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

2 Influential People The theory of the Influence of Variables on Boolean Functions [KKL,BL,R,M] and related issues, has been introduced to tackle social choice problems. This area 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 [KKL,BL,R,M] and related issues, has been introduced to tackle social choice problems. This area 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 Where to go for Dinner? The alternatives Diners would cast their vote in an (electronic) envelope. The system would decide – not necessarily by majority… It turns out someone –in the Florida wing- has the ability to flip some votes Power influence

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

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  P([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 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 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 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]) 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 For a long-code, the legal code-words are all monotone dictatorships How about the Hadamard code? How about the Hadamard code?

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

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: a Boolean function f with small high-frequency weight in a p-biased measure is an [ ,j]-junta Thm: 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 [M,R,BL,KKL,F] high-frequency weight [KKL,B] noise-sensitivity [BKS] Parameters: average-sensitivity [M,R,BL,KKL,F] high-frequency weight [KKL,B] noise-sensitivity [BKS]

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 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 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) It now makes sense to consider the degree of f or to break it according to the various degrees of the monomials..

17 High/Low Frequencies 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 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 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 Fourier Expansion Prop: Prop:

21 Influence /Variation Def: the variation of I on f: Prop: the following are equivalent definitions to the variation of I on f: Influence i (f) = variation i (f) = variation {i} (f)

22 Proof Recall Recall Therefore Therefore

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

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

25 Low-frequencies Variation and a.s. 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 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 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 The KKL/Freidgut Framework 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 [n] J

29 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

30 KKL/Freidgut Lemma: Proof: Now, lets bound each argument: Prop[KKL]: Proof: characters of size  k contribute to the average-sensitivity at least (since) [n] J

31 KKL/Freidgut 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 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 g of degree k

33 Beckner/Nelson/Bonami Corollary Proof:

34 Freidgut’s Proof 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 ! 

35 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) Concentration of measure: 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] (!) [FKN, ext.] if f is close to linear then f is close to shallow (  a constant function or a dictatorship)

36 Almost Linear  Almost Shallow Thm([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!

37 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). [n] J I1I1 I2I2 IrIr I

38 So what? f I [x] is close to linear f I [x] is close to linear By [FKN], f I [x] is shallow for any x Still, f I [x] could be a different dictatorship for different x’s, hence the variation of each i  I might be low!! P([n]) J I1I1 I2I2 IrIr I

39 Dictatorship and its Singleton Prop: for a dictatorship h,  coordinate i s.t.(where p is the bias). Prop: for a dictatorship h,  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

40 Main 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

41 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: 

42 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

43 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.

44 Lemma: Lemma: Proof: denote, then Proof: denote, 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

45 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: max n aiai no more than 1 no more than n aiai 1/a max 1

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

47 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

48 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):

49 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

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

51 Where to go for Dinner? The alternatives Diners would cast their vote in an (electronic) envelope. The system would decide – not necessarily by majority… It turns out someone –in the Florida wing- has the ability to flip some votes Power influence Of course they’ll have to discuss it over dinner….

52 Discussion Tests that look at only 2 or 3 places cannot produce a large gap between probability of acceptance of a dictatorship and that of a function not so close to a junta Tests that look at only 2 or 3 places cannot produce a large gap between probability of acceptance of a dictatorship and that of a function not so close to a junta Nevertheless, if requiring the function to have additional properties, such as local-maximality, one may be able to design a test with a large gap Nevertheless, if requiring the function to have additional properties, such as local-maximality, one may be able to design a test with a large gap

53 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

54 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

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. 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 ?

56 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 ?

57 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

58 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.

59 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.

60 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

61 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 

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 Next value must be far from {-1,1} ! Next value must be far from {-1,1} ! A contradiction! (to ) A contradiction! (to ) 1 ?

63 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 ?

64 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

65 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|)

66 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|)

67 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

68 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

69 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.

70 The End

71 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.

72 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…

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74 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)]).

75 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= 

76 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.