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The circuit evaluation problem is the computational problem of calculating the output of a given Boolean circuit given an input.

For a circuit C and input string w, the value of C on w can be written as$C\left(w\right)$.

Suppose,.

Consider the theorem,

Suppose$t\text{:}M鈫�M$ be a function, where $t\left(m\right)猢�m$.

If $w鈭�\text{TIME}\left(t\left(m\right)\right)$, then the complexity of circuit A is given by $O\left(t^2\left(m\right)\right)$.

According to the above theorem,

On input w, the production of a circuit takes place by the reduction. The process of reduction simulates the Turning Machine for w in polynomial time. The w itself can be taken as an input to the circuit.

The above theorem shows the way of reduction of a language w

(Which is in P) to$\text{CIRCUIT}-\text{VALUE}$.

A $\log space$is used to carry out the reduction because the circuit produced by it contains a repetitive and a simple structure.

Hence, it shows that CIRCUIT-VALUE is P complete and circuit produced by it has a repetitive structure. Therefore, if $A{猢�}_{L}B$ and B is in NC then A is in NC.