Some Minimality Results on Biresidual and Biseparable Automata

Residual finite state automata (RFSA) are a subclass of non-deterministic finite automata (NFA) with the property that every state of an RFSA defines a residual language of the language accepted by the RFSA. Recently, a notion of a biresidual automaton (biRESA) an RFSA such that its reversal automaton is also an RFSA was introduced by Latteux; Roos, and Terlutte, who also showed that a subclass of biRFSAs called biseparable automata consists of unique state-minimal NFAs for their languages. In this paper, we present some new minimality results concerning biRFSAs and biseparable automata. We consider two lower bound methods for the number of states of NFAs the fooling set and the extended fooling set technique and present two results related to these methods. First, we show that the lower bound provided by the fooling set technique is tight for and only for biseparable automata. And second, we prove that the lower bound provided by the extended fooling set technique is tight for any language accepted by a biRFSA. Also, as a third result of this paper, we show that any reversible canonical biRFSA is a transition-minimal is an element of-NFA. To prove this result, the theory of transition-minimal is an element of-NFAs by S. John is extended.