Minimal NFA and biRFSA Languages

In this paper, we define the notion of biRFSA which is a residual finate state automaton (RFSA) whose the reverse is also an RFSA. The languages recognized by such automata are called biRFSA languages. We prove that the canonical RFSA of a biRFSA language is a minimal NFA for this language and that each minimal NFA for this language is a sub-automaton of the canonical RFSA. This leads to a characterization of the family of biRFSA languages. In the second part of this paper, we define the family of biseparable automata. We prove that every biseparable NFA is uniquely minimal among all NFAs recognizing a same language, improving the result of H. Tamm and E. Ukkonen for bideterministic automata.