Theory of Computability Giorgi Japaridze Theory of Computability Circuit complexity Section 9.3

(sometimes more) is designated the output gate. Boolean circuits 9.3.a Giorgi Japaridze Theory of Computability A Boolean circuit is a collection of gates and inputs connected by wires. Cycles aren’t permitted. Gates take thee forms: AND, OR, and NOT. One of the gates (sometimes more) is designated the output gate. Every circuit C with n inputs computes some n-ary Boolean function, i.e. a function of the type {0,1}n{0,1}. The following animation illustrates a Boolean circuit C at work. We see that C(010)=1. 1 inputs x1 x2 x3 ¬ 1 ∧ ∧ ∨ ¬ 1 1 output ∧ 1

What functions are computed by the following circuits? Examples 9.3.b Giorgi Japaridze Theory of Computability What functions are computed by the following circuits? x1 x2 x3 ∧ ∧ ∧ ∨ x1 x2 x3 ∨ ∨ ∨ ∧

function). “Depth minimal” is defined similarly. Main definitions 9.3.c Giorgi Japaridze Theory of Computability A circuit family C is an infinite list (C0,C1,C2,…) of circuits where Cn has n input variables. C is said to decide a language A over {0,1} if, for every bitstring w of length n, wA iff Cn(w)=1. The size of a circuit is the number of its gates. The depth of a circuit is the length (number of wires) of the longest path from an input variable to the output gate. A circuit is size minimal if no smaller circuit is equivalent to it (i.e. computes the same function). “Depth minimal” is defined similarly. The size complexity of a circuit family (C0,C1,C2,…) is the function f: NN, where f(n) is the size of Cn. Depth complexity is defined similarly. The circuit size complexity of a language is the size complexity of a minimal circuit family for that language. Circuit depth complexity is defined similarly.

A has circuit size complexity O(t2(n)). Main theorems 9.3.d Giorgi Japaridze Theory of Computability Theorem 9.30 Let t: NN be a function, where t(n) ≥ n. If ATIME(t(n)), then A has circuit size complexity O(t2(n)). Corollary Every problem in P has a polynomial circuit size complexity. Lots of efforts have gone into attempts to show that certain NP-problems have no polynomial size circuits (and hence, in view of the above corollary, P≠NP). However, the progress has been very slow. A circuit is said to be satisfiable if there is an assignment for its variables such that the circuit outputs 1. Theorem 9.33 The circuit satisfiability problem is NP-complete.