 
Summary: Com S 633: Randomness in Computation
Lecture 2 Scribe: Dave Doty
Unless stated otherwise, let = f0; 1g be the binary alphabet.
1 Circuits
Example 1.1 (Hardwiring). Let L be such that L 0 and L is undecidable. Then
L 2 P=poly because, on input x 2 n , we hardwire the value 0 n into a circuit that outputs 1
if x = 0 n and 0 otherwise.
Example 1.2 (Hardwiring). Let L be such that, for all n 2 N, jL \ n j n 2 . Then
L 2 P=poly because we can hardwire all strings in L \ n into the circuit using only about
O(n n 2 ) = O(n 3 ) gates.
So far, we have discussed four notions of \eÆcient computation":
Deterministic Uniform: P
Probabilistic Uniform: BPP
Deterministic Nonuniform: P=poly
Probabilistic Nonuniform: BPSIZE(poly)
The last is dened by probabilistic circuits, each of which can be thought of as a circuit
C taking two strings of input bits, the input x 2 , and the random bits r 2 . If
L 2 BPSIZE(poly), then there exists a family of probabilistic circuits C = [C 0 ; C 1 ; : : :] and
polynomials p; s such that, for all n 2 N, jC n j s(n), and, for all x 2 n ,
x 2 L =) Pr
