Languages polylog-time reducible to dot-depth 1/2

Abstract

We study (i) regular languages that are polylog-time reducible to languages of dot-depth 1/2 and (ii) regular languages that are polylog-time decidable. For both classes we provide

• forbidden-pattern characterizations, and

• characterizations in terms of regular expressions.

This implies that both classes are decidable. In addition, we show that a language is in class (ii) if and only if the language and its complement are in class (i). Our observations have three consequences.

• Gap theorems for balanced regular-leaf-language definable classes $\mathcal{C}$ and $\mathcal{D}$:

  • Either $\mathcal{C}$ is contained in NP, or $\mathcal{C}$ contains coUP.

  • Either $\mathcal{D}$ is contained in P, or $\mathcal{D}$ contains UP or coUP.

  We also extend both theorems such that no promise classes are involved. Formerly, such gap theorems were known only for the unbalanced approach.

• Polylog-time reductions can tremendously decrease dot-depth complexity (despite that these reductions cannot count). We construct languages of arbitrary dot-depth that are reducible to languages of dot-depth 1/2.

• Unbalanced star-free leaf languages can be much stronger than balanced ones. We construct star-free regular languages $L_n$ such that $L_n$'s balanced leaf-language class is NP, but the unbalanced leaf-language class of $L_n$ contains level n of the unambiguous alternation hierarchy. This demonstrates the power of unbalanced computations.