Randomized Complexity Classes
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1 Randomized Complexity Classes
2 We allow TM to toss coins/thow dice etc. We wite M(x,R) fo output of M on input x, coin tosses R Def: L R <=> poly-time andomized M : x L => R [M(x,R)=1] 1/ x L => R [M(x,R)=1] = 0 Def: L B <=> poly-time andomized M : x L => R [M(x,R)=1] /3 x L => R [M(x,R)=1] 1/3 Execise: Fo R, can eplace 1/ with 1/n c, o 1-1/ m fo m = n c, fo any c Fo B, can eplace (/3,1/3) = (1/ + 1/n c, 1/-1/n c ) o (1-1/ m, 1/ m ).
3 Execise: The following ae equivalent: 1) L R co-r ) Thee is a andomized poly-time machine M fo L : x, R, M(x,R) {L(x),?}, x, R [M(x,R) =? ] 1/ 3) Thee is a andomized machine M fo L : x, R, M(x,R) = L(x) the expected unning time of M on x is poly(n) This class is known as Z.
4 Claim: Z R B oof: By definition. Claim: R oof:? N
5 Claim: Z R B oof: By definition. Claim: R N oof: The witness is the andom sting Big open question, is = Z = R = B? Supisingly, this is believed to be the case
6 Claim: B /poly oof: Let L B. Let M(x,R) be a andomized poly-time TM deciding L. Make the eo < -n. Note that fo evey x, R [ L(x) M(x,R) ] < -n So by the pobabilistic method,?????????????????????????????????????????????????????????
7 Claim: B /poly oof: Let L B. Let M(x,R) be a andomized poly-time TM deciding L. Make the eo < -n. Note that fo evey x, R [ L(x) M(x,R) ] < -n So by the pobabilistic method, thee exists some sting R* : L(x) = M(x,R*) x. The cicuit coesponding to M(x,R*) is the desied cicuit. Upshot: Randomness is only useful fo TM, not fo cicuits.
8 Claim: B
9 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} We'll show the answe to this question is equivalent to x L We then show this question can be asked in
10 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <=?
11 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A?
12 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A?
13 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A / < x L, we show we can cove. Idea pick the shifts at andom and show [do not cove] <?
14 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A / < x L, we show we can cove. Idea pick the shifts at andom and show [do not cove] < 1: s1,, s [ y {0,1} : y U s + A]?
15 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A / < x L, we show we can cove. Idea pick the shifts at andom and show [do not cove] < 1: s1,, s [ y {0,1} : y U s + A] y s1,,s [y U s + A] =?
16 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A / < x L, we show we can cove. Idea pick the shifts at andom and show [do not cove] < 1: s1,, s [ y {0,1} : y U s + A] y s1,,s [y U s + A] = y ( s [ y s + A])?
17 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A / < x L, we show we can cove. Idea pick the shifts at andom and show [do not cove] < 1: s1,, s [ y {0,1} : y U s + A] y s1,,s [y U s + A] = y ( s [ y s + A]) y (1/ ) <1 So M(x,R) = 1 <=>?
18 Claim: B oof: Let M(x,R) toss R = coins, and have eo < 1/ Fix x and ask: Can we cove {0,1} with shifts of A := { R {0,1} : M(x,R) = 1 }? Fo s {0,1}, the s-shift is s+a := { s XOR a : a A } {0,1} x L, we show we cannot cove. Note A <= /. s 1,, s : s 1 +A U s +A U U s +A A / < x L, we show we can cove. Idea pick the shifts at andom and show [do not cove] < 1: s1,, s [ y {0,1} : y U s + A] y s1,,s [y U s + A] = y ( s [ y s + A]) y (1/ ) <1 So M(x,R) = 1 <=> s 1,, s : y {0,1}, y U s + A <=> s 1,, s : y {0,1}, V i=1 M(x, y + s i )=1
19 Coollay: = N => = B. oof:?
20 Coollay: = N => = B. oof: = N => = H, and so B H =
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