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. (1) Gap theorems for balanced regular-leaf-language definable classes C and D : (a) Either C is contained in NP, or C contains coUP. (b) Either D is contained in P, or 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. (2) 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. (3) 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.
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