Comparing Notions ofFull Derandomization

Lance FortnowNEC Research Institute

With thanks toDieter van Melkebeek

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.

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.

Derandomization Notions

I. P = NP.II. Pseudorandom generators exist.III. Circuit approximation is easy.IV. P = BPP.V. P = RP.VI. P = ZPP.

Hypothesis II The following are equivalent

Efficient Pseudorandom generators. Efficient Hitting Set generators. E requires 2(n) size circuits.

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

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).

Proofs of Equivalences Efficient Pseudorandom Generators

imply Efficient Hitting Set Generators. Range of pseudorandom generator is a

hitting set.

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.

Proofs of Equivalence E requires 2(n) size circuits implies

efficient pseudorandom generators exist. Impagliazzo-Wigderson ‘97

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.

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.

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.

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.

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.

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.

Proofs of Equivalences Circuit Approximation implies

finding accepting inputs of circuits that accept many inputs.

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

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.

Proofs of Equivalences Circuit Approximation implies

finding accepting inputs of circuits that accept many inputs.

Inputs of C beginning with 1

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

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 …

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.

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.

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.

The Other Hypotheses

III. Promise-BPP is easy impliesIV. P = BPP impliesV. P = RP impliesVI. P = ZPP.

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.

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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|.

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.

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.

Slight Derandomization Babai-Fortnow-Nisan-Wigderson

If BPP is not infinitely often in subexponential time then EXP = MA.

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.

Collapse of NEXP Impagliazzo-Kabanets-Wigderson

If NEXP has polynomial-size circuits then NEXP = MA.

Collapse of NEXP Impagliazzo-Kabanets-Wigderson

If NEXP has polynomial-size circuits then NEXP = EXP.

Collapse of NEXP Impagliazzo-Kabanets-Wigderson

If NEXP has polynomial-size circuits and EXP = AM then NEXP = EXP.

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.

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.

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.