Hierarchies and reducibilities on regular languages related to modulo counting

Abstract

We discuss some known and introduce some new hierarchies and reducibilities on regular languages, with the emphasis on the quantifier-alternation and difference hierarchies of the quasi-aperiodic languages. The non-collapse of these hierarchies and decidability of some levels are established. Complete sets in the levels of the hierarchies under the polylogtime and some quantifier-free reducibilities are found. Some facts about the corresponding degree structures are established. As an application, we characterize the regular languages whose balanced leaf-language classes are contained in the polynomial hierarchy. For any discussed reducibility we try to give motivations and open questions, in a hope to convince the reader that the study of these reducibilities is interesting for automata theory and computational complexity.

Fine hierarchies and m-reducibilities in theoretical computer science

“…, x k ) of elements of P such that x 0 ≤ · · · ≤ x k and x i ∈ K iff x i+1 ∈ K , for each i < k. Such a chain is 1-alternating chain if x 0 ∈ K , otherwise it is called a 0-alternating chain. The next easy fact is from [145]. Proposition 3.4.…”

“…In the particular case when (P; ≤) is a well poset it is also possible [145] to give a similar characterization of the class BC (L) which is sometimes also of use. By ω-alternating chain for a set K ⊆ P we mean an ω-sequence Now we formulate two facts related to the structural properties introduced in Section 2.10.…”

“…This fact is probably new here, but it is checked in the same way as the known analogous facts in DST [92] and computability theory [119] which we state in the corresponding sections below. The second fact from [122,145] characterizes the ambiguous levels of the DH over bases with the separation property. It is useful in establishing the discreteness properties of the DH's.…”

“…It is known [155,136,145] that the class FO τ (FO τ d ) coincides with the class of so called regular quasi-aperiodic (respectively, d-quasi-aperiodic) languages. In [36,145] it was observed that FO τ = d FO τ d .…”

“…This makes these hierarchies similar to the hierarchies of sentences in Section 2.1, only this time one considers sentences modulo equivalence in finite models of a theory. More recently, people began to consider also fine hierarchies of regular languages [147,149,51,128,136,145] and reducibilities inducing nontrivial degree structures on the regular sets [10,167,158,48,136,145].…”