Semantical counting circuits

Counting functions can be defined syntactically or semantically depending on whether they count the number of witnesses in a non-deterministic or in a deterministic computation on the input. In the Turing machine based model, these two ways of defining counting were proven to be equivalent for many important complexity classes. In the circuit based model, it was done for #P, but for low-level complexity classes such as #AC 0 and #NC 1 only the syntactical definitions were considered. We give appropriate semantical definitions for these two classes and prove them to be equivalent to the syntactical ones. We also consider semantically defined probabilistic complexity classes corresponding to AC 0 and NC 1 and prove that in the case of unbounded error, they are identical to their syntactical counterparts.