aa r X i v : . [ m a t h . P R ] J un A TAME SEQUENCE OF TRANSITIVE BOOLEAN FUNCTIONS
MALIN P. FORSSTRÖM
Abstract.
Given a sequence of Boolean functions p f n q n ě , f n : t , u n Ñ t , u , and a sequence p X p n q q n ě of continuous time p n -biased random walks X p n q “ p X p n q t q t ě on t , u n , let C n be the(random) number of times in p , q at which the process p f n p X t qq t ě changes its value. In [6], theauthors conjectured that if p f n q n ě is non-degenerate, transitive and satisfies lim n Ñ8 E r C n s “ 8 ,then p C n q n ě is tight. We give an explicit example of a sequence of Boolean functions whichdisproves this conjecture. Introduction
The aim of this paper is to present an example of a sequence of Boolean functions, which showthat a conjecture made in [6], in its full generality, is false. To be able to present this conjecture,we first give some background.For each n ě , fix some p n P p , q and let X p n q “ p X p n q t q t ě be the continuous time random walkon the n -dimensional hypercube defined as follows. For each i P r n s : “ t , , . . . , n u independently,let p X p n q t p i qq t ě be the continuous time Markov chain on t , u which at random times, distributedaccording to a rate one Poisson process, is assigned a new value, chosen according to p ´ p n q δ ` p n δ ,independently of the Poisson process. The unique stationary distribution of p X p n q t q t ě , denoted by π n , is the measure pp ´ p n q δ ` p n δ q b n on t , u n . Throughout this paper, we will always assumethat X p n q is chosen with respect to this measure. When t ą is small, the difference between X p n q and X p n q t is often thought of as noise, describing a small proportion ´ e ´ t « t of the bits beingmiscounted or corrupted.A function f n : t , u n ÞÑ t , u will be referred to as a Boolean function. Some classical examplesof Boolean functions are the so called Dictator function f n p x q “ x p q , the Majority function f n p x q “ sgn `ř ni “ p x p i q ´ { q ˘ and the Parity function sgn `ś ni “ p x p i q ´ { q ˘ (see e.g. [10, 5]). Since it issometimes not natural to require that a sequence of Boolean functions is defined for each n P N ,we only require that a sequence of Boolean functions is defined for n in an infinite sub-sequence of N . Such sub-sequences of N will be denoted by N “ t n , n , . . . u , where ď n ă n ă . . . . Tosimplify notation, whenever we consider the limit of a sequence p x n i q i ě and the dependency on N is clear, we will abuse notation and write lim n Ñ8 x n instead of lim i Ñ8 x n i . Also, we will write p x n q n P N instead of p x n i q i ě .One of the main objectives of [6] was to introduce notation which describes possible behavioursof p f n p X p n q t qq t ě . Some of these definitions which will be relevant for this paper is given in thefollowing definition. Definition 1.1.
Let p f n q n P N , f n : t , u n Ñ t , u , be a sequence of Boolean functions. For n P N ,let C n “ C n p f n q denote the (random) number of times in p , q at which p f n p X p n q t qq t ě has changedits value, i.e. let C n : “ lim N Ñ8 ř N ´ i “ p f n p X p n q i { N q ‰ f n p X p n qp i ` q{ N qq . The sequence p f n q n P N is saidto be(i) lame if lim n Ñ8 P p C n “ q “ , ii) tame if p C n q n ě is tight, that is for every ε ą there is k ě and n ě such that P p C n ě k q ă ε @ n P N : n ě n , (iii) volatile if C n ñ 8 in distribution.In [6], the authors showed that a sequence of Dictator functions is tame and that a sequenceof Parity functions is volatile, while a sequence of Majority functions is neither tame nor volatile.More generally, the authors also showed that any noise sensitive sequence of Boolean functions (seee.g. [2, 5, 10]) is volatile, while any sequence of Boolean functions which is lame or tame is noisestable [2, 5, 10]. As noted in [1] and [4], there are many sequences of functions which are both noisestable and volatile, and hence the opposite does not hold.Given a sequence p p n q n ě , p n P p , q , a sequence of Boolean functions p f n q n P N is said to be non-degenerate if ă lim inf n Ñ8 P p f n p X p n q q “ q ď lim inf n Ñ8 P p f n p X p n q q “ q ă . A Booleanfunction f n : t , u n Ñ t , u is said to be transitive if for all i, j P r n s : “ t , , . . . , n u there isa permutation σ of r n s which is such that (i) σ p i q “ j and (ii) for all x P t , u n , if we define σ p x q : “ p x p σ p k qqq k Pr n s , then f n p x q “ f n p σ p x qq . To simplify notation, we will abuse notation slightlyand say that a sequence of Boolean functions p f n q n ě is transitive if f n is transitive for each n ě .In [6], the authors show that a sufficient, but not necessary, condition for a non-degenerate sequence p f n q n P N of Boolean functions to be tame, is that sup n E r C n s ă 8 . It is natural to ask if this conditionis also necessary for some natural subset of the set of all sequences of Boolean functions. This isthe motivation for the following conjecture. Conjecture 1.2 (Conjecture 1.21 in [6]) . For any sequences p p n q n ě and N , if p f n q n P N is transitive,non-degenerate and lim n Ñ8 E r C n s “ 8 , then p f n q n P N is not tame. The main objective of this paper is to show that this conjecture, in its full generality, is false.This result will follow as an immediate conjecture of the following theorem, which is our main result.
Theorem 1.3.
For any decreasing sequence p p n q n ě which satisfies p ˆ n „ p n whenever | n ´ ˆ n | “ o p n, ˆ n q , lim n Ñ8 np n “ 8 and lim n Ñ8 np rn “ for some r ě , there is a sequence of positive integers N and a sequence p f n q n P N of Boolean functions, f n : t , u n Ñ t , u , which (w.r.t p p n q n ě ) is(a) non-degenerate,(b) transitive,(c) tame, and satisfies(d) lim n Ñ8 E r C n s “ 8 . In the proof of Theorem 1.3 we give explicit examples of sequences of functions which satisfiesthe above conditions, and hence contradicts Conjecture 1.2, for sequences p p n q n ě which satisfiesthe assumptions above. We remark however that this example cannot directly be extended to thecase ! p n ! . In particular, the conjecture might hold with some additional restriction on thesequence p p n q n ě . Remark . The assumption that lim n Ñ8 np n “ 8 is very natural. To see this, note that thenumber of jumps of p X t q t ě in p , q is given by np n p ´ p n q , and hence if lim sup n Ñ8 np n ă 8 ,then any sequence p f n q n ě of Boolean functions is tame and satisfies lim sup n Ñ8 E r C n s ă 8 . Remark . With slightly more work, one can modify the example given in the proof of Theorem 1.3to get a sequence of functions which, in addition to satisfying (a), (b), (c) and (d) of Theorem 1.3,is also monotone in the sense that if x, x P t , u n , n P N , if x p i q ď x p i q for all i P r n s , then f p x q ď f p x q . Remark . The same idea which is used in the proof of Theorem 1.3 work in general to disproveConjecture 1.2 whenever one can find a non-degenerate, tame and transitive sequence of Boolean unctions. In particular, this implies that the assumption that E r C n s “ 8 can be dropped fromConjecture 1.2.With the previous remark in mind, we suggest the following modified conjecture. Conjecture 1.7. If p p n q n ě satisfies lim n Ñ8 np rn “ 8 for all r ą , and p f n q n ě is transitive andnon-degenerate (w.r.t. p p n q n ě ), then p f n q n ě is not tame. Proof of the main result
Definition 2.1 (Easily convinced tribes) . Fix r ě and n ě , and let ℓ n ě and k n be positiveintegers with the property that ℓ n k n “ n . Partition r n s into k n sets S p n q , S p n q , . . . , S p n q k n , each ofsize ℓ n , and for x P t , u n , let g n p x q “ g p k n ,ℓ n ,r q n p x q be equal to one exactly when there is some j P t , , . . . , k n u such that ř i P S p n q j x p i q ě r .Since Definition 2.1 requires that k n ℓ n “ n and that ℓ n ě , g p k n ,ℓ n ,r q n is only well defined when n is not a prime, and we will in general only want to consider sub-sequences N of N which have theproperty that k n and ℓ n can be chosen such that they satisfy certain growth conditions.We will now show that we can choose r , p p n q n ě , N , p ℓ n q n P N and p k n q n P N so that (a), (b) and(d) of Theorem 1.3 hold. Lemma 2.2.
For any r ě and any decreasing sequence p p n q n ě which satisfies p n P p , q , p ˆ n „ p n whenever | n ´ ˆ n | “ o p n ^ ˆ n q , lim n Ñ8 np n “ 8 and lim n Ñ8 np rn “ , there is N and sequences p ℓ n q n P N , p k n q n P N of positive integers such that p g n q n P N is(a) non-degenerate,(b) transitive, and(c) tame.Proof. Assume that there are sequences L , p ℓ n q n P L and p k n q n P N such that(i) r ă inf ℓ n ,(ii) lim n Ñ8 p n ℓ n “ , and(iii) p rn ℓ rn k n — .We first show that this assumption implies that the conclusions of the lemma hold, and then showthat we can find sequences L , p ℓ n q n P L and p k n q n P N with these properties. Proof of (a).
Note first that P p g n p X p n q q “ q “ ˆ r ´ ÿ i “ ˆ ℓ n i ˙ p in p ´ p n q ℓ n ´ i ˙ k n . (1)For integers ă r ă ℓ such that r ă ℓ , define T : R Ñ R by T p x q : “ ř r ´ i “ ` ℓi ˘ x i p ´ x q l ´ i . Then $’’’’’’&’’’’’’% T p x q “ ř r ´ i “ ` ℓi ˘ x i p ´ x q l ´ i T p x q “ ´ ` ℓr ˘ ¨ rx r ´ p ´ x q ℓ ´ r T p x q “ ´ ` ℓr ˘ ¨ r p r ´ q x r ´ p ´ x q ℓ ´ r ` ` ℓr ˘ ¨ r p ℓ ´ r q x r ´ p ´ x q ℓ ´ r ´ . . .T p m q p x q “ ´ ` ℓr ˘ ř m ^ r ^ ℓ ´ ri “ ` ri ˘ p r q i x r ´ i p ℓ ´ r q m ´ i p ´ x q ℓ ´ r ´ i p´ q m ´ i and hence $’&’% T p q “ T p m q p q “ if j “ , , . . . , r ´ T p r q p q “ ´p ℓ q r . oreover, if we assume that x P p , q and that ℓx ă , then for all ξ P p , x q we have that | T p r ` q p ξ q| “ ˇˇˇ ´ ˆ ℓr ˙ r ÿ i “ ˆ ri ˙ p r q i ξ r ´ i p ℓ ´ r q r ` ´ i p ´ ξ q ℓ ´ r ´ i p´ q r ` ´ i ˇˇˇ ď ˆ ℓr ˙ r ÿ i “ ˆ ri ˙ p r q i ξ r ´ i p ℓ ´ r q r ` ´ i ď ˆ ℓr ˙ r ÿ i “ ˆ ri ˙ p r q i ξ r ´ i ℓ r ´ i ` ď ˆ ℓr ˙ r ÿ i “ ˆ ri ˙ p r q i ¨ ℓ ď ℓ r ` r . Applying Taylor’s theorem to the right hand side of (1), and noting that r ă p r ` q ! , we obtain P ` f n p X p n q q “ ˘ “ ´ ´ p ℓ n q r p rn ` C n p r ` n ℓ r ` n ¯ k n where | C n | ă for all n . By using the inequalities e ´ x ď ´ x ď e ´ x , valid for all x P r , { q , thedesired conclusion follows by applying (iii). Proof of (b).
Fix some n P L and i, i P r n s . We need to show that there is a permutation σ whichis such that σ p i q “ i and f n p σ p x qq “ f n p x q for all x P t , u n . We now divide into two cases. First,if i, i P S p n q m for some m P r k n s , then we can set σ “ p ii q . On the other hand, if there are distinct m, m P r k n s such that i P S p n q m “ t i, i , . . . , i ℓ n u and i P S p n q m “ t i , i , . . . , i ℓ n u , then we can set σ “ p ii q ś ℓ n j “ p i j i j q . This concludes the proof of (b). Proof of (c).
Fix some n P L and note that whenever g n p X p n q q “ , the distribution of the smallesttime t ą at which g n p X p n q t q “ stochastically dominates an exponential distribution with rate r .From this the desired conclusion follows.To complete the proof of Lemma 2.2 it now remains only to show that there are sequences N , p ℓ n q n P L and p k n q n P N such that (i), (ii) and (iii) hold. To this end, for each n ě let ℓ n : “ p np rn q ´ {p r ´ q and k n : “ n { ℓ n “ p np n q r {p r ´ q . Then one easily verifies that r ă inf ℓ n , lim n Ñ8 p n ℓ n “ , and p rn ℓ rn k n — . However, in general, neither ℓ n nor k n need to be integers. Tofix this problem, define $’&’% ˆ n : “ r ℓ n s r k n s , ˆ ℓ ˆ n : “ r ℓ n s , and ˆ k ˆ n : “ r k n s . Let N Ď N be an infinite sequence on which the mapping n ÞÑ ˆ n it is a bijection, and let ˆ N beits image. We will show that the desired properties hold for ˆ N , p p ˆ n q ˆ n P ˆ N , p ˆ ℓ ˆ n q ˆ n P ˆ N and p ˆ k ˆ n q ˆ n P ˆ N . Tothis end, note first that inf ˆ n P ˆ N ˆ ℓ ˆ n “ inf n P N r ℓ n s ą inf n P N r ℓ n s ą inf n P N ℓ n ą r, and hence (i) holds. Next, since ˆ n ě n for each n P N and p p n q n ě is decreasing, we have that p ˆ n ď p n for all n P N . Using this observation, we obtain lim ˆ n Ñ8 ˆ ℓ ˆ n p ˆ n ď lim ˆ n Ñ8 ˆ ℓ ˆ n p n “ lim n Ñ8 r ℓ n s p n “ lim n Ñ8 ` ℓ n p n ` p r ℓ n s ´ ℓ n q p n ˘ “ , and hence (ii) holds. Finally, to see that (iii) holds, note that for any n P N , | n ´ ˆ n | “ ˆ ℓ ˆ n ´ ˆ k ˆ n ´ ℓ n k n “ r k n sr ℓ n s ´ ℓ n k n “ p r k n s ´ k n qp r ℓ n s ´ ℓ n q ` ℓ n p r k n s ´ k n q ` k n p r ℓ n s ´ ℓ n q ă ℓ n ` k n ` . ince both ℓ n Ñ 8 and k n Ñ 8 by definition, | n ´ ˆ n | “ o p ˆ n q , and hence by assumption, p n „ p ˆ n .This implies in particular that for ˆ n P ˆ N , we have p r ˆ n ˆ ℓ r ˆ n ˆ k ˆ n „ p rn ˆ ℓ r ˆ n ˆ k ˆ n “ p rn ˆ r ℓ n s r r k n s “ p rn ` ℓ n ´ p ℓ n ´ r ℓ n s q ˘ r ` k n ´ p k n ´ r k n s q ˘ Using the assumption that p n ℓ n Ñ , it follows that p r ˆ n ˆ ℓ r ˆ n ˆ k ˆ n „ p rn ` ℓ n ´ p ℓ n ´ r ℓ n s q ˘ r k n “ p rn ℓ rn k n r ÿ i “ ˆ ri ˙ˆ r ℓ n s ´ ℓ n ℓ n ˙ r ´ i „ p rn ℓ rn k n — . This concludes the proof. (cid:3)
Remark . Using essentially the same argument as in the proof of Theorem 1.3(c), one can showthat for any r ě , N , p ℓ n q n P N and p k n q n P N , we have lim n Ñ8 E r C n p g n qs ă 8 . This implies inparticular that for any such sequences, p g n q n P N does not satisfy (c) in Theorem 1.3, and hence doesnot provides a counter-example to Conjecture 1.2.We now want to modify the sequence p g n q n P N slightly to obtain a sequence p f n q n P N of Booleanfunctions which in addition to satisfying (a), (b) and (c) of Lemma 2.2 also satisfies lim n Ñ8 E r C n p f n qs “8 . To this end, we first define a degenerate sequence of Boolean functions with this property. Definition 2.4.
For each n ě , let a n ą and H n : “ np n ` a n a np n p ´ p n q . For x P t , u n , let } x } : “ ř ni “ x i and define h n p x q : “ I p} x } ě H n q . Lemma 2.5. If lim n Ñ8 np n “ 8 and a n “ a log p np n q , then p h n q n ě is(a) degenerate,(b) transitive(c) lame, and satisfies(d) lim n Ñ8 E “ C n p h n q ‰ “ 8 .Proof. Note first that the assumptions on p p n q n ě and p a n q n ě together imply that lim n Ñ8 a n “ 8 and ? np n e ´ a n Ñ 8 . Proof of (a).
By definition, we have E “ } X p n q } ‰ “ np n and Var ` } X p n q } ˘ “ np n p ´ p n q . UsingChebyshev’s inequality, we thus obtain P ` h n p X p n q q “ ˘ “ P ´ } X p n q } ě E “ } X p n q } ‰ ` a n b Var ` } X p n q } ˘¯ ď a ´ n . Since a n Ñ 8 , this implies that p h n q n ě is degenerate, which is the desired conclusion. Proof of (b).
Since for any x P t , u n , h n p x q depends on x only through } x } , p h n q n ě is transitive. Proof of (c).
Recall that whenever np n p ´ p n q Ñ 8 , ˆ } X p n q t } ´ np n a np n p ´ p n q ˙ t ě D ñ p Z t q t ě , where p Z t q t ě is a so-called Ornstein-Uhlenbeck process with infinitesimal mean and variance givenby µ p z q “ ´ z and σ p x q “ respectively (see e.g. pp. 170–173 in [7]). Given z P R , let τ z denotethe first time t ě at which Z t “ z , given that Z is chosen according to the stationary distributionof p Z t q t ě . By Corollary 1 in [8] (see also [3]), when z ą is large, we have E π r τ z s „ { ˆ h p z q and Var π p τ z q „ { ˆ h p z q , where ˆ h p z q “ z exp p´ z { q{? π . By the Paley-Zygmund inequality, thisimplies that for any finite time t ą , lim z Ñ8 P p τ z ą t q “ . This implies in particular that p h n q n ě is lame whenever if a n Ñ 8 , completing the proof of (c). roof of (d). By Proposition 1.19 in [6], for each n ě we have E r C n p h n qs “ n ÿ i “ I p p n q i p h n q where I p p n q i p h n q is the so-called influence of the i th bit on h n at p n , defined as the probability thatresampling the i th bit of X p n q according to p ´ p n q δ ` p n δ changes the value of h n p X p n q q . Usingthis result, we obtain E r C n p h n qs “ nI p p n q p h n q “ n ˜ P ` } X p n q } “ H n ´ ˘ ¨ n ´ p H n ´ q n ¨ p n ` P ` } X p n q } “ H n ˘ ¨ H n n ¨ p ´ p n q ¸ “ n ˜ˆ nH n ´ ˙ p H n ´ n p ´ p n q n ´ H n ` ¨ n ´ p H n ´ q n ¨ p n ` ˆ nH n ˙ p H n n p ´ p n q n ´ H n ¨ H n n ¨ p ´ p n q ¸ “ H n ˆ nH n ˙ p H n n p ´ p n q n ´ H n ` . Using Stirling’s formula, it follows that E r C n p h n qs „ ? np n ¨ e ´ a n ? π . In particular, if ? np n e ´ a n Ñ 8 , then lim n Ñ8 E r C n p h n qs “ 8 . This completes the proof of (d). (cid:3) We are now ready to give a proof of our main result.
Proof of Theorem 1.3.
Fix some r ě and sequences N , p ℓ n q n P N , p k n q n P N and p a n q n ě so thatthe assumptions of Lemmas 2.2 and Lemma 2.5 both hold. For n P N and x P t , u n , let H n : “ np n ` a n a np n p ´ p n q and define f n p x q : “ g n p x q I p} x } ă H n ´ q ` I p} x } ě H n q “ g n p x q I p} x } ă H n ´ q ` h n p x q . (2)Note that f n p x q and h n p x q agree whenever } x } ě H n ´ . Combining Lemma 2.2 and Lemma 2.5,the desired conclusion now immediately follows. (cid:3) References [1] Forsström, M. P.,
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Department of Mathematics, KTH Royal Institute of Technology, 100 44Stockholm, Sweden.
E-mail address : [email protected]@kth.se