Abstract

The lectures are devoted to boolean circuit lower bounds. We consider circuits with gates ∧; ∨; ¬. Suppose L ∈ {0;1} ∗ is a language. Let Ln = L ∩ {0;1} n. We say that L is computed by a family of circuits C1;C2;::: if on an input x = (x1;:::;xn), Cn(x) is 1 when x ∈ Ln and is 0 when x = ∈ Ln. For a circuit C, we de ne size(C) to be the number of edges in the graph representing C, and depth(C) to be the length of the longest path from an input to the output. We say that L ∈ P/poly if there exists a family C1;C2;::: computing L such that size(Cn) = n O(1). It is easy to see that if a Turing machine computes L in time T(n), then there exists a family of circuits C1;C2;::: computing L so that size(Cn) ≤ (T(n)) 2. If a Turing machine is given a circuit C and an input x, then it can compute C(x) in time size(C). However, there exist languages that are computable by families of circuits but are not computable by Turing machines. The simplest example is L = Halting problem, Cn(x) = 1 i L(n) = 1 for |x | = n. But in some sense, this is not an interesting example. Theorem 1 (Karp and Lipton). If NP ⊆ P/poly, then PH collapses: PH = 2 = 2. Exercise 1. Prove the Karp{Lipton Theorem. What we would really like to prove is that NP ⊆ P/poly implies P = NP but this is currently

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