1. Comparing Notions of Full Derandomization Lance Fortnow NEC Research Institute With thanks to Dieter van Melkebeek

2. Derandomization Impagliazzo-Wigderson ’97 If E requires 2 (n) size circuits then P = BPP. Andreev-Clementi-Rolim ’98 If efficient hitting set generators exist then P = BPP.

3. 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.

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

5. Hypothesis II The following are equivalent Efficient Pseudorandom generators. Efficient Hitting Set generators. E requires 2 (n) size circuits.

6. 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 n  n s.t. for all circuits C of size n,

7. Hypothesis II The following are equivalent Efficient Pseudorandom generators. Efficient Hitting Set generators. E requires 2 (n) size circuits. Hitting Set Generator H maps 1 n 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(1 n ).

8. Proofs of Equivalences Efficient Pseudorandom Generators imply Efficient Hitting Set Generators. Range of pseudorandom generator is a hitting set.

9. 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(1 n ). Let S be the set of prefixes of elements of H(1 n ) of size k(n). S is in E. If S had 2 o(k(n)) size circuits we could build C of size n that avoids strings whose prefixes are in S.

10. Proofs of Equivalence E requires 2 (n) size circuits implies efficient pseudorandom generators exist. Impagliazzo-Wigderson ‘97

11. 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.

12. 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.

13. Hypothesis III The following are equivalent Circuit Approximation is Easy Given C and 1 n 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.

14. 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.

15. 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.

16. 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.

17. Proofs of Equivalences Circuit Approximation implies finding accepting inputs of circuits that accept many inputs.

18. 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

19. 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/n 2 and take larger.

20. Proofs of Equivalences Circuit Approximation implies finding accepting inputs of circuits that accept many inputs. Inputs of C beginning with 1

21. 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

22. 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 …

23. 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.

24. Proofs of Equivalences Promise RP is easy implies Promise BPP is easy. Lautemann’s 1983 proof that BPP is in  2 actually gives Promise-BPP in Promise-RP Promise-RP.

25. 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.

26. The Other Hypotheses III. Promise-BPP is easy implies IV. P = BPP implies V. P = RP implies VI. P = ZPP.

27. The Other Hypotheses III. Promise-BPP is easy implies IV. P = BPP implies V. P = RP implies VI. P = ZPP. Impagliazzo-Naor ’88 Generic Oracles make P = BPP but Promise-BPP is not easy.

28. The Other Hypotheses III. Promise-BPP is easy implies IV. P = BPP implies V. P = RP implies VI. P = ZPP. Muchnik and Vereschagin ’96 Relativized world where P = RP  BPP

29. The Other Hypotheses III. Promise-BPP is easy implies IV. P = BPP implies V. P = RP implies VI. P = ZPP. Muchnik and Vereschagin ’96 Relativized world where P = ZPP  RP

30. All of the Hypotheses Baker-Gill-Solovay ’75 Oracle where P = NP and all hypotheses are true. Heller ’84 and Kurtz ’85 Oracle where ZPP = EXP and all hypotheses fail in strong way.

31. Relationship of II and III Pseudorandom generators imply circuit approximation. Andreev-Clementi-Rolim ’98 Hitting set generators imply Promise-BPP is easy. Kabanets and Cai ’00 Hypotheses equivalent if one can compute minimum circuit size.

32. 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.

33. 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.

34. 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.

35. 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.

36. Oracle Construction Let L(M A ) be complete for E. Stage n: Pick random y n of length 5n for all n. Promise x in L(M A )  in A. This gives us E has linear size circuits with advice y n.

37. Stage n continued For all circuits C and current A If C A accepts some input then encode that input at If C A accepts no input then encode at all strings of A queried on by C A (x) on at least 1/(2|c|) of inputs x.

38. Why this works We have y 1 hardwired. If we know y k and C A accepts at least half the inputs we will either Find an x such that C A (x) accepts. Find a y j for some j > k. We repeat until we find an x since C cannot query y j for j > |C|.

39. 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.

40. 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.

41. Slight Derandomization Babai-Fortnow-Nisan-Wigderson If BPP is not infinitely often in subexponential time then EXP = MA.

42. 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.

43. Collapse of NEXP Impagliazzo-Kabanets-Wigderson If NEXP has polynomial-size circuits then NEXP = MA.

44. Collapse of NEXP Impagliazzo-Kabanets-Wigderson If NEXP has polynomial-size circuits then NEXP = EXP.

45. Collapse of NEXP Impagliazzo-Kabanets-Wigderson If NEXP has polynomial-size circuits and EXP = AM then NEXP = EXP.

46. 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.

47. 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.

48. Future Directions How does Promise-ZPP is easy fit in? Connections to other hypotheses? If for every n there is an x with high n j time-bounded Kolmogorov complexity and low n k time bounded Kolmogorov complexity then efficient pseudorandom generators exist.