CMPT 407 - Complexity Theory Lecture

A Boolean circuit C on n inputs x1, . . . , xn is a directed acyclic graph (DAG) with n nodes of in-degree 0 (the inputs x1, . . . , xn), one node of out-degree 0 (the output), and every node of the graph except the input nodes is labeled by AND, OR, or NOT; it has in-degree 2 (for AND and OR), or 1 (for NOT). The Boolean circuit C computes a Boolean function f(x1, . . . , xn) in the obvious way: the value of the function is equal to the value of the output gate of the circuit when the input gates are assigned the values x1, . . . , xn. The size of a Boolean circuit C, denoted |C|, is defined to be the total number of nodes (gates) in the graph representation of C. The depth of a Boolean circuit C is defined as the length of a longest path (from an input gate to the output gate) in the graph representation of the circuit C. A Boolean formula is a Boolean circuit whose graph representation is a tree. Given a family of Boolean functions f = {fn}n≥0, where fn depends on n variables, we are interested in the sizes of smallest Boolean circuits Cn computing fn. Let s(n) be a function such that |Cn| ≤ s(n), for all n. Then we say that the Boolean function family f is computable by Boolean circuits of size s(n). For a function s : N → N, we define the complexity class SIZE(s) to be the set of all Boolean function families f = {fn}n≥0 for which there exists a circuit family {Cn}n≥0 such that Cn computes fn, and, for all sufficiently large n, we have |Cn| ≤ s(n). In the above definition, if s(n) is a polynomial, then we say that f is computable by polysize circuits. It is not difficult to see that every language in P is computable by polysize circuits. Note that given any language L over the binary alphabet, we can define the Boolean function family {fn}n≥0 by setting fn(x1, . . . , xn) = 1 iff x1 . . . xn ∈ L. More precisely, we have the following simulation of time-bounded TMs by circuits: