Fixed Point Characterization of Infinite Behavior of Finite-State Systems

Innnite behavior of nondeterministic nite state automata running over innnite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of xed point terms interpreted in powerset algebras. These terms involve the least and greatest xed point operators and disjunction as the only logical operation. A tight correspondence is established between a hierarchy of Rabin indices of automata and a hierarchy induced by alternation of the least and greatest xed point operators. It is shown that, in the powerset algebra of trees constructed from a set of functional symbols, the xed point hierarchy is innnite unless all the symbols are unary (i.e. trees are words). It is also shown that an interpretation of a closed xed point term in any powerset algebra can be factorized through the interpretation of this term in the powerset algebra of trees, from which it is deduced that the question whether a term denotes always ; can be answered in polynomial time.