Recently, we introduced a new automata model, so-called colored finite automata (CFAs) whose accepting states are multi-colored (i.e., not conventional black-and-white acceptance) in order to classify their input strings into two or more languages, and solved the specific complexity problems concerning color-unmixedness of nondeterministic CFA. That is, so-called UV, UP, and UE problems are shown to be NLOG-complete, P, and NP-complete, respectively. In this paper, we apply the concept of colored accepting mechanism to pushdown automata and show that the corresponding versions of the above-mentioned complexity problems are all undecidable. We also investigate the case of unambiguous pushdown automata and show that one of the problems turns out to be permanent true (the others remain undecidable).

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.