A taxonomy of problems with fast parallel algorithms
Information and Control
The complexity of Boolean functions
The complexity of Boolean functions
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
A catalog of complexity classes
Handbook of theoretical computer science (vol. A)
Circuit definitions of nondeterministic complexity classes
SIAM Journal on Computing
A very hard log-space counting class
Theoretical Computer Science - Special issue on structure in complexity theory
Gap-definable counting classes
Journal of Computer and System Sciences
Descriptive complexity of #P functions
Journal of Computer and System Sciences
Recursion theoretic characterizations of complexity classes of counting functions
Theoretical Computer Science
Nondeterministic NC1 computation
Journal of Computer and System Sciences - Eleventh annual conference on structure and complexity 1996
Verifying the determinant in parallel
Computational Complexity
On Counting AC0 Circuits with Negative Constants
MFCS '98 Proceedings of the 23rd International Symposium on Mathematical Foundations of Computer Science
Bounded Depth Arithmetic Circuits: Counting and Closure
ICAL '99 Proceedings of the 26th International Colloquium on Automata, Languages and Programming
On TC/sup 0/, AC/sup 0/, and Arithmetic Circuits
CCC '97 Proceedings of the 12th Annual IEEE Conference on Computational Complexity
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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 and #L, but for low-level complexity classes such as #AC0 and #NC1 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. This enables us to show that #AC0 is included in the family of counting 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 complexity classes corresponding to AC0 and NC1 and prove that in the case of unbounded error, they are identical to their syntactical counterparts.