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Com S 633: Randomness in Computation Lecture 2 Scribe: Dave Doty

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 \e∆cient computation":
 Deterministic Uniform: P
 Probabilistic Uniform: BPP
 Deterministic Nonuniform: P=poly
 Probabilistic Nonuniform: BPSIZE(poly)
The last is de ned 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


Source: Aduri, Pavan - Department of Computer Science, Iowa State University


Collections: Computer Technologies and Information Sciences