Umans Complexity Theory Lectures Lecture 17: Natural Proofs.

UmansComplexity Theory Lectures

Lecture 17: Natural Proofs

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!

• two natural approaches– simulation + diagonalization (uniform)

– circuit lower bounds (non-uniform)

• no success for either approach as applied to date

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

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?

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

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

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.

Natural Proofs

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

efficiently distinguish

pseudorandom functions

truly random functions– but cryptographic assumption implies

existence of pseudorandom functions for which this is impossible

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)

Proof (continued)

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

G(x) = (y1, y2)

• Graphically:

Proof (continued)

• A function F:{0,1}k → {0,1}2n

(set n = k x

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

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

Proof (continued)

• |Prx[T(f