Counting with Counterfree Automata

We study the power of balanced regular leaf-languages. First, we investigate (i) regular languages that are polylog-time reducible to languages in 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. Both classes are decidable. The intersection of class (i) with their complement is exactly class (ii). We apply our observations and obtain 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. Also we 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 they cannot count). We exploit a weak type of counting possible with counterfree automata, and construct languages of arbitrary dot-depth that are reducible to languages in dot-depth 1/2. 3. Unbalanced starfree leaf-languages can be much stronger than balanced ones. We construct starfree regular languages Ln such that the balanced leaf-language class of Ln is contained in NP, but the unbalanced leaf-language class of Ln contains level n of the unambiguous alternation hierarchy. This demonstrates the power of unbalanced computations.