lec10 - Worst-case vs. Average-case Theorem (NW): if E...

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1 CS151 Complexity Theory Lecture 10 April 28, 2011 April 28, 2011 2 Worst-case vs. Average-case Theorem (NW): if E contains 2 Ω(n) -unapp- roximable functions then BPP = P . • How reasonable is unapproximability assumption? • Hope: obtain BPP = P from worst-case complexity assumption – try to fit into existing framework without new notion of “unapproximability” April 28, 2011 3 Worst-case vs. Average-case Theorem (Impagliazzo-Wigderson, Sudan-Trevisan-Vadhan) If E contains functions that require size 2 Ω(n) circuits, then E contains 2 Ω(n) –unapp- roximable functions. • Proof: – main tool: error correcting code April 28, 2011 4 Codes and Hardness 0 1 0 0 1 0 1 0 m : 0 1 0 0 1 0 1 0 Enc(m): 0 0 0 1 0 0 1 1 0 0 0 1 0 R: 0 1 0 0 0 D C f:{0,1} log k {0,1} f ’:{0,1} log n {0,1} small circuit C approximating f’ decoding procedure i 2 {0,1} log k small circuit that computes f exactly f(i) April 28, 2011 5 Encoding • use a (variant of) Reed-Muller code concatenated with the Hadamard code – q (field size), t (dimension), h (degree) encoding procedure : – message m 2 {0,1} k – subset S μ F q of size h – efficient 1-1 function Emb: [k] ! S t – find coeffs of degree h polynomial p m : F q t ! F q for which p m (Emb(i)) = m i for all i (linear algebra) so, need h t ≥ k April 28, 2011 6 Encoding encoding procedure (continued): – Hadamard code Had:{0,1} log q ! {0,1} q • = Reed-Muller with field size 2, dim. log q, deg. 1 • distance ½ by Schwartz-Zippel – final codeword: (Had(p m ( x ))) x 2 F q t • evaluate p m at all points, and encode each evaluation with the Hadamard code
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2 April 28, 2011 7 Encoding 0 1 0 0 1 0 1 0 m : Emb: [k] ! S t S t F q t p m degree h polynomial with p m (Emb(i)) = m i 5 7 2 9 2 1 0 3 8 3 6 0 1 0 0 1 0 1 0 . . . . . . evaluate at all x 2 F q t encode each symbol with Had:{0,1} log q ! {0,1} q April 28, 2011 8 Decoding • small circuit C computing R, agreement ½ + Decoding step 1 – produce circuit C’ from C • given x 2 F q t outputs “guess” for p m ( x ) • C’ computes {z : Had(z) has agreement ½ + with x-th block}, outputs random z in this set 0 1 0 0 1 0 1 0 Enc(m): 0 0 0 1 0 1 1 0 0 0 1 0 R: 0 1 0 0 April 28, 2011 9 Decoding Decoding step 1 (continued): – for at least /2 of blocks, agreement in block is at least ½ + /2 – Johnson Bound: when this happens, list size is S = O(1/ 2 ), so probability C’ correct is 1/S – altogether: • Pr x [C’(x) = p m (x)] ≥ ( 3 ) • C’ makes q queries to C • C’ runs in time poly(q) April 28, 2011 10 Decoding • small circuit C’ computing R’, agreement ’ = ( 3 ) Decoding step 2 – produce circuit C’’ from C’ • given
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lec10 - Worst-case vs. Average-case Theorem (NW): if E...

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