Umans Complexity Theory Lectures Lecture 17: Natural Proofs.

UmansComplexity Theory Lectures

Lecture 17: Natural Proofs

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Approaches to open problems

• Almost all major open problems we have seen entail proving lower bounds – P ≠ NP - P = BPP *– L ≠ P - NP = AM *– P ≠ PSPACE– NC proper– BPP ≠ EXP– PH proper– EXP * P/poly

• we know circuit lower bounds imply derandomization

• more difficult (and recent): derandomization implies circuit lower bounds!

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• two natural approaches– simulation + diagonalization (uniform)

– circuit lower bounds (non-uniform)

• no success for either approach as applied to date

Why?

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in a precise, formal sense these approaches are

too powerful !

• if they could be used to resolve major open problems, a side effect would be:– proving something that is false, or– proving something that is believed to be false

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Circuit lower bounds

• Relativizing techniques are out…

• but most circuit lower bound techniques do not relativize

• exponential circuit lower bounds known for weak models:– e.g. constant-depth poly-size circuits

• But, utter failure (so far) for more general models. Why?

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Natural Proofs• Razborov and Rudich defined the following “natural” format

for circuit lower bounds:– identify property P of functions f:{0,1}* {0,1} – P = n Pn is a natural property if:

• (useful) n fn Pn implies f does not have poly-size circuits [i.e. fn Pn implies circuit size >> poly(n)]

• (constructive) can decide “fn Pn?” in poly time given the truth table of fn

• (large) at least (½)O(n) fraction of all 22n functions on n bits are in Pn

– show some function family g = {gn} is in Pn

All known circuit lower bound techniques are natural for a suitably parameterized version of the definition

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Definition of a One-Way-Permutation

• A One-Way-Permutation 2n-OWF:– Is a function fk:{0,1}k → {0,1}k

that is computed by a polynomial size circuit family, but not invertible by any 2k size circuit family.

Example: factoring believed to be 2n-OWF

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Natural ProofsNatural Proof Theorem (RR): if there is a 2n-OWF, then

there is no natural property P.- General version of Natural Proof Theorem also rules out natural

properties useful for proving many other separations, under similar cryptographic assumptions

Converse of Natural Proof Theorem (RR):

If there is a natural property P then there is no 2n-OWF.

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Natural Proofs

• Proof:– Main Idea: A Natural Property Pn can

efficiently distinguish

pseudorandom functions

from

truly random functions– but cryptographic assumption implies

existence of pseudorandom functions for which this is impossible

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Proof (continued)• Recall:

– Assuming there is a 2n-OWF: A One-Way-Permutation

fk:{0,1}k → {0,1}k

that is not invertible by 2k size circuits,

• Then we can construct a Pseudo Random Generator (PRG) G:{0,1}k → {0,1}2k

– no circuit C of size s = 2kfor which

|Prx[C(G(x)) = 1] – Prz[C(z) = 1]| > 1/s

(BMY construction with slightly modified parameters)

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Proof (continued)

• Think of G as G:{0,1}k → {0,1}k X {0,1}k

G(x) = (y1, y2)

• Graphically:

G

x

y1 y2

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Proof (continued)

• A function F:{0,1}k → {0,1}2n

(set n = k x

G G

G

G

G G

G

G G G G G G G G

height n-log k

Given x, i, can compute i-th output bit in time npoly(k)

Each x defines a poly-time

computable function fx

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Proof (continued)

• Useful: fx in poly-time : for all x: fx Pn

• Constructive: exists (truth table) circuit T:{0,1}2n → {0,1} of size 2O(n) for which

|Prx[T(fx) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

• Large: Prg[g Pn] ≥ (1/2)O(n) for random fns g on n bits.

Use Natural Proof Properties:

(useful) n fn Pn f does not have poly-size circuits

(constructive) “fn Pn?” in poly time given truth table of fn

(large) at least (½)O(n) fraction of all 22n fns. on n-bits in Pn

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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G

Distribution D0: pick roots of red subtrees independently from {0,1}k

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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G


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Proof (continued)

• |Prx[T(f

x) = 1] – Prg[T(g) = 1]| ≥ (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G

Distribution D2n/k-1: pick roots of red subtrees independently from {0,1}k

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Proof (continued)

– For some i: |Pr[T(Di) = 1] - Pr[T(Di-1) = 1]| ≥ (1/2)O(n)/2n = (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G

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Proof (continued)

– For some i: |Pr[T(Di) = 1] - Pr[T(Di-1) = 1]| ≥ (1/2)O(n)/2n = (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G

Fix values at roots of all other subtrees to preserve difference

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Proof (continued)

– For some i: |Pr[T( Di’ ) = 1] - Pr[T( Di-1’ ) = 1]| ≥ (1/2)O(n)/2n = (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G

Di’: distribution Di after fixing

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Proof (continued)

– For some i: |Pr[T( Di’ ) = 1] - Pr[T( Di-1’ ) = 1]| ≥ (1/2)O(n)/2n = (1/2)O(n)

x

G G

G

G

G G

G

G G G G G G G G

Di-1’: distribution Di-1 after fixing

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Proof (continued)

|Pr[T( Di’ ) = 1] - Pr[T( Di-1’ ) = 1]| ≥ (1/2)O(n)/2n = (1/2)O(n)

– C(y1,y2)=T( )

|Prx[C(G(x)) = 1] - Pry1, y2[C(y1, y2) = 1]| ≥ (1/2)O(n)

y1 y2

T( Di’ ) T( Di-1’ )

G G

G

G

G G G G G G G G

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Proof (continued)

– recall: no circuit C of size s = 2kfor which:

|Prx[C(G(x)) = 1] – Pry1, y2[C(y1, y2) = 1]| > 1/s

– we have C of size 2O(n) for which:|Prx[C(G(x)) = 1] - Pry1, y2

[C(y1, y2) = 1]| ≥ (1/2)O(n)

– with n = k, arbitrary constant– set such that 2O(n) < s– contradiction.

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Natural ProofsConclusion:•To prove circuit lower bounds, we must either:

– Violate largeness: seize upon an incredibly specific feature of hard functions (one not possessed by a random function ! )

– Violate constructivity: identify a feature of hard functions that cannot be computed efficiently from the truth table

•no “non-natural property” known for all but the very weakest models…

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“We do not conclude that researchers should give up on proving serious lower bounds…”Quite the contrary, by classifying a large number of techniques that are unable to do the job, we hope to focus research in a more fruitful direction. Pessimism will only be warranted if a long period of time passes without the discovery of a non-naturalizing lower bound proof.”

Rudich and Razborov

1994

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“We do not conclude that researchers should give up on proving serious lower bounds. Quite the contrary, by classifying a large number of techniques that are unable to do the job, we hope to focus research in a more fruitful direction…” Pessimism will only be warranted if a long period of time passes without the discovery of a non-naturalizing lower bound proof.”

Rudich and Razborov

1994

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“We do not conclude that researchers should give up on proving serious lower bounds. Quite the contrary, by classifying a large number of techniques that are unable to do the job, we hope to focus research in a more fruitful direction. Pessimism will only be warranted if a long period of time passes without the discovery of a non-naturalizing lower bound proof.”

Rudich and Razborov

1994

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Moral

• To resolve central questions:– avoid relativizing arguments

• use PCP theorem and related results• focus on circuits, etc…

– avoid constructive arguments– avoid arguments that yield lower bounds for

random functions

Umans Complexity Theory Lectures Lecture 17: Natural Proofs.

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