Finite Automata with Colored Accepting States and Their Unmixedness Problems

Abstract

Some textbooks of formal languages and automata theory implicitly state the structural equality of the binary n-dimensional de Bruijn graph and the state diagram of minimum state deterministic finite automaton which accepts regular language (0+1)^*1(0+1)^n-1. By introducing special finite automata whose accepting states are refined with two or more colors, we extend this fact to both k-ary versions. That is, we prove that k-ary n-dimensional de Brujin graph and the state diagram for minimum state deterministic colored finite automaton which accepts the (k-1)-tuple of the regular languages (0+1+…+k-1)^*1(0+1+…+k-1)^n-1,...,and(0+1+…+k-1)^*(k-1)(0+1+…+k-1)^n-1 are isomorphic for arbitrary k more than or equal to 2. We also investigate the properties of colored finite automata themselves and give computational complexity results on three decision problems concerning color unmixedness of nondeterminisitic ones.