Assume that we are interested in deterministic circuits of size $f(n)$. Here, $n$ represents the number of inputs to the circuit.The standard way of defining probabilistic/nondeterministic circuits is to augment the input with inputs. Let's assume that the circuit is still of size $f(n)$. What is $n$ in this case? $n$ is the number of deterministic inputs. $n$ is the number of deterministic PLUS probabilistic/nondeterministic inputs.My confusion is that some references consider case 1, while others consider case 2.
On Defining Probabilistic/Nondeterministic Circuits
cc.complexity theory;circuit complexity;nondeterminism;probabilistic circuits
I found the following text in [1] (cited below); it shows that---as Ryan pointed out---there must be some function limiting the number of nondeterministic/random inputs; unless it can be judged from other things without ambiguity:We develop nondeterministic $\mathbf{NC}$ ($\mathbf{NC}$) circuits by uniformly adding guessing gates to families of $\mathbf{LOGSPACE}$-uniform $\mathbf{NC}$ circuits. Let $\mathbf{NNC}(f(n))$ be the class of languages accepted by $\mathbf{LOGSPACE}$-uniform families of $\mathbf{NC}$ circuits with $O(f(n))$ nondeterministic gates or guess gates, where $n$ is the length of the input. We will refer to a circuit from such a family as a uniform $\mathbf{NNC}$ circuit, and such a circuit accepts an input if it outputs a 1. The guessing gates give the circuit a set of guessing inputs, $y$, in addition to the ordinary inputs, $x$. Note that $|x|=n$ and $|y|=f(n)$. An $\mathbf{NNC}$ circuit is said to accept $x$ if and only if there is some string of guessing bits $y$ that causes the circuit to output a 1. Thus, $f(n)$ can be thought of as the maximum number of guess bits used in computations on inputs of length $n$.[1] Wolf, M. J. 1994. Nondeterministic circuits, space complexity and quasigroups. Theor. Comput. Sci. 125, 2 (Mar. 1994), 295-313. DOI= http://dx.doi.org/10.1016/0304-3975(92)00014-I