in the remaining cases. We conjecture that our lower bounds on the state complexity of reversal on binary prefix-free languages are tight if n≥12. Our experimental results support this conjecture.