The Polynomial Hierarchy And Randomized Computations Complexity ©D. Moshkovitz

Introduction Objectives: To introduce the polynomial-time hierarchy (PH) To introduce BPP To show the relationship between the two Overview: satisfiability and PH probabilistic TMs and BPP BPP2 Complexity ©D. Moshkovitz

Deciding Satifiability We’ve already seen, that deciding whether a formula is satisfiable… x1 …xn(x1x2x8)… (x6x3) x1x2x3… [(x1x2x8)…(x6x3)] only existential quantifier existential & universal quantifiers PSPACE-complete NP-complete Complexity ©D. Moshkovitz

Technical Note x1x2…xk is the same as x=<x1,x2,…,xk> Thus, allowing several adjacent quantifiers of the same type does not change the problem. Complexity ©D. Moshkovitz

i alternating quantifiers The Hierarchy Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type: x1x2 x3… R(x1,x2,x3,…) i alternating quantifiers Complexity ©D. Moshkovitz

i alternating quantifiers The Hierarchy Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type: x1x2x3… R(x1,x2,x3,…) i alternating quantifiers Complexity ©D. Moshkovitz

PH (Polynomial-time Hierarchy) Definition: PH = i i Complexity ©D. Moshkovitz

Simple Observations “base”: 1=NP “connection between  and ”: i=coi “hierarchy”: ii+1 and ii+1 “upper bound”: PHPSPACE Complexity ©D. Moshkovitz

Can the Hierarchy Collapse? Proposition: If NP=coNP, then PH=NP. Proof Idea: By induction on i, i=NP. Complexity ©D. Moshkovitz

Probabilistic Turing Machines Probabilistic TMs have an “extra” tape: the random tape “standard” TMs probabilistic TMs M(x) Prr[M(x,r)] content of input tape content of random tape Complexity ©D. Moshkovitz

Does It Really Capture The Notion of Randomized Algorithms? It doesn’t matter if you toss all your coins in advance or throughout the computation… Complexity ©D. Moshkovitz

BPP (Bounded-Probability Polynomial-Time) Definition: BPP is the class of all languages L which have a probabilistic polynomial time TM M, s.t x Prr[M(x,r) = L(x)]  2/3 L(x)=1  xL such TMs are called ‘Atlantic City’ Complexity ©D. Moshkovitz

random strings for which M is right BPP Illustrated Note: TMs which are right for most x’s (e.g for PRIMES: always say ‘NO’) are NOT acceptable! For any input x, all random strings random strings for which M is right Complexity ©D. Moshkovitz

We can get better amplifications, but this will suffice here... Claim: If LBPP, then there exists a probabilistic polynomial TM M’, and a polynomial p(n) s.t x{0,1}n Prr{0,1}p(n)[M’(x,r)L(x)] < 1/(3p(n)) We can get better amplifications, but this will suffice here... Complexity ©D. Moshkovitz

Proof Idea Repeat Output the majority answer M(x,r) Repeat Pick r uniformly at random Simulate M(x,r) Output the majority answer 0111001 Yes 1011100 Yes 0001001 No 1100000 Yes 0010011 No 0110001 Yes Complexity ©D. Moshkovitz

ignore the random input Relations to P and NP ? P  BPP  NP ignore the random input Complexity ©D. Moshkovitz

Does BPPNP? We may have considered saying: “Use the random string as a witness” Why is that wrong? Because non-members may be recognized as members Complexity ©D. Moshkovitz

Make sure you understand why the theorem follows “Some Comfort” Theorem (Sipser,Lautemann): BPP2 Underlying observation: LBPP  there exists a poly. probabilistic TM M, s.t for any n and x{0,1}n let m=p(n) s.t xL  s1,…,sm{0,1}m r{0,1}m 1imM(x,rsi)=1 Make sure you understand why the theorem follows Complexity ©D. Moshkovitz

Yes-instance {0, 1}m Complexity ©D. Moshkovitz

No-instance {0, 1}m Complexity ©D. Moshkovitz

false for less than 1/3m of the r’s Our Starting Point m bits n bits M x r xL? LBPP By amplification, there’s a poly-time machine M which uses m random coins errs w.p < 1/3m false for less than 1/3m of the r’s Complexity ©D. Moshkovitz

Proving the Underlying Observation We will follow the Probabilistic Method Prr[r has property P] > 0   r with property P Complexity ©D. Moshkovitz

Yes-Instances Accepted Let xL. We want s1,…,sm{0,1}m s.t r{0,1}m 1imM(x,rsi)=1 So we’ll bound the probability over si’s that it doesn’t hold. Complexity ©D. Moshkovitz

Bounding The Probability Random si’s Do Not Satisfy This union-bound si’s independent r: s is random  rs is random xL Complexity ©D. Moshkovitz

No-Instances Rejected Let xL. Let s1,…,sm{0,1}m . We want r{0,1}m s.t 1imM(x,rsi)=0 So we’ll bound the probability over r that it doesn’t hold. Complexity ©D. Moshkovitz

Bounding The Probability Random r Does Not Satisfy This union-bound xL Complexity ©D. Moshkovitz

Q.E.D! It follows that: LBPP  there’s a poly. prob. TM M, s.t for any x there is m s.t xL  s1,…,sm r 1imM(x,rsi)=1 Thus, L2  BPP2 Complexity ©D. Moshkovitz

 Summary We defined the polynomial-time hierarchy Saw NP  PH  PSPACE NP=coNP  PH=NP (“the hierarchy collapses”) Complexity ©D. Moshkovitz

 Summary We presented probabilistic TMs We defined the complexity class BPP We saw how to amplify randomized computations We proved P  BPP  2 Complexity ©D. Moshkovitz

 Summary We also presented a new paradigm for proving existence utilizing the algebraic tools of probability theory Prr[r has property P] > 0   r with property P The probabilistic method Complexity ©D. Moshkovitz