Residual Finite State Automata

We introduce a subclass of non deterministic finite automata (NFA) that we call Residual Finite State Automata (RFSA): a RFSA is a NFA all the states of which define residual languages of the language it recognizes. We prove that for every regular language L, there exists a unique RFSA that recognizes L and which has both a minimal number of states and a maximal number of transitions. Moreover, this canonical RFSA may be exponentially smaller than the equivalent minimal DFA but it also may have the same number of states as the equivalent minimal DFA, even if minimal equivalent NFA are exponentially smaller. We provide an algorithm that computes the canonical RFSA equivalent to a given NFA. We study the complexity of several decision and construction problems linked to the class of RFSA: most of them are PSPACE-complete.