Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

An arithmetic circuit is just like a boolean circuit, except that the gates are either multiplication or addition gates, and the inputs are either variables or constants. Formally, it is a directed acyclic graph where every vertex has in-degree (aka fan-in) 2 or 0, each vertex (aka gate) of in-degree 0 is labeled by a variable or a field element, and every other vertex is labeled either + or ×. The fan-out of a gate is its out-degree. We say that a circuit computes the polynomial p(X1, . . . , Xn) if there is gate in the circuit whose evaluation gives the polynomial p(X1, . . . , Xn). Note that although every function from Fn 2 → F2 can be represented as a polynomial, and multiplication and addition (over F2) can be used to simulate any boolean function on 2 bits, it is harder to design a small arithmetic circuit for computing a particular polynomial than it is to design a boolean circuit that computes the corresponding function. This is because many polynomials can evaluate to the same function on F2 → F2. For example, the polynomial (X + Y )(X + Z) evaluates to the same function as the polynomial X + Y + Z + Y Z, so if you want to compute the function X + Y + Z + Y Z, you can do it with three gates, even though computing the polynomial requires 4 gates. A boolean circuit can choose to evaluate the “easiest” polynomial, and so be of smaller size. Indeed, this restriction allows us to prove stronger lower bounds on arithmetic circuits, and gives us more techniques to attack them. Suppose we want to compute the polynomial Xd. This can be done by repeatedly squaring X with a circuit of size log d. It is easy to see that this construction is tight: each additional gate can at most double the degree of the polynomials computed by the circuit, so at least log d multiplications are needed to get degree d. What about if we want a circuit that simultaneously computes each of the polynomials Xd 1 , X d 2 , X d 3 , . . . , X d n? One way to do this is to compute each one separately, for a total size of n log d. Is this the best one can do?