Counting functions can be defined syntactically or seman- tically 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 and #L, but for low-level complexity classes such as #AC⁰ and #NC¹ only the syntactical defi- nitions were considered. We give appropriate semantical definitions for these two classes and prove them to be equivalent to the syntactical ones. This enables us to show that #AC⁰ is included in the family of count- ing functions computed by polynomial size and constant width counting branching programs, therefore completing a result of Caussinus et al [CMTV98]. We also consider semantically defined probabilistic complex- ity classes corresponding to AC⁰ and NC¹ and prove that in the case of unbounded error, they are identical to their syntactical counterparts.