Adleman Theorem and Sipser Gács Lautemann Theorem. CS422 Computation Theory CS422 Complexity Theory

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1 Adleman Theorem and Sipser Gács Lautemann Theorem CS422 Computation Theory CS422 Complexity Theory

2 Complexity classes, N, co N, SACE, EXTIME, H, R, RE, ALL, Complexity theorists: "Which is contained in which?" First thing we learned: N SACE EXTIME This lecture Adleman's theorem: B /poly Sipser Gács Lautemann theorem: B Σ 2 Π 2

3 robabilistic algorithm Nondeterministic WHILE+ program: "guess x i " Change meaning: choose x i as 0 or 1 with equal probability Example: Miller Rabin primality test Given input n, pick a at random and do something with n, a If false, return "composite" If true, return "probably prime"; might be wrong Correct with probability 3Τ4 Repetition to improve probability of correctness

4 Las Vegas algorithm Might not give an answer (or only gives answer after long time) If it gives an answer, it must be correct Alternatively: it either gives correct answer or says "don't know" Example: (Randomized) quicksort Choosing good pivots leads to O n lg n time Choosing bad pivots leads to O n 2 time Z (Zero-error robabilistic olynomial time) Has Las Vegas algorithm with polynomial expected runtime

5 Monte Carlo algorithm Always gives an answer, but might be wrong One-sided error: One answer always correct True-biased: "True" is always correct, but "false" might be wrong False-biased: The other way around Examples: Miller-Rabin, Schwartz-Zippel (polynomial identity testing) Two-sided error: Both answers might be wrong Las Vegas algorithm can be converted to Monte Carlo roof: Homework roblem 13h But not the other way around robabil ities Act ual Yes No Yes Given No

6 Monte Carlo-based classes Worst case running time must be polynomial Act ual Given Yes No Yes 1 0 No 0 1 Act ual R Yes Given No Yes Τ 1 2 Τ 1 2 No 0 1 co R Act ual Yes Given No Yes 1 0 No Τ 1 2 Τ 1 2 Act ual N Yes Given No Yes > 0 < 1 No 0 1 Act ual B Yes Given No Yes 2/3 Τ 1 3 No Τ 1 3 2/3 Act ual Yes Given No Yes > Τ 1 2 < Τ 1 2 No Τ 1 2 Τ 1 2 Act ual ALL Yes Given No Yes Τ 1 2 Τ 1 2 No Τ 1 2 Τ 1 2

7 Monte Carlo-based classes Inclusions? R Randomized olynomial time B Bounded-error robabilistic olynomial time robabilistic olynomial time Obvious: Z R B R N Not obvious, but true: Z = R co R SACE Unknown: = B? B and N???

8 Advice String that depends on length of input, but not input itself Don't care how to obtain the advice; assume we have it Example: Boolean circuits Example: Miller Rabin primality test For any fixed length of n, exists a small set of a that guarantees correctness Advice: these values of a /f n : decidable in polynomial time (without randomness) given advice of length f n for input of length n

9 Advice How powerful? ALL = set of all decision problems E.g. ADDITION, 3-SAT, HALTING TALLY = set of all decision problems in unary E.g. H 1 = 2 n n H where H is halting problem /1 TALLY roof: Homework roblem 13c EX/2 n = ALL roof: Homework roblem 13d /poly: how large?

10 robabilistic method Using probability to show something exists Example: 2Τ3 students are smart and 2Τ3 students are handsome not smart 1Τ3 and not handsome 1Τ3 not smart or not handsome = 2 3 smart and handsome = 1 3 > 0 So there is some student that is smart and handsome Introduced by aul Erdős 1947: R k, k k 2 2 Τ for k 3 (a result in graph theory)

11 Markov's inequality X is non-negative random variable with expected value μ Then X αμ 1 α roof: if more, then E X αμ X αμ Useful for probabilistic method > μ, contradiction Can be used to prove Z R Turn Las Vegas algorithm to Monte Carlo algorithm Let X be the running time of the Las Vegas algorithm Exercise: finish the rest of the proof (homework roblem 13h)

12 (Special case of) Chernoff bound X 1, X 2,, X m be independent random variables that take value 1 with probability p and 0 otherwise Let X = X 1 + X X m ; expected value μ = pm For any δ 0, X 1 + δ μ e μ 3 min δ,δ2 roof: look it up, won't be part of this course Idea: Markov's inequality, then minimize stuff with calculus Also useful for probabilistic method

13 Adleman's theorem: B /poly Idea: put the randomness in the advice roblem: we need the same advice for all inputs of a given length Idea: find it with probabilistic method!

14 Adleman's theorem: B /poly Consider L B Assume it has algorithm M deciding L with error probability 1Τ3 Not necessarily true: we can only say 1Τ3, not = 1Τ3 Exercise: generalize the proof so it works in general Let n be input length; let M be an algorithm that runs M for 36n times and takes majority answer Claim: M decides L with error probability e n

15 Adleman's theorem: B /poly Define X i = 1 if the i-th run gives wrong answer Note p = X i = 1 = 1 3 Define m = 36n and X = X 1 + X X 36n Note M gives wrong answer if X 18n Note μ = 12n Define δ = 1, so min δ, δ2 = and 1 + δ μ = 18n

16 Adleman's theorem: B /poly Chernoff bound: X 1 + δ μ e μ 3 min δ,δ2 lug in 1 + δ μ = 18n, μ = 12n, min δ, δ 2 = 1 4 M gives wrong answer = X 18n e n

17 Adleman's theorem: B /poly M gives wrong answer Fix x, pick r randomly e n M x, r answers incorrectly e n x: M x, r answers incorrectly 2 n e n < 1 x: M x, r answers correctly = 1 above > 0 robabilistic method: exists r such that M answers correctly on all inputs x (of length n) r has length 36n poly n = poly n, so give r as advice

18 Review: olynomial Hierarchy L Σ k if there exists L such that input x L iff y 1 y 2 y 3 y 4 y k, all of poly-length, and x, y 1, y 2,, y k L L Π k if there exists L such that input x L iff y 1 y 2 y 3 y 4 y k, all of poly-length, and x, y 1, y 2,, y k L

19 Sipser Gács Lautemann theorem: B Σ 2 Π 2 rove B Σ 2 first Idea: if L B, find L such that x L iff y 1 y 2 all of poly-length and x, y 1, y 2 L Idea: L is the verifier for L, and y 1, y 2 provide the randomness

20 Sipser Gács Lautemann theorem: B Σ 2 Assume we're working with bit-string model Let M decide L: given input x and random bits r, it gives answer M x, r = "run M on input x, random bits r" Take M with error probability 2 n Run repeatedly to achieve the bound (homework roblem 13g) A x = set of all poly-length r such that M x, r accepts p n = length of r when x has length n; is polynomial R = set of all bit-strings of length p n Observation: if x L, then A x is large; if x L, then A x is small If x L, A x R 1 2 n ; if x L, A x R 2 n

21 Sipser Gács Lautemann theorem: B Σ 2 Idea: "Translating" (XORing) bit strings t = translation vector of same length A x t = set of all r such that r t A x Let k = p n ; pick translation vectors t 1, t 2,, t k k Let S = ሪ i=1 A x t i Set of all r such that r t i A x for some i Claim: if x L, there exist t 1, t 2,, t k such that R = S Claim: if x L, there don't exist such t 1, t 2,, t k

22 Sipser Gács Lautemann theorem: B Σ 2 Claim: if x L, there exist t 1, t 2,, t k such that S = R ick t i 's randomly For each i, A x t i R 1 2 n (r A x t i ) 1 2 n for any fixed i r A x t i 2 n for any fixed i i: r A x t i 2 n k Note r S iff i: r A x t i

23 Sipser Gács Lautemann theorem: B Σ 2 Claim: if x L, there exist t 1, t 2,, t k such that S = R ick t i 's randomly, r S There are 2 k possible r r: r S 2 k 2 nk < 1 r: r S = 1 above > 0 2 nk robabilistic method: exist t 1, t 2,, t k such that r S for all r

24 Sipser Gács Lautemann theorem: B Σ 2 Claim: if x L, there don't exist t 1, t 2,, t k such that S = R ick r randomly For each i, A x t i R 2 n r A x t i 2 n for any fixed i i: r A x t i k 2 n Note r S iff i: r A x t i Since k = poly n, for large n we have r S k 2 n < 1 robabilistic method: exists r such that r S

25 Sipser Gács Lautemann theorem: B Σ 2 k Let S = ሪ i=1 A x t i Claim: if x L, there exist t 1, t 2,, t k such that R = S Claim: if x L, there don't exist such t 1, t 2,, t k Define y 1 = t 1, t 2,, t k, y 2 = r Define L = set of all x such that for some i, M x, r t i accepts If x L, there exists y 1 such that for all y 2, x, y 1, y 2 L If x L, for all y 1 there exists y 2 such that x, y 1, y 2 L This proves B Σ 2

26 Sipser Gács Lautemann theorem: B Σ 2 Π 2 B Σ 2 If L B then L c B roof: exercise If L Σ 2 then L c Π 2 roof: also exercise So B Π 2