1. 1 Noise-Insensitive Boolean-Functions are Juntas Guy Kindler & Muli Safra Slides prepared with help of: Adi Akavia

2. 2 Influential People The theory of the influence of variables on Boolean functions [BL, KKL] and related issues, has been introduced to tackle social choice problems, furthermore has motivated a magnificent sequence of works, related to economics [K], percolation [BKS], Hardness of approximation [DS] Revolving around the Fourier/Walsh analysis of Boolean functions… The theory of the influence of variables on Boolean functions [BL, KKL] and related issues, has been introduced to tackle social choice problems, furthermore has motivated a magnificent sequence of works, related to economics [K], percolation [BKS], Hardness of approximation [DS] Revolving around the Fourier/Walsh analysis of Boolean functions… And the real important question: And the real important question:

3. 3 Where to go for Dinner? Who has suggestions: Each cast their vote in an (electronic) envelope, and have the system decided, not necessarily according to majority… It turns out someone –in the Florida wing- has the power to flip some votes Power influence

4. 4 Voting Systems n agents, each voting either “for” (T) or “against” (F) – a Boolean function over n variables f is the outcome n agents, each voting either “for” (T) or “against” (F) – a Boolean function over n variables f is the outcome The values of the agents (variables) may each, independently, flip with probability The values of the agents (variables) may each, independently, flip with probability It turns out: one cannot design an f that would be robust to such noise -that is, would, on average, change value w.p. < O(1) - unless taking into account only very few of the votes It turns out: one cannot design an f that would be robust to such noise -that is, would, on average, change value w.p. < O(1) - unless taking into account only very few of the votes

5. 5 Dictatorship Def: a Boolean function P([n])  {-1,1} is a monotone e-dictatorships --denoted f e -- if:

6. 6 Juntas Def: a Boolean function f:P([n])  {-1,1} is a j-Junta if  J  [n] where |J|≤ j, s.t. for every x  [n]: f(x) = f(x  J) Def: f is an [ , j]-Junta if  j-Junta f’ s.t. Def: f is an [ , j, p]-Junta if  j-Junta f’ s.t. We would tend to omit p p-biased, product distribution

7. 7 Long-Code In the long-code L:[n]  {0,1} 2 n each element is encoded by an 2 n -bits In the long-code L:[n]  {0,1} 2 n each element is encoded by an 2 n -bits This is the most extensive binary code, having one bit for every subset in P([n]) This is the most extensive binary code, having one bit for every subset in P([n])

8. 8 Long-Code Encoding an element e  [n]: Encoding an element e  [n]: E e legally-encodes an element e if E e = f e E e legally-encodes an element e if E e = f e F F F F T T T T T T

9. 9 Long-Code  Monotone-Dictatorship The truth-table of a Boolean function over n elements, can be considered as a 2 n bits long string (each corresponding to one input setting – or a subset of [n]) For a long-code, the legal code-words are all monotone dictatorships How about the Hadamard code? The truth-table of a Boolean function over n elements, can be considered as a 2 n bits long string (each corresponding to one input setting – or a subset of [n]) For a long-code, the legal code-words are all monotone dictatorships How about the Hadamard code?

10. 10 Long-code Tests Def (a long-code test): given a code- word w, probe it in a constant number of entries, and Def (a long-code test): given a code- word w, probe it in a constant number of entries, and accept w.h.p if w is a monotone dictatorship accept w.h.p if w is a monotone dictatorship reject w.h.p if w is not close to any monotone dictatorship reject w.h.p if w is not close to any monotone dictatorship

11. 11 Efficient Long-code Tests For some applications, it suffices if the test may accept illegal code-words, nevertheless, ones which have short list-decoding: Def(a long-code list-test): given a code-word w, probe it in 2/3 places, and accept w.h.p if w is a monotone dictatorship, accept w.h.p if w is a monotone dictatorship, reject w.h.p if w is not even approximately determined by a short list of domain elements, that is, if  a Junta J  [n] s.t. f is close to f’ and f’(x)=f’(x  J) for all x reject w.h.p if w is not even approximately determined by a short list of domain elements, that is, if  a Junta J  [n] s.t. f is close to f’ and f’(x)=f’(x  J) for all x Note: a long-code list-test, distinguishes between the case w is a dictatorship, to the case w is far from a junta.

12. 12 General Direction These tests may vary These tests may vary The long-code list-test a, in particular the biased case version, seem essential in proving improved hardness results for approximation problems The long-code list-test a, in particular the biased case version, seem essential in proving improved hardness results for approximation problems Other interesting applications Other interesting applications Hence finding simple, weak as possible, sufficient-conditions for a function to be a junta is important. Hence finding simple, weak as possible, sufficient-conditions for a function to be a junta is important.

13. 13 Background Thm (Friedgut): a Boolean function f with small average-sensitivity is an [ ,j]-junta Thm (Friedgut): a Boolean function f with small average-sensitivity is an [ ,j]-junta Thm (Bourgain): a Boolean function f with small high- frequency weight is an [ ,j]-junta Thm (Bourgain): a Boolean function f with small high- frequency weight is an [ ,j]-junta Thm (Kindler&Safra): a Boolean function f with small high-frequency weight in a p-biased measure is an [ ,j]-junta Thm (Kindler&Safra): a Boolean function f with small high-frequency weight in a p-biased measure is an [ ,j]-junta Corollary: a Boolean function f with small noise- sensitivity is an [ ,j]-junta Corollary: a Boolean function f with small noise- sensitivity is an [ ,j]-junta Parameters: average-sensitivity [BL,KKL,F] high-frequency weight [KKL,B] noise-sensitivity [BKS] Parameters: average-sensitivity [BL,KKL,F] high-frequency weight [KKL,B] noise-sensitivity [BKS]

14. 14 [n] x I I z Noise-Sensitivity How often does the value of f changes when the input is perturbed? x I I z

15. 15 Def( ,p,x [n] ): Let 0< <1, and x  P([n]). Then y~ ,p,x, if y = (x\I)  z where Def( ,p,x [n] ): Let 0< <1, and x  P([n]). Then y~ ,p,x, if y = (x\I)  z where I~  [n] is a noise subset, and I~  [n] is a noise subset, and z~  p I is a replacement. z~  p I is a replacement. Def( -noise-sensitivity): let 0< <1, then [ When p=½ equivalent to flipping each coordinate in x w.p. /2.] [n] x I I z Noise-Sensitivity

16. 16 Fourier/Walsh Transform Write f:{-1, 1} n  {-1, 1} as a polynomial What would be the monomials? For every set S  [n] we have a monomial which is the product of all variables in S (the only relevant powers are either 0 or 1) ????? For every set S  [n] we have a monomial which is the product of all variables in S (the only relevant powers are either 0 or 1) ????? Make sense now to consider the degree of f or to break it according to the various degrees of the monomials..

17. 17 High/Low Frequencies and their Weights Def: the high-frequency portion of f: Def: the low-frequency portion of f: Def: the high-frequency-weight is: Def: the low-frequency-weight is:

18. 18 Low High-Frequency Weight Prop: the -noise-sensitivity can be expressed in Fourier transform terms as Prop: the -noise-sensitivity can be expressed in Fourier transform terms as Prop: Low ns  Low high-freq weight Proof: By the above proposition, low noise-sensitivity implies nevertheless, f being {-1, 1} function, by Parseval formula (that the norm 2 of the function and its Fourier transform are equal) implies Proof: By the above proposition, low noise-sensitivity implies nevertheless, f being {-1, 1} function, by Parseval formula (that the norm 2 of the function and its Fourier transform are equal) implies

19. 19 Average and Restriction Def: Let I  [n], x  P([n]\I), the restriction function is Def: the average function is Note: [n] I x y I x y y y y y

20. 20 Fourier Expansion Prop: Prop: Prop????: Prop????: Corollary: Corollary:

21. 21 Variation Def: the variation of f: Prop: the following are equivalent definitions to the variation of f:

22. 22 Proof Recall Recall Therefore Therefore

23. 23 Proof – Cont. Recall Recall Therefore (by Parseval): Therefore (by Parseval):

24. 24 Proof First, let’s show : First, let’s show :

25. 25 Low-freq Variation and Low-freq Average-Sensitivity Def: the low-frequency variation is: Def:the average sensitivity is Def:the average sensitivity is And in Fourier representation: Def: the low-frequency average sensitivity is:

26. 26 Biased Walsh Product [Talagrand] Def: In the p-biased product distribution  p, the probability of a subset x is The usual Fourier basis  is not orthogonal with respect to the biased inner-product, The usual Fourier basis  is not orthogonal with respect to the biased inner-product, Hence, we use the Biased Walsh Product: Hence, we use the Biased Walsh Product:

27. 27 Main Result Theorem:  constant  >0 s.t. any Boolean function f:P([n])  {-1,1} satisfying is an [ ,j]-junta for j=O(  -2 k 3  2k ). Corollary: fix a p-biased distribution  p over P([n]). Let >0 be any parameter. Set k=log 1- (1/2). Then  constant  >0 s.t. any Boolean function f:P([n])  {-1,1} satisfying is an [ ,j]-junta for j=O(  -2 k 3  2k ).

28. 28 Where to go for Dinner? Who has suggestions: Each cast their vote in an (electronic) envelope, and have the system decided, not necessarily according to majority… It turns out someone –in the Florida wing- has the power to flip some votes Power influence Of course they’ll have to discuss it over dinner….

29. 29 First Attempt: Following Freidgut’s Proof Thm: any Boolean function f is an [ ,j]-junta for Proof: 1. Specify the junta where, let k=O(as(f)/  ) and fix  =2 -O(k) 2. Show the complement of J has small variation P([n]) J

30. 30 Proving [n]\J has small variation Prop: Let f be a Boolean function, s.t. variation J (f)   /2, then f is an [ ,|J|]-junta. Proof: define a junta f’ as follows: f’(x)=f(x  J)???????? then f’ is a |J|-junta, and hence

31. 31 Following Freidgut - Cont Lemma: Proof: Now, lets bound each argument: Prop: Proof: characters of size  k contribute to the average-sensitivity at least (since) P([n]) J

32. 32 Following Freidgut - Cont Lemma: Proof: Now, lets bound each argument: Prop: Proof: characters of size  k contribute to the average-sensitivity at least (since) P([n]) J

33. 33 Following Freidgut - Cont Prop: Proof: we do not know whether as(f) is small!  this way with only as  k ! True only since this is a {-1,0,1} function. So we cannot proceed this way with only as  k ! 

34. 34 If k were 1 Easy case (!?!): If we’d have a bound on the non- linear weight, we should be done. The linear part is a set of independent characters (the singletons) In order for those to hit close to 1 or -1 most of the time, they must avoid the law of large numbers, namely be almost entirely placed on one singleton [by Chernoff like bound] Thm[FKN, ext.]: Assume f is close to linear, then f is close to shallow (  a constant function or a dictatorship)

35. 35 How to Deal with Dependency between Characters Recall Recall (theorem’s premise) (theorem’s premise) Idea: Let Partition [n]\J into I 1,…,I r, for r >> k Partition [n]\J into I 1,…,I r, for r >> k w.h.p f I [x] is close to linear (low freq characters intersect I expectedly by  1 element, while high-frequency weight is low). w.h.p f I [x] is close to linear (low freq characters intersect I expectedly by  1 element, while high-frequency weight is low). P([n]) J I1I1 I2I2 IrIr I

36. 36 So what? f I [x] is close to linear f I [x] is close to linear By FKN f I [x] is either a constant-function or a dictatorship, for any x Still, f I [x] could be a different dictatorship for every x, hence the variation of each i  I might be low P([n]) J I1I1 I2I2 IrIr I

37. 37 almost linear  almost shallow Theorem([FKN]):  global constant M, s.t.  Boolean function f,  shallow Boolean function g, s.t. Hence, ||f I [x] >1 || 2 is small  f I [x] is close to shallow! Hence, ||f I [x] >1 || 2 is small  f I [x] is close to shallow!

38. 38 Dictatorship and its Singleton Prop: if f I [x] is a dictatorship, then  coordinate i s.t.(where p is the bias). Prop: if f I [x] is a dictatorship, then  coordinate i s.t.(where p is the bias). Corollary (from [FKN]):  global constant M, s.t.  Boolean function h, either or Corollary (from [FKN]):  global constant M, s.t.  Boolean function h, either or {1} {2} {i}{n} {1,2} {1,3}{n-1,n}S{1,..,n} weight Characters Total weight of no more than 1-p

39. 39 f I [x] Mostly Constant Lemma:  >0, s.t. for any  and any function g:P([m])   Lemma:  >0, s.t. for any  and any function g:P([m])   Def: Let D I be the set of x  P(I), s.t. f I [x] is a dictatorship Def: Let D I be the set of x  P(I), s.t. f I [x] is a dictatorship Next we show, that |D I | must be small, hence for most x, f I [x] is constant. Next we show, that |D I | must be small, hence for most x, f I [x] is constant.

40. 40 Lemma: Lemma: Proof: let, then Proof: let, then |D I | must be small Prev lemma Each S is counted only for one index i  I. (Otherwise, if S was counted for both i and j in I, then |S  I|>1!) Parseval

41. 41 Simple Prop Prop: let {a i } i  I be sub-distribution, that is,  i  I a i  1, 0  a i, then  i  I a i 2  max i  I {a i }. Prop: let {a i } i  I be sub-distribution, that is,  i  I a i  1, 0  a i, then  i  I a i 2  max i  I {a i }. Proof: Proof: 1 2 3 max n aiai no more than 1 no more than 1 1 1 2 3 n aiai 1/a max 1

42. 42 |D I | must be small - Cont Therefore (since), Therefore (since), Hence Hence

43. 43 Obtaining the Lemma It remains to show that indeed: It remains to show that indeed: Prop1: Prop1: Prop2: Prop2: {  S } S {  S } S are orthonormal, and RecallRecall HoweverHowever

44. 44 Obtaining the Lemma – Cont. Prop3: Prop3: Proof: separate by freq: Proof: separate by freq: Small freq: Small freq: Large freq: Large freq: Corollary(from props 2,3): Corollary(from props 2,3):

45. 45 Obtaining the Lemma – Cont. Recall: by corollary from [FKN], Eitheror Recall: by corollary from [FKN], Eitheror Hence Hence By Corollary By Corollary Combined with Prop1 we obtain: Combined with Prop1 we obtain: |D I | is small

46. 46 prop1 Corollary (from[FKN]): eitheror prop2 |D I | must be small

47. 47 Important Lemma Lemma:  >0, s.t. for any  and any function g:P([m])  , the following holds: Lemma:  >0, s.t. for any  and any function g:P([m])  , the following holds: Low-freq high-freq

48. 48 Beckner/Nelson/Bonami Inequality Def: let T  be the following operator on f Thm: for any p≥r and  ≤((r-1)/(p-1)) ½ Corollary: for f s.t. f >k =0

49. 49 Beckner/Nelson/Bonami Corollary Proof:

50. 50 Probability Concentration Simple Bound: Simple Bound: Proof: Proof: Low-freq Bound: Let g:P([m])   be of degree k and  >0, then  >0 s.t. Low-freq Bound: Let g:P([m])   be of degree k and  >0, then  >0 s.t. Proof: recall the corollary: Proof: recall the corollary: 

51. 51 Lemma’s Proof Now, let’s prove the lemma: Now, let’s prove the lemma: Bounding low and high freq separately:  , Bounding low and high freq separately:  , simple bound Low-freq bound

52. 52 Shallow Function Def: a function f is linear, if only singletons have non-zero weight Def: a function f is linear, if only singletons have non-zero weight Def: a function f is shallow, if f is either a constant or a dictatorship. Def: a function f is shallow, if f is either a constant or a dictatorship. Claim: Boolean linear functions are shallow. Claim: Boolean linear functions are shallow. 0123kn0123kn weight Character size

53. 53 Boolean Linear  Shallow Claim: Boolean linear functions are shallow. Claim: Boolean linear functions are shallow. Proof: let f be Boolean linear function, we next show: Proof: let f be Boolean linear function, we next show: 1.  {i o } s.t. (i.e. ) 2. And conclude, that eitheror i.e. f is shallow

54. 54 Claim 1 Claim 1: let f be boolean linear function, then  {i o } s.t. Claim 1: let f be boolean linear function, then  {i o } s.t. Proof: w.l.o.g assume Proof: w.l.o.g assume for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} then. then. Next value must be far from {-1,1}, Next value must be far from {-1,1}, A contradiction! (boolean function) A contradiction! (boolean function) Therefore Therefore 1 ?

55. 55 Claim 1 Claim 1: let f be boolean linear function, then  {i o } s.t. Claim 1: let f be boolean linear function, then  {i o } s.t. Proof: w.l.o.g assume Proof: w.l.o.g assume for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} then. then. But this is impossible as f(x 00 ),f(x 10 ),f(x 01 ), f(x 11 )  {-1,1}, hence their distances cannot all be >0 ! But this is impossible as f(x 00 ),f(x 10 ),f(x 01 ), f(x 11 )  {-1,1}, hence their distances cannot all be >0 ! Therefore. Therefore. 1 ?

56. 56 Claim 2 Claim 2: let f be boolean function, s.t. Then eitheror Claim 2: let f be boolean function, s.t. Then eitheror Proof: consider f(  ) and f(i 0 ): Proof: consider f(  ) and f(i 0 ): Then Then but f is boolean, hence but f is boolean, hence therefore therefore 1 0

57. 57 Linearity and Dictatorship Prop: Let f be a balanced linear boolean function then f is a dictatorship. Proof: f(  ),f(i 0 )  {-1,1}, hence Prop: Let f be a balanced boolean function s.t. as(f)=1, then f is a dictatorship. Proof:, but f is balanced, (i.e. ), therefore f is also linear.

58. 58 Proving FKN: almost-linear  close to shallow Theorem: Let f:P([n])   be linear, Theorem: Let f:P([n])   be linear, Let Let let i 0 be the index s.t. is maximal let i 0 be the index s.t. is maximalthen Note: f is linear, hence w.l.o.g., assume i 0 =1, then all we need to show is: We show that in the following claim and lemma. Note: f is linear, hence w.l.o.g., assume i 0 =1, then all we need to show is: We show that in the following claim and lemma.

59. 59 Corollary Corollary: Let f be linear, and then  a shallow boolean function g s.t. Corollary: Let f be linear, and then  a shallow boolean function g s.t. Proof: let, let g be the boolean function closest to l. Then, this is true, as Proof: let, let g be the boolean function closest to l. Then, this is true, as is small (by theorem), is small (by theorem), and additionallyis small, since and additionallyis small, since

60. 60 Claim 1 Claim 1: Let f be linear. w.l.o.g., assume then  global constant c=min{p,1-p} s.t. Claim 1: Let f be linear. w.l.o.g., assume then  global constant c=min{p,1-p} s.t. {} {1} {2} {i}{n} {1,2} {1,3}{n-1,n}S{1,..,n} weight Characters Each of weight no more than c 

61. 61 Proof of Claim1 Proof: assume Proof: assume for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} then then Next value must be far from {-1,1} ! Next value must be far from {-1,1} ! A contradiction! (to ) A contradiction! (to ) 1 ?

62. 62 Proof of Claim1 Proof: assume. Proof: assume. for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} for any z  {3,…,n}, consider x 00 =z, x 10 =z  {1}, x 01 =z  {2}, x 11 =z  {1,2} then. then. Hence Hence Therefore, for a random x this holds w.p. at least c, and therefore-- a contradiction. Therefore, for a random x this holds w.p. at least c, and therefore-- a contradiction. they cannot all be near {-1,1}! 1 ?

63. 63 Lemma Lemma: Let g be linear, let assume, then Lemma: Let g be linear, let assume, then Corrolary: The theorem follows from the combination of claim1 and the lemma: Corrolary: The theorem follows from the combination of claim1 and the lemma: Let m be the minimal index s.t. Let m be the minimal index s.t. Consider Consider If m=2: the theorem is obtained (by lemma) If m=2: the theorem is obtained (by lemma) Otherwise -- a contradiction to minimality of m : Otherwise -- a contradiction to minimality of m : note

64. 64 Lemma’s Proof Lemma’s Proof: Note Lemma’s Proof: Note Hence, all we need to show is that Hence, all we need to show is that Intuition: Intuition: Note that |g| and |b| are far from 0 (since |g| is  -close to 1, and c  -close to b). Note that |g| and |b| are far from 0 (since |g| is  -close to 1, and c  -close to b). Assume b>0, then for almost all inputs x, g(x)=|g(x)| (as ) Assume b>0, then for almost all inputs x, g(x)=|g(x)| (as ) Hence E[g]  E[|g(x)|], and Hence E[g]  E[|g(x)|], and therefore var(g)  var(|g|) therefore var(g)  var(|g|)

65. 65 E 2 [g] - E 2 [|g|] = 2E 2 [|g|1 {f<0} ]  o(  ) (by Azuma’s inequality) E 2 [g] - E 2 [|g|] = 2E 2 [|g|1 {f<0} ]  o(  ) (by Azuma’s inequality) We next show var(g)  var(|g|): We next show var(g)  var(|g|): By the premise By the premise however however therefore therefore Proof-map: |g|,|b| are far from 0 g(x)=|g(x)| for almost all x E[g]  E[|g|] var(g)  var(|g|)

66. 66 Variation Lemma Lemma(variation):  >0, and r>>k s.t. Lemma(variation):  >0, and r>>k s.t. Corollary: for most I and x, f I [x] is almost constant Corollary: for most I and x, f I [x] is almost constant P([n]) J I1I1 I2I2 IrIr I

67. 67 By union bound on I 1,…,I r : By union bound on I 1,…,I r : (set) (set) Let f’(x) = sign( A J [f](x  J) ). f’ is the boolean function closest to A J [f], therefore Let f’(x) = sign( A J [f](x  J) ). f’ is the boolean function closest to A J [f], therefore Hence f is an [ ,j]-junta. Hence f is an [ ,j]-junta. Using Idea2 P([n]) J I1I1 I2I2 IrIr I

68. 68 variation-Lemma - Proof Plan Lemma(variation):  >0, and r>>k s.t. Sketch for proving the variation lemma: 1. w.h.p f I [x] is almost linear 2. w.h.p f I [x] is close to shallow 3. f I [x] cannot be close to dictatorship too often.

69. 69 The End

70. 70 XOR Test Let  be a random procedure for choosing two disjoint subsets x,y s.t.:  i  [n], i  x\y w.p 1/3, i  y\x w.p 1/3, and i  x  y w.p 1/3. Let  be a random procedure for choosing two disjoint subsets x,y s.t.:  i  [n], i  x\y w.p 1/3, i  y\x w.p 1/3, and i  x  y w.p 1/3. Def(XOR-Test): Pick ~ , Def(XOR-Test): Pick ~ , Accept if f(x)  f(y), Accept if f(x)  f(y), Reject otherwise. Reject otherwise.

71. 71 Example Claim: Let f be a dictatorship, then f passes the XOR-test w.p. 2/3. Claim: Let f be a dictatorship, then f passes the XOR-test w.p. 2/3. Proof: Let i be the dictator, then Pr ~  [f(x)  f(y)]=Pr ~  [i  x  y]=2/3 Proof: Let i be the dictator, then Pr ~  [f(x)  f(y)]=Pr ~  [i  x  y]=2/3 Claim: Let f’ be a  -close to a dictatorship f, then f’ passes the XOR- test w.p. 2/3 – 2/3  (  -  2 ). Claim: Let f’ be a  -close to a dictatorship f, then f’ passes the XOR- test w.p. 2/3 – 2/3  (  -  2 ). Proof: see next slide… Proof: see next slide…

72. 72

73. 73 Local Maximality Def: f is locally maximal with respect to a test, if  f’ obtained from f by a change on one input x 0, that is, Pr ~  [f(x)  f(y)]  Pr ~  [f’(x)  f’(y)] Def: f is locally maximal with respect to a test, if  f’ obtained from f by a change on one input x 0, that is, Pr ~  [f(x)  f(y)]  Pr ~  [f’(x)  f’(y)] Def: Let  x be the distribution of all y such that ~ . Def: Let  x be the distribution of all y such that ~ . Claim: if f is locally maximal then f(x) = -sign(E y~  (x) [f(y)]). Claim: if f is locally maximal then f(x) = -sign(E y~  (x) [f(y)]).

74. 74 Claim: Claim: Proof: immediate from the Fourier- expansion, and the fact that y  x=  Proof: immediate from the Fourier- expansion, and the fact that y  x= 

75. 75 Conjecture: Let f be locally maximal (with respect to the XOR-test), assume f passes the XOR-test w.p  1/2 + , for some constant  >0, then f is close to a junta. Conjecture: Let f be locally maximal (with respect to the XOR-test), assume f passes the XOR-test w.p  1/2 + , for some constant  >0, then f is close to a junta.