Jump complexity of finite automata with translucent letters

Abstract

We define the jump complexity of a finite automaton with translucent letters as a function that computes the smallest upper bound on the number of jumps needed by the automaton in order to accept each word of length n, for any positive integer n. We prove that a sufficient condition for a finite automaton with translucent letters to accept a regular language is to have a jump complexity bounded by a constant. Along the same lines, we show that there are languages which require a jump complexity in Ω(n) of any finite automaton with translucent letters accepting one of these languages. We also show that there exist nondeterministic finite automata with translucent letters of jump complexity in O(log⁡n) and O(n) that accept non-regular languages. Several open problems and directions for further developments are finally discussed.