Comparing Notions of Full Derandomization

1
Comparing Notions ofFull Derandomization

• Lance Fortnow
• NEC Research Institute
• With thanks to
• Dieter van Melkebeek

2
Derandomization

• Impagliazzo-Wigderson 97
• If E requires 2?(n) size circuitsthen P BPP.
• Andreev-Clementi-Rolim 98
• If efficient hitting set generators exist then P
  BPP.

3
Derandomization

• E requires 2?(n) size circuits.
• Efficient hitting set generators exist.
• These assumptions are equivalent.
• Are they equivalent to P BPP?
• How about Promise-BPP is easy?
• Main Result
• There exist a relativized world where Promise-BPP
  is easy but E has small circuits.

4
Derandomization Notions

1. P NP.
2. Pseudorandom generators exist.
3. Circuit approximation is easy.
4. P BPP.
5. P RP.
6. P ZPP.

5
Hypothesis II

• The following are equivalent
• Efficient Pseudorandom generators.
• Efficient Hitting Set generators.
• E requires 2?(n) size circuits.

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Hypothesis II

• The following are equivalent
• Efficient Pseudorandom generators.
• Efficient Hitting Set generators.
• E requires 2?(n) size circuits.
• Pseudorandom Generator
• A function G?k log n??n s.t. for all circuits C
  of size n,

7
Hypothesis II

• The following are equivalent
• Efficient Pseudorandom generators.
• Efficient Hitting Set generators.
• E requires 2?(n) size circuits.
• Hitting Set Generator
• H maps 1n to a polynomial-list of strings such
  that if C is size n and accepts at least half of
  its inputs then one of those inputs is in H(1n).

8
Proofs of Equivalences

• Efficient Pseudorandom Generators imply Efficient
  Hitting Set Generators.
• Range of pseudorandom generator is a hitting set.

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Proofs of Equivalence

• Hitting set generators imply E requires 2?(n)
  size circuits ISW,ACR
• Let k(n) 1log of the size of the hitting set
  generated by H(1n).
• Let S be the set of prefixes of elements of H(1n)
  of size k(n).
• S is in E. If S had 2o(k(n)) size circuits we
  could build C of size n that avoids strings whose
  prefixes are in S.

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Proofs of Equivalence

• E requires 2?(n) size circuits implies efficient
  pseudorandom generators exist.
• Impagliazzo-Wigderson 97

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P NP and Hypothesis II

• P NP ? Hitting Set Generators
• Probabilistic methods guarantee existence of
  hitting sets.
• Minimum generator in polynomial-time hierarchy.
• Relative to a random oracle, P ? NP and
  Pseudorandom generators exist.

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Hypothesis III

• The following are equivalent
• Circuit Approximation is Easy
• Promise-BPP is easy
• Promise-RP is easy
• Efficiently find accepting inputs of circuits
  that accept many inputs.

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Hypothesis III

• The following are equivalent
• Circuit Approximation is Easy
• Given C and 1n can compute in poly(c,n) time, a
  value v within 1/n of accepting probability of C.
• Promise-BPP is easy
• Promise-RP is easy
• Efficiently find accepting inputs of circuits
  that accept many inputs.

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Hypothesis III

• The following are equivalent
• Circuit Approximation is Easy
• Promise-BPP is easy
• For Probabilistic Polytime M there is L in P,
• If Pr(M(x) accepts)gt2/3 then x in L.
• If Pr(M(x) accepts)lt1/3 then x not in L.
• Promise-RP is easy
• Efficiently find accepting inputs of circuits
  that accept many inputs.

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Hypothesis III

• The following are equivalent
• Circuit Approximation is Easy
• Promise-BPP is easy
• Promise-RP is easy
• For Probabilistic Polytime M there is L in P,
• If Pr(M(x) accepts)gt1/2 then x in L.
• If Pr(M(x) accepts) 0 then x not in L.
• Efficiently find accepting inputs of circuits
  that accept many inputs.

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Hypothesis III

• The following are equivalent
• Circuit Approximation is Easy
• Promise-BPP is easy
• Promise-RP is easy
• Efficiently find accepting inputs of circuits
  that accept many inputs.
• Given C accepting at least half of inputs, can in
  polytime find an accepting input.

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Proofs of Equivalences

• Circuit Approximation impliesfinding accepting
  inputs of circuits that accept many inputs.

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Proofs of Equivalences

• Circuit Approximation impliesfinding accepting
  inputs of circuits that accept many inputs.

Inputs of C beginning with 1
Inputs of C beginning with 0
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Proofs of Equivalences

• Circuit Approximation impliesfinding accepting
  inputs of circuits that accept many inputs.

Inputs of C beginning with 1
Inputs of C beginning with 0
Approximate the size of each one within factor of
1/n2 and take larger.
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Proofs of Equivalences

• Circuit Approximation impliesfinding accepting
  inputs of circuits that accept many inputs.

Inputs of C beginning with 1
21
Proofs of Equivalences

• Circuit Approximation impliesfinding accepting
  inputs of circuits that accept many inputs.

Inputs of C beginning with 11
Inputs of C beginning with 10
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Proofs of Equivalences

• Circuit Approximation impliesfinding accepting
  inputs of circuits that accept many inputs.

Inputs of C beginning with 11
Inputs of C beginning with 10
Repeat
23
Proofs of Equivalences

• Finding accepting inputs of circuits that accept
  many inputs implies Promise-RP is easy.
• Convert Promise-RP machine M to a circuit whose
  inputs are random coins to M.

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Proofs of Equivalences

• Promise RP is easy impliesPromise BPP is easy.
• Lautemanns 1983 proof thatBPP is in ?2 actually
  givesPromise-BPP in Promise-RPPromise-RP.

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Proofs of Equivalences

• Promise BPP is easy impliesCircuit Approximation
  is easy
• Consider probabilistic machine M that chooses m
  random inputs to C and accepts if j accepts.
• M will accept w.h.p if accepting probability of C
  is gt j/m a little.
• M will reject w.h.p if accepting probability of C
  is lt j/m a little.

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The Other Hypotheses

1. Promise-BPP is easy implies
2. P BPP implies
3. P RP implies
4. P ZPP.

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The Other Hypotheses

• Promise-BPP is easy implies
• P BPP implies
• P RP implies
• P ZPP.
• Impagliazzo-Naor 88
• Generic Oracles make P BPP butPromise-BPP is
  not easy.

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The Other Hypotheses

• Promise-BPP is easy implies
• P BPP implies
• P RP implies
• P ZPP.
• Muchnik and Vereschagin 96
• Relativized world whereP RP ? BPP

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The Other Hypotheses

• Promise-BPP is easy implies
• P BPP implies
• P RP implies
• P ZPP.
• Muchnik and Vereschagin 96
• Relativized world whereP ZPP ? RP

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All of the Hypotheses

• Baker-Gill-Solovay 75
• Oracle where P NP andall hypotheses are true.
• Heller 84 and Kurtz 85
• Oracle where ZPP EXP andall hypotheses fail in
  strong way.

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Relationship of II and III

• Pseudorandom generators imply circuit
  approximation.
• Andreev-Clementi-Rolim 98
• Hitting set generators implyPromise-BPP is easy.
• Kabanets and Cai 00
• Hypotheses equivalent if one can compute minimum
  circuit size.

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Our Result

• There exists a relativized world where E has
  linear-size circuits and we can efficiently find
  accepting inputs of circuits that accept many
  inputs.
• Corollary
• There exists relativized world where Hypothesis
  II is false and III is true.

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Relativization

• Result relative to set A means all machines can
  query A at unit cost.
• All results mentioned in this talk hold relative
  to all sets A.
• Any proof that Hypothesis II and III are
  equivalent would require different techniques.

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Differences of II and III

• 1-sided vs. 2-sided error nonissue.
• Hypothesis II
• Generators must work against all circuits.
• Hypothesis III
• Given circuit can find accepting input.

35
Oracle Construction Issues

• Idea Use circuit to point to its own accepting
  input.
• Cannot encode every circuit orP NP and
  Hypothesis II is true.
• Just want to encode accepting inputs of circuits
  that accept many inputs.
• We do not know as we construct which circuits to
  encode.

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Oracle Construction

• Let L(MA) be complete for E.
• Stage n
• Pick random yn of length 5n for all n.
• Promise x in L(MA) ? ltx,yngt in A.
• This gives us E has linear size circuits with
  advice yn.

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Stage n continued

• For all circuits C and current A
• If CA accepts some input then encode that input
  at ltyn,C,gt
• If CA accepts no input then encode at ltyn,C,gt
  all strings of A queried on by CA(x) on at least
  1/(2c) of inputs x.

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Why this works

• We have y1 hardwired.
• If we know yk and CA accepts at least half the
  inputs we will either
• Find an x such that CA(x) accepts.
• Find a yj for some j gt k.
• We repeat until we find an x since C cannot query
  yj for j gt C.

39
Relativization

• All of the equivalences and implications
  discussed relativize, i.e., hold if all machines
  involved have access to the same oracle.
• Most combinatorial and algebraic techniques in
  complexity theory relativize.

40
Hard Sets Implies PRGs

• Klivans-van Melkebeek 99
• If f is computable in exponential time relative
  to A and no subexponential size circuit family
  with B gates can compute f then there exists an
  efficient pseudo-random generator computable with
  an oracle for A secure against circuits with
  oracle gates for B.

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Slight Derandomization

• Babai-Fortnow-Nisan-Wigderson
• If BPP is not infinitely often in subexponential
  time then EXP MA.

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Slight Derandomization

• Babai-Fortnow-Nisan-Wigderson
• If BPP is not infinitely often in subexponential
  time then EXP has polynomial-size circuits.
• Babai-Fortnow-Lund, Nisan
• If EXP has polynomial-size circuits then EXP MA.

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Collapse of NEXP

• Impagliazzo-Kabanets-Wigderson
• If NEXP has polynomial-size circuits then NEXP
  MA.

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Collapse of NEXP

• Impagliazzo-Kabanets-Wigderson
• If NEXP has polynomial-size circuits then NEXP
  EXP.

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Collapse of NEXP

• Impagliazzo-Kabanets-Wigderson
• If NEXP has polynomial-size circuits and EXP AM
  then NEXP EXP.

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Collapse of NEXP

• Impagliazzo-Kabanets-Wigderson
• If NEXP has polynomial-size circuits and EXP AM
  then NEXP EXP.
• Babai-Fortnow-Lund, Nisan
• If EXP has polynomial-size circuits then EXP MA
  ? AM.

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Limited Derandomization

• Impagliazzo-Wigderson 98
• If EXP ? BPP then BPP is infinitely often
  heuristically in subexponential time.
• Open if this relativizes.
• Uses special random-self-reducible and downward
  reducible properties of the permanent.
• Same properties used in first interactive proofs
  of the permanent.

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Future Directions

• How does Promise-ZPP is easy fit in?
• Connections to other hypotheses?
• If for every n there is an x with high nj
  time-bounded Kolmogorov complexity and low nk
  time bounded Kolmogorov complexity then efficient
  pseudorandom generators exist.