Comparing Notions of Full Derandomization
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Comparing Notions ofFull Derandomization
Lance FortnowNEC Research Institute
With thanks toDieter van Melkebeek
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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.
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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.
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Derandomization Notions
I. P = NP.II. Pseudorandom generators exist.III. Circuit approximation is easy.IV. P = BPP.V. P = RP.VI. P = ZPP.
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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 nn s.t. for all circuits
C of size n,
n
yGCrCnkn yr
11)((Pr1)(Pr
log
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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).
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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) = 1+log 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)>2/3 then x in L.• If Pr(M(x) accepts)<1/3 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)>1/2 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 implies
finding accepting inputs of circuits that accept many inputs.
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Proofs of Equivalences Circuit Approximation implies
finding 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 implies
finding 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 implies
finding accepting inputs of circuits that accept many inputs.
Inputs of C beginning with 1
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Proofs of Equivalences Circuit Approximation implies
finding 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 implies
finding accepting inputs of circuits that accept many inputs.
Inputs of C beginning with 11
Inputs of C beginning with 10
Repeat …
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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 implies
Promise BPP is easy. Lautemann’s 1983 proof that
BPP is in 2 actually givesPromise-BPP in Promise-RPPromise-RP.
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Proofs of Equivalences Promise BPP is easy implies
Circuit 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 > j/m + a little.• M will reject w.h.p if accepting probability of
C is < j/m – a little.
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The Other Hypotheses
III. Promise-BPP is easy impliesIV. P = BPP impliesV. P = RP impliesVI. P = ZPP.
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The Other Hypotheses
III. Promise-BPP is easy impliesIV. P = BPP impliesV. P = RP impliesVI. P = ZPP. Impagliazzo-Naor ’88
Generic Oracles make P = BPP butPromise-BPP is not easy.
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The Other Hypotheses
III. Promise-BPP is easy impliesIV. P = BPP impliesV. P = RP impliesVI. P = ZPP. Muchnik and Vereschagin ’96
Relativized world whereP = RP BPP
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The Other Hypotheses
III. Promise-BPP is easy impliesIV. P = BPP impliesV. P = RP impliesVI. 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 and
all 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.
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Oracle Construction Issues Idea: Use circuit to point to its own
accepting input. Cannot encode every circuit or
P = 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) <x,yn> 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 <yn,C,…>
If CA accepts no input then encode at <yn,C,…> all strings of A queried on by CA(x) on at least 1/(2|c|) 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 > k.
We repeat until we find an x since C cannot query yj for j > |C|.
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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.
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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.