Theorem 2.2 in Nondeterministic circuits, space complexity and quasigroups, by Wolf, 1994 (a technical report version is available here without fee), proves that NP = NNC, where NNC is the class of languages decidable by an L-uniform family of nondeterministic NC circuits. A nondeterministic NC circuit is an NC circuit with a polynomial number of nondeterministic input bits. The simulation essentially guesses a tableau of an arbitrary NP machine then verifies that each configuration proceeds from the previous one according to the transition function of the machine. My interpretation of this proof is that the NC circuit is in fact an NC1 circuit, since it simply takes the conjunction of a polynomial number of subcircuits, each of which checks that a local window in the tableau is valid.Assuming that interpretation is correct, it should follow that NP = NL, since NC1 is in L, so the same argument will apply. What is the error in my interpretation of this result? Why doesn't the strategy guess a tableau, verify that each configuration follows from the previous work?
Is $\mathsf{NP}$ in $\mathsf{NNC}^1$?
cc.complexity theory;complexity classes