Counting and empty alternating pushdown automata

We show that the class of context free languages CFL is not equal to ⊕1−PDA (= ⊕CFL), the class of languages, which are recognized by a nondeterministic oneway push-down automaton equipped with parity acceptance. Furthermore we show that LOG(⊕CFL) = ⊕AuxPDApt contains all languages, which can be recognized by a uniform weak unambiguous AC1-circuit introduced in [LR90a]. Therefore, it contains all languages, which can be recognized by a CREW -PRAM. We show, that L#Aux PDApt is contained in uniform TC1, which sharpens a result in [Vin91], where inclusion in NC2 was shown. At last we show, that AC, SAC and P can be characterized as the logarithmic closure of certain types of languages. These languages are either those given by linear alternating grammars or they are languages which are recognized by alternating pushdown automata with the restriction that their storage has to be empty if they make an alternating transition. In particular, we consider the cases when the depth of alternation is bounded by a constant or a polylogarithmic function.