Value Constraint and Monotone circuit

1. Overview This paper talks about that monotone circuit is P-Complete. Decision problem that include P-Complete is mapping that classify input with a similar property. Therefore equivalence relation of input value is important for computation. But monotone circuit cannot compute the equivalence relation of the value because monotone circuit can compute only monotone function. Therefore, I make the value constraint explicitly in the input and monotone circuit can compute equivalence relation. As a result, we can compute P-Complete problem with monotone circuit. We can reduce implicit value constraint to explicit with logarithm space. Therefore, monotone circuit is P-Complete. 2. Monotone circuit and Equivalence relation First, I show that a monotone circuit cannot compute equivalence relation. Decision problem that include P-Complete is mapping of {0, 1} * → {0, 1} and classify {0, 1} * to two sets. It is important to classify {0, 1} as the element of {0, 1} *. But monotone circuit cannot classify {0, 1}. Theorem 1. Monotone circuit cannot classify equivalence relation. Proof. I prove it using reduction to absurdity. We assume that monotone circuit can classify equivalence relation. From assumptions, this monotone circuit output same value when input value is (0, 0) , (1, 1). But monotonicity of monotone circuit 0 ∧ 0 = 0 ∨ 0 = 0,1 ∧ 1 = 1 ∨ 1 = 1 makes different output with inputs (0, 0) , (1, 1), and contradicts a condition to classify equivalence relation. Therefore, this theorem was shown than reduction to absurdity. The reason why a monotone circuit has such a limit is that monotone circuit do not have a Not gate. Monotone circuit cannot compute value exclusivity, therefore monotone circuit cannot classify equivalence relation. 3. Flatten problem Next, I show that how to compute equivalence relation with monotone circuit. To reduce implicit value constraint explicitly, we can compute equivalence relation with monotone circuit.