Extremal minimality conditions on automata

In this paper we investigate the minimality problem of DFAs by varying the set of final states. In other words, we are interested on how the choice of the final states can affect the minimality of the automata. The state-pair graph is a useful tool to investigate such a problem. The choice of a set of final states for the automaton A defines a coloring of the closed components of the state-pair graph and the minimality of A corresponds to a property of these colored components. A particular attention is devoted to the analysis of some extremal cases such as, for example, the automata that are minimal for any choice of the subset of final states F from the state set Q of the automaton (uniformly minimal automata), the automata that are minimal for any proper subset F of Q (almost uniformly minimal automata) and the automata that are never minimal, under any assignment of final states (never-minimal automata). More generally, we seek to characterize those families of automata and show that some of them are related to well-known objects in a different context (e.g. multiple-entry automata and Fischer covers of irreducible sofic shifts in Symbolic Dynamics). Next, we study the complexity of the related decision problems and show, in some cases, how to derive a polynomial algorithm. Finally, we pay particular attention to the relationship between the problem to decide if an automaton is never-minimal and the "syntactic monoid problem". (C) 2012 Elsevier B.V. All rights reserved.