Predicate Characterizations in the Polynomial-Size Hierarchy

Abstract

The polynomial-size hierarchy is the hierarchy of ‘minicomplexity’ classes which correspond to two-way alternating finite automata with polynomially many states and finitely many alternations. It is defined by analogy to the polynomial-time hierarchy of standard complexity theory, and it has recently been shown to be strict above its first level.

It is well-known that, apart from their definition in terms of polynomial-time alternating Turing machines, the classes of the polynomial-time hierarchy can also be characterized in terms of polynomial-time predicates, polynomial-time oracle Turing machines, and formulas of second-order logic. It is natural to ask whether analogous alternative characterizations are possible for the polynomial-size hierarchy, as well.

Here, we answer this question affirmatively for predicates. Starting with the first level of the hierarchy, we experiment with several natural ways of defining what a ‘polynomial-size predicate’ should be, so that existentially quantified predicates of this kind correspond to polynomial-size two-way nondeterministic finite automata. After reaching an appropriate definition, we generalize to every level of the hierarchy.