1. Umans Complexity Theory Lectures Lecture 17: Natural Proofs

2. 2 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!

3. 3 Approaches to open problems two natural approaches –simulation + diagonalization (uniform) –circuit lower bounds (non-uniform) no success for either approach as applied to date Why ?

4. 4 Approaches to open problems 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

5. 5 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?

6. 6 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 P n is a natural property if: (useful)  n f n  P n implies f does not have poly- size circuits [i.e. f n  P n implies circuit size >> poly(n)] (constructive) can decide “f n  P n ?” in poly time given the truth table of f n (large) at least (½) O(n) fraction of all 2 2 n functions on n bits are in P n –show some function family g = {g n } is in P n All known circuit lower bound techniques are natural for a suitably parameterized version of the definition

7. 7 Definition of a One-Way- Permutation A One-Way-Permutation 2 n  -OWF: –Is a function f k :{0,1} k → {0,1} k that is computed by a polynomial size circuit family, but not invertible by any 2 k  size circuit family. Example: factoring believed to be 2 n  -OWF

8. 8 Natural Proofs Natural Proof Theorem (RR): if there is a 2 n  -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 2 n  -OWF.

9. 9 Natural Proofs Proof: –Main Idea: A Natural Property P n can efficiently distinguish pseudorandom functions from truly random functions –but cryptographic assumption implies existence of pseudorandom functions for which this is impossible

10. 10 Proof (continued) Recall: –Assuming there is a 2 n  -OWF: A One-Way- Permutation f k :{0,1} k → {0,1} k that is not invertible by 2 k  size circuits, Then we can construct a Pseudo Random Generator (PRG) G:{0,1} k → {0,1} 2k –no circuit C of size s = 2 k  for which |Pr x [C(G(x)) = 1] – Pr z [C(z) = 1]| > 1/s (BMY construction with slightly modified parameters)

11. 11 Proof (continued) Think of G as G:{0,1} k → {0,1} k X {0,1} k G(x) = (y 1, y 2 ) Graphically: G x y1y1 y2y2

12. 12 Proof (continued) A function F:{0,1} k → {0,1} 2 n (set n = k   x GG G G GG G GGGGGGGG height n-log k Given x, i, can compute i-th output bit in time n poly(k) Each x defines a poly-time computable function f x

13. 13 Proof (continued) Useful: f x in poly-time : for all x: f x  P n Constructive: exists (truth table) circuit T:{0,1} 2 n → {0,1} of size 2 O(n) for which |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) Large: Pr g [g  P n ] ≥ (1/2) O(n) for random fns g on n bits. Use Natural Proof Properties: (useful)  n f n  P n  f does not have poly-size circuits (constructive) “f n  P n ?” in poly time given truth table of f n (large) at least (½) O(n) fraction of all 2 2 n fns. on n-bits in P n

14. 14 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 0 : pick roots of red subtrees independently from {0,1} k 

15. 15 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 1 : pick roots of red subtrees independently from {0,1} k 

16. 16 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 2 : pick roots of red subtrees independently from {0,1} k 

17. 17 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 3 : pick roots of red subtrees independently from {0,1} k 

18. 18 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 4 : pick roots of red subtrees independently from {0,1} k 

19. 19 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 5 : pick roots of red subtrees independently from {0,1} k 

20. 20 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 6 : pick roots of red subtrees independently from {0,1} k 

21. 21 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 7 : pick roots of red subtrees independently from {0,1} k 

22. 22 Proof (continued) |Pr x [T(f x ) = 1] – Pr g [T(g) = 1]| ≥ (1/2) O(n) x GG G G GG G GGGGGGGG Distribution D 2 n /k-1 : pick roots of red subtrees independently from {0,1} k 

23. 23 Proof (continued) –For some i: |Pr[T(D i ) = 1] - Pr[T(D i-1 ) = 1]| ≥ (1/2) O(n) /2 n = (1/2) O(n) x GG G G GG G GGGGGGGG 

24. 24 Proof (continued) –For some i: |Pr[T(D i ) = 1] - Pr[T(D i-1 ) = 1]| ≥ (1/2) O(n) /2 n = (1/2) O(n) x GG G G GG G GGGGGGGG  Fix values at roots of all other subtrees to preserve difference

25. 25 Proof (continued) –For some i: |Pr[T( D i ’ ) = 1] - Pr[T( D i-1 ’ ) = 1]| ≥ (1/2) O(n) /2 n = (1/2) O(n) x GG G G GG G GGGGGGGG  D i ’: distribution D i after fixing

26. 26 Proof (continued) –For some i: |Pr[T( D i ’ ) = 1] - Pr[T( D i-1 ’ ) = 1]| ≥ (1/2) O(n) /2 n = (1/2) O(n) x GG G G GG G GGGGGGGG  D i-1 ’: distribution D i-1 after fixing

27. 27 Proof (continued) |Pr[T( D i ’ ) = 1] - Pr[T( D i-1 ’ ) = 1]| ≥ (1/2) O(n) /2 n = (1/2) O(n) –C(y 1,y 2 )=T( ) |Pr x [C(G(x)) = 1] - Pr y 1, y 2 [C(y 1, y 2 ) = 1]| ≥ (1/2) O(n) y1y1 y2y2 T( D i ’ ) T( D i-1 ’ ) GG G G GGGGGGGG 

28. 28 Proof (continued) –recall: no circuit C of size s = 2 k  for which: |Pr x [C(G(x)) = 1] – Pr y 1, y 2 [C(y 1, y 2 ) = 1]| > 1/s –we have C of size 2 O(n) for which: |Pr x [C(G(x)) = 1] - Pr y 1, y 2 [C(y 1, y 2 ) = 1]| ≥ (1/2) O(n) –with n = k ,  arbitrary constant –set  such that 2 O(n) < s –contradiction.

29. 29 Natural Proofs Conclusion: 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…

30. 30 “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

31. 31 “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

32. 32 “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

33. 33 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