Comparing Notions of Full Derandomization

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

2. 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.

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 • P = NP. • Pseudorandom generators exist. • Circuit approximation is easy. • P = BPP. • P = RP. • 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 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).

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(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.

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 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.

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 impliesfinding accepting inputs of circuits that accept many inputs.

18. Proofs of Equivalences • Circuit Approximation impliesfinding 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 impliesfinding 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.

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

21. Proofs of Equivalences • Circuit Approximation impliesfinding 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 impliesfinding 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 impliesPromise BPP is easy. • Lautemann’s 1983 proof thatBPP is in 2 actually givesPromise-BPP in Promise-RPPromise-RP.

25. Proofs of Equivalences • Promise BPP is easy impliesCircuit 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 • Promise-BPP is easy implies • P = BPP implies • P = RP implies • P = ZPP.

27. The Other Hypotheses • Promise-BPP is easy implies • P = BPP implies • P = RP implies • P = ZPP. • Impagliazzo-Naor ’88 • Generic Oracles make P = BPP butPromise-BPP is not easy.

28. The Other Hypotheses • Promise-BPP is easy implies • P = BPP implies • P = RP implies • P = ZPP. • Muchnik and Vereschagin ’96 • Relativized world whereP = RP  BPP

29. The Other Hypotheses • Promise-BPP is easy implies • P = BPP implies • P = RP implies • P = ZPP. • Muchnik and Vereschagin ’96 • Relativized world whereP = ZPP  RP

30. 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 andall hypotheses fail in strong way.

31. 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.

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 orP = 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(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.

37. 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.

38. 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|.

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 nj time-bounded Kolmogorov complexity and low nk time bounded Kolmogorov complexity then efficient pseudorandom generators exist.