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

2. (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

3. 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 ∨ ∨ ∨ ∧

4. 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.

5. A has circuit size complexity O(t2(n)).
Main theorems
9.3.d
Giorgi Japaridze Theory of Computability
Theorem 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 The circuit satisfiability problem is NP-complete.