Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.
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Transcript of Complexity ©D. Moshkovitz 1 And Randomized Computations The Polynomial Hierarchy.
Complexity©D. Moshkovitz
1
And Randomized Computations
The Polynomial Hierarchy
Complexity©D. Moshkovitz
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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
– BPP2
Complexity©D. Moshkovitz
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Deciding Satifiability
We’ve already seen, that deciding whether a formula is satisfiable…
x1x2x3… [(x1x2x8)…(x6x3)]x1 …xn(x1x2x8)… (x6x3)
only existential quantifier existential & universal quantifiers
Complexity©D. Moshkovitz
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Technical Note
x1x2…xk is the same as x=<x1,x2,…,xk>
x1x2…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
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The Hierarchy
Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:
x1x2 x3… R(x1,x2,x3,…)
i alternating quantifiers
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The Hierarchy
Definition (i): i is the class of all languages reducible to deciding the sat. of a formula of type:
x1x2x3… R(x1,x2,x3,…)
i alternating quantifiers
Complexity©D. Moshkovitz
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PH (Polynomial-time Hierarchy)
Definition:
PH = i i
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Simple Observations
• “base”: 1=NP
• “connection between and ”: i=coi
• “hierarchy”: ii+1 and ii+1
• “upper bound”: PHPSPACE
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Can the Hierarchy Collapse?
Proposition: If NP=coNP, then PH=NP.
Proof Idea: By induction on i, i=NP.
Complexity©D. Moshkovitz
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Probabilistic Turing Machines
• Probabilistic TMs have an “extra” tape: the random tape
M(x) Prr[M(x,r)]content of input
tape
“standard” TMs probabilistic TMs
content of random
tape
Complexity©D. Moshkovitz
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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…
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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 xL
such TMs are called ‘Atlantic City’
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BPP Illustrated
For any input x,
all random strings
random strings for which M is
right
Note: TMs which are right for most x’s (e.g for PRIMES: always say
‘NO’) are NOT acceptable!
Complexity©D. Moshkovitz
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Amplification
Claim: If LBPP, 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
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Proof Idea
• Repeat– Pick r uniformly at
random– Simulate M(x,r)
• Output the majority answer
r M(x,r)
0111001 Yes
1011100 Yes
0001001 No
1100000 Yes
0010011 No
0110001 Yes
Complexity©D. Moshkovitz
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Relations to P and NP
P BPP NP
ignore the random input
?
Complexity©D. Moshkovitz
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Does BPPNP?
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
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“Some Comfort”
Theorem (Sipser,Lautemann): BPP2
Underlying observation:LBPP there exists a poly. probabilistic TM M,
s.t for any n and x{0,1}n let m=p(n) s.t
xL s1,…,sm{0,1}m r{0,1}m 1imM(x,rsi)=1
Make sure you understand why the theorem follows
Complexity©D. Moshkovitz
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{0, 1}m
Yes-instance
Complexity©D. Moshkovitz
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No-instance
{0, 1}m
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Our Starting Point
• LBPP• By amplification, there’s a
poly-time machine M which– uses m random coins – errs w.p < 1/3m
M
x r
xL?
n bits
m bits
false for less than 1/3m of the
r’s
Complexity©D. Moshkovitz
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Proving the Underlying Observation
We will follow the Probabilistic Method
Prr[r has property P] > 0 r with property P
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Yes-Instances Accepted
• Let xL.
• We want s1,…,sm{0,1}m s.t
r{0,1}m 1imM(x,rsi)=1
• So we’ll bound the probability over si’s that it doesn’t hold.
Complexity©D. Moshkovitz
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Bounding The Probability Random si’s Do Not Satisfy This
m1 m R
mm
is ,...,s {0,1}i 1
Pr r 0,1 , M x,r s 0
m1 m R
mm
is ,...,s {0,1}i 1
Pr r 0,1 , M x,r s 0
union-bound m
1 m Rm
m
is ,...,s {0,1}i 1r {0,1}
Pr M x,r s 0
m
1 m Rm
m
is ,...,s {0,1}i 1r {0,1}
Pr M x,r s 0
m1 m Rm
m
is ,...,s {0,1}i 1r {0,1}
Pr M(x,r s ) 0
m1 m Rm
m
is ,...,s {0,1}i 1r {0,1}
Pr M(x,r s ) 0
si’s
independent
mR
mm
s {0,1}i 1
2 Pr M x,s 0
mR
mm
s {0,1}i 1
2 Pr M x,s 0
r: s is random rs is random
mm 1
2 13m
mm 1
2 13m
xL
Complexity©D. Moshkovitz
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No-Instances Rejected
• Let xL.
• Let s1,…,sm{0,1}m .
• We want r{0,1}m s.t
1imM(x,rsi)=0
• So we’ll bound the probability over r that it doesn’t hold.
Complexity©D. Moshkovitz
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Bounding The Probability Random r Does Not Satisfy This
mR
m
ir {0,1}i 1
Pr M x,r s 1
mR
m
ir {0,1}i 1
Pr M x,r s 1
union-bound m
m
ir {0,1}i 1
Pr M x,r s 1
m
m
ir {0,1}i 1
Pr M x,r s 1
1
m 13m
1
m 13m
xL
Complexity©D. Moshkovitz
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Q.E.D!
It follows that:LBPP there’s a poly. prob. TM M,
s.t for any x there is m s.t
xL s1,…,sm r
1imM(x,rsi)=1
Thus, L2
BPP2
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Summary
• We defined the polynomial-time hierarchy – Saw NP PH PSPACE– NP=coNP PH=NP (“the
hierarchy collapses”)
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Summary
• We presented probabilistic TMs– We defined the complexity class BPP– We saw how to amplify randomized
computations
– We proved P BPP 2
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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