Variations on the Sensitivity Conjecture

We present a selection of known as well as new variants of the Sensitivity Conjecture and point out some weaker versions that are also open.

Study of sensitivity of Boolean functions originated from Cook and Dwork [10] and Reischuk [24]. They showed an Ω(log s(f )) lower bound on the number of steps required to compute a Boolean function f on a CREW PRAM. A CREW PRAM, abbreviated from Consecutive Read Exclusive Write Parallel RAM, is a collection of synchronized processors computing in parallel with access to a shared memory with no write conflicts. The minimum number of steps required to compute a function f on a CREW PRAM is denoted by CREW(f ). After Cook, Dwork and Reischuk introduced sensitivity, Nisan [20] found a way to modify the definition of sensitivity to characterize CREW(f ) exactly. He introduced a related notion called block sensitivity. A block B is a subset of [n] = {1, 2, . . . , n}. Let e B ∈ {0, 1} n denote the characteristic vector of B, i.e., the i th bit of e B is 1 if i ∈ B and 0 otherwise. We say that a block B is sensitive for f on x if f (x ⊕ e B ) = f (x). The block sensitivity of f on x, denoted by bs(f, x), is the maximum number of pairwise disjoint sensitive blocks of f on x.
The rest of the paper is organized as follows. In Section 2 we describe complexity measures of Boolean functions polynomially related to block sensitivity. In Section 3 we review progress on the Sensitivity Conjecture. In Section 4 we present alternative formulations of the Sensitivity Conjecture and point out weaker versions that are also open. Along the way we encounter important examples of Boolean functions. We present these functions in Section 5.

Measures Related to Block Sensitivity
Block sensitivity is polynomially related to several other complexity measures of Boolean functions, which we describe in this section. A deterministic decision tree on n variables x 1 , . . . , x n is a rooted binary tree, whose internal nodes are labeled with variables, and the leaves are labeled 0 or 1. Edges are also labeled 0 or 1. To evaluate such a tree on input x, start at the root and query the corresponding variable, then move to the next node along the edge labeled with the outcome of the query. Repeat until a leaf is reached, at which point the label of the leaf is declared to be the output of the evaluation. A decision tree computes a Boolean function f if it agrees with f on all inputs. Definition 2.1. The deterministic decision tree complexity of a Boolean function f , denoted by D(f ), is the depth of a minimum-depth decision tree that computes f . One way to extend the deterministic decision tree model is to add randomness to the computation. In this extended model, each node v has an associated bias p v ∈ [0, 1]. Evaluation proceeds as before, except when deciding which edge to follow after querying x i at node v, we follow an edge corresponding to the outcome of the query with probability p v and the other edge with probability 1 − p v . Definition 2.2. The bounded-error randomized decision tree complexity of a Boolean function f , denoted by R 2 (f ), is the depth of a minimum-depth randomized decision tree computing f with probability at least 2/3 for all x ∈ {0, 1} n .
A certificate of a Boolean function f on input x, is a subset S ⊂ [n], such that (∀y ∈ {0, 1} n )(x| S = y| S ⇒ f (x) = f (y)). The certificate complexity of a Boolean function f on input x, denoted by C(f, x), is the minimum size of a certificate of f on x.
Definition 2.3. The certificate complexity of a Boolean function f , also known as non-deterministic decision tree complexity and denoted by C(f ), is the maximum of C(f, x) over all choices of x.
The degree of a Boolean function f , denoted by deg(f ), is the degree of the unique multilinear polynomial that represents f.
The approximate degree of a Boolean function f , denoted by deg(f ), is the minimum degree of a polynomial that approximately represents f.
We denote the quantum decision tree complexity with bounded error of a Boolean function f by Q 2 (f ). Discussion of quantum complexity is outside the scope of this note. For an introduction to quantum complexity see a survey by de Wolf [11]. Theorem 2.7. The following complexity measures of Boolean functions are all polynomially related: Table 1 presents a quick summary of the known polynomial relations between complexity measures that play a prominent role in this note. An entry from the table shows the smallest known exponent of a polynomial in the corresponding measure from the column that gives an upper bound on the corresponding measure from the row, as well as the exponent of the biggest known gap between two measures. An entry also contains references to papers, where the result can be found. References of the form [*] indicate that the result is immediate from the definitions of complexity measures. For example, entry 3 [19](log 3 6 [22]) in the second row and third column means that D(f ) = O(deg(f ) 3 ) (see [19]) and there is a Boolean function f , for which D(f ) = Ω(deg(f ) log 3 6 ) (see [22]). For a thorough treatment of polynomial relation between various complexity measures of Boolean functions (including variants of quantum query complexity) see a survey by Buhrman and de Wolf [6]. Using Theorem 2.7, one immediately obtains many equivalent formulations of the "sensitivity versus block sensitivity" conjecture. The purpose of this note is to point out some nontrivial variations on this conjecture that, to our knowledge, have not been stated explicitly in the literature. We also propose several weaker versions of the Sensitivity Conjecture, which might provide starting points.
We introduce the following pictorial notation to indicate relations between the statements appearing in this note and the Sensitivity Conjecture.
-a consequence of the Sensitivity Conjecture.
-implies the Sensitivity Conjecture, but the reverse implication is not known. These might be good candidates for refutation.
-equivalent to the Sensitivity Conjecture.
-conditionally equivalent to the Sensitivity Conjecture.

Progress on the Sensitivity Conjecture
The progress on the Sensitivity Conjecture has been limited. Simon [29] proved that for any Boolean function that depends on all n variables, sensitivity is at least 1 2 log n − 1 2 log log n + 1 2 . An immediate corollary is that for any Boolean function f , bs(f ) = O s(f )4 s(f ) . Kenyon and Kutin [16] proved that sensi- 1 The construction appeared in [5] before the notion of block sensitivity was introduced. The analysis of C(f ) and bs(f ) of the example appears in [1]. 2 The result is due to [4,15,30]. 3 The example is due to Kushilevitz and appears in footnote 1 on p. 560 of the Nisan-Wigderson paper [22]. See Example 5.4 in Section 5 of this paper. 4 These gaps are demonstrated by a commonly known AND-of-ORs function, see Example 5.2 in Section 5 of this paper. tivity is polynomially related to ℓ-block sensitivity for any constant ℓ (ℓ-block sensitivity considers only the sensitive blocks of size at most ℓ). The best known upper bound on block sensitivity in terms of sensitivity is exponential and appears in the work of Kenyon and Kutin [16] on ℓ-block sensitivity: In the 80s, Rubinstein [26] exhibited a function with sensitivity Θ( √ n) and block sensitivity Θ(n) (see Example 5.1 in Section 5). Gaps between sensitivity and some other complexity measures are surveyed by Buhrman and de Wolf [6].
In the light of Rubinstein's example, the best possible upper bound on block sensitivity in terms of sensitivity could be quadratic. Nisan and Szegedy [21] asked the following question: Turán [31] proved that any property of n-vertex graphs (viewed as a Boolean function on n 2 variables) has sensitivity Ω(n). Turán asked if every transitive function on n variables has sensitivity at least Ω( √ n). Chakraborty [7] answered this question in the negative by constructing a transitive function with sensitivity Θ(n 1/3 ) and block sensitivity Θ(n 2/3 ) (see Example 5.5 in Section 5). We propose the following modification of Turán's question: An example of a different behavior of transitive Boolean functions with the property f (0) = f (1) is due to Rivest and Vuillemin [25]. They proved that if n is a prime power, and f (0) = f (1), then D(f ) = n. From their proof it can be immediately inferred that in fact deg(f ) = n. A conjecture appearing in the Gotsman-Linial paper [13] states that deg(f ) = O(s(f ) 2 ), which, if true, would answer Question 3.2 positively for n that are prime powers.

Sensitivity vs Other Complexity Measures
Unlike the complexity measures mentioned in Section 2, the complexity measures in this section are not polynomially related to block sensitivity and yet proving a polynomial relation of these measures to sensitivity turns out to be equivalent to proving a polynomial relation between block sensitivity and sensitivity, i.e., the Sensitivity Conjecture itself.
Let F (x, y) be a Boolean function. Consider a setting in which Alice has a Boolean string x and Bob has a Boolean string y, and their goal is to compute the value of F (x, y) by communicating as few bits as possible. Alice and Bob agree on a communication protocol beforehand. Having received inputs, they communicate in accordance with the protocol. At the end of the communication one of the parties declares the value of the function F . The cost of the protocol is the number of bits exchanged on the worst-case input. For more information on communication complexity see [17]. Given a Boolean function on n variables, we will typically consider F (x,

Log-rank vs Sensitivity
For a Boolean function F of two arguments, rank(F ) denotes the rank of the corresponding matrix M x,y = F (x, y) over R.
In this section we present some implications of a recent result by Sherstov. It appears as Theorem 6.4 in [27].  Proof.

Proof.
⇒ Similar to the same direction in Theorem 4.1.3. ⇐ For a Boolean function f , define F (x, y) = f (x ∧ y). It is easy to check that s(f ) ≤ s(F ) ≤ 2s(f ), and also that The result now follows from Theorem 4.1.3.
Definition 4.1.6. The sign-rank of Boolean function F of two arguments, denoted by rank ± , is defined as The notion of sign-rank was introduced by Paturi and Simon [23] to give a characterization of the unbounded error probabilistic communication complexity.
Since rank ± (F ) ≤ rank(F ) for every F , we propose a possibly weaker version of Conjecture 4.1.4 stated for the sign-rank.

Parity Decision Trees
Parity decision trees are similar to decision trees; the difference is that instead of querying only one variable at a time, one may query the sum modulo 2 of an arbitrary subset of variables (see [32] for a brief introduction to parity decision trees). The parity decision tree complexity of a Boolean function f is denoted by In this section, we explore the relationship between D ⊕ and sensitivity. Note that parity decision trees are strictly more powerful than decision trees. For instance, parity of n bits requires a decision tree of depth n whereas a parity decision tree of depth 1 suffices. This seemingly weaker conjecture is actually equivalent to the Sensitivity Conjecture.

Proof.
⇒ Observe that the OR function (and consequently the AND function) has full mod 2 degree. It follows that h has full mod 2 degree, which shows that D ⊕ (h) = n, since for any Boolean function f , deg ⊕ (f ) ≤ D ⊕ (f ).
Similar to the question stated in the survey by Buhrman and de Wolf [6] whether D(f ) ≤ O(bs(f ) 2 ), we ask the following: Remark 4.2.5. A positive answer to the above question would imply that deg(f ) ≤ bs(f ) 2 , improving the current best known bound deg(f ) ≤ bs(f ) 3 (see [3]).

Analytic Setting
In the previous sections, we considered Boolean functions from {0, 1} n to {0, 1}. For the purpose of studying the Fourier spectrum of Boolean functions, it is convenient to use range {+1, −1}, replacing 0 with +1 and 1 with −1. This operation preserves the complexity measures up to an additive constant. For a brief introduction to Fourier Analysis on the Boolean cube, see, for instance, the survey by de Wolf [12].    ⇐ Let α = min S: f (S) =0 | f (S)|. Since S f (S) 2 = 1 (Parseval's Identity), the number of non-zero Fourier coefficients is at most α −2 . Consider matrix M with entries M x,y = f (x ⊕ y). It is easy to check that for each S ⊂ [n], the vector (χ S (y)) y∈{0,1} n is an eigenvector to M with a corresponding eigenvalue 2 n f (S). Since the χ S form an orthogonal set of vectors, the 2 n f (S) are all the eigenvalues of M .
Hence, α ≥ 2 − poly(s(f )) implies that rank of f (x ⊕ y) is at most 2 poly(s(f )) . The proof is complete by Corollary 4.1.5.
The following consequence of the Sensitivity Conjecture appears to be open.   F (x, y) be a Boolean function. Suppose Alice has a Boolean string x and Bob has a Boolean string y. The bounded-error randomized communication complexity with shared randomness of F , denoted by RC 2 (F ), is the least cost of a randomized protocol that computes F correctly with probability at least 2/3 on every input, when Alice and Bob are given the same random bits.
Next we prove that Conjecture 4.3.5 is equivalent to the Sensitivity Conjecture under the following variant of the Log-rank Conjecture due to Grolmusz [14]. To prove the equivalence we will need the following result by Sherstov(see [27], Theorem 5.1). Proof. Assume Conjecture 4.3.7. Now, we want to prove that Sensitivity Conjecture ⇐⇒ Conjecture 4.3.5.

Shi's Characterization of Sensitivity
In this section we present some applications of Shi's work [28], which contains an interesting characterization of the sensitivity of Boolean functions.
A polynomial representing a Boolean function f : {0, 1} n → {0, 1} (see Section 2) provides a multilinear extension of f from R n to R, which (abusing notation) we denote by the same letter f .
Let ℓ = (a, b) denote the line segment in [0, 1] n that starts at point a and ends at point b. Theorem 4.4.2 (Shi [28]). For every Boolean function f , s(f ) = sup ℓ ||f ′ ℓ || ∞ . Proof. It is easy to check that it suffices to consider the lines that join two points of the Boolean cube.
For x ∈ [0, 1] n , let x (i,1) (x (i,0) ) denote a vector whose i th coordinate is 1 (0) and the other coordinates match with those of x. Let a, b ∈ {0, 1} n and ℓ = (a, b) be the line joining a and b.
Thus we have: where D t denotes the following probability distribution on Boolean cube: for each k, Pr(p k = 1) = (1 − t)a k + b k . Notice that the right hand side of (1) is at most s(f ). For the other direction let a ∈ {0, 1} n and b be obtained from a by flipping each bit. It is easy to check that: = s(f, a).
Choosing a vector a with maximum sensitivity completes the proof.
Combining Theorem 4.4.2 with Conjecture 4.3.3 puts the original "sensitivity versus block sensitivity" problem into an analytic setting.  Observe that, unlike all previous equivalence results, Theorem 4.4.4 gives a complexity measure polynomially related to s(f ) rather than bs(f ). It follows that the following conjecture is equivalent to the Sensitivity Conjecture.

Subgraphs of the n-cube
Let Q n denote the n-cube graph, i.e., V (Q n ) = {0, 1} n and two vertices are adjacent if the corresponding vectors differ in exactly one position. Denote the maximum degree of graph G by ∆(G). For a subgraph H of a graph G define Γ(H) = max{∆(H), ∆(G − H)}. Gotsman and Linial [13] proved the following remarkable equivalence. The proof of Theorem 4.5.1 translates a Boolean function with a polynomial gap between degree and sensitivity into a graph with the same polynomial gap between Γ and n, and vice versa. For example, observe that Rubinstein's function (see Example 5.1 in Section 5) has sensitivity Θ(n) and full degree, which can be easily verified by a direct computation of f ([n]). Therefore, Rubinstein's function can be used to obtain a graph G with the surprising property Γ(G) = Θ( √ n). Chung et al. [9] independently constructed a graph G with Γ(G) < √ n + 1. Their example can be also obtained from Theorem 4.5.1 by applying the reduction in the proof of A ⇒ B' to the AND-of-ORs function (see Example 5.2 in Section 5), but note that the Gotsman-Linial theorem was not available at the time when Chung et. al. gave their construction.
It immediately follows that the following conjecture is equivalent to the Sensitivity Conjecture, by taking h to be an inverse polynomial in Theorem 4.5.1.

Two-colorings of Integer Lattices
Two points a, b ∈ Z d are called neighbors if ||a − b|| 2 = 1. A two-coloring C of Z d with colors red and blue is non-trivial if the origin is colored red, and there is a point colored blue on each of the coordinate axes. Sensitivity of a point a ∈ Z d under coloring C, denoted by S(a, C), is the number of neighbors of a that are colored differently from a. Aaronson [2] stated the following question, a positive answer to which would imply the Sensitivity Conjecture. For completeness, we also present a reduction. Claim 4.6.3 (Aaronson). A positive answer to Question 4.6.2 implies that Sensitivity Conjecture.
Proof. Given a Boolean function f on n variables, let x be an input, on which f achieves the highest block sensitivity b = bs(f ). Let S 1 , . . . , S b be pairwise disjoint sensitive blocks of f on x, and let R = [n] − ( i S i ). Let γ i : Z → {0, 1} |Si| represent a Gray code with γ i (0) = x| Si . Consider the following mapping φ : Z b → {0, 1} n : a point a ∈ Z b is mapped to a Boolean string y ∈ {0, 1} n with y| Si = γ i (a i ) and y| R = x| R . Finally, obtain coloring C of Z b by composing f with φ. Clearly, C is non-trivial and s(C) ≤ 2s(f ), hence bs(f ) = b ≤ poly(s(C)) ≤ poly(s(f )).

Some Boolean Functions
In this section we present some interesting examples of Boolean functions. They provide lower or upper bounds for various complexity measures, and some of them appear in more than one context.
The following function was exhibited by Rubinstein [26]. It was discussed in Section 3 and Section 4.5 of this note.
Example 5.1 (Rubinstein's function). Rubinstein's function is defined on n = k 2 variables, which are divided into k blocks with k variables each. The value of the function is 1 if there is at least one block with exactly two consecutive 1s in it, and it is 0 otherwise.
Block sensitivity of Rubinstein's function on k 2 variables is Θ(k 2 ) (hence, certificate complexity and decision tree complexity is also Θ(k 2 )) and sensitivity is Θ(k). It has full degree as can be verified by a direct computation of Fourier coefficient of the set [k 2 ].
The following folklore example was discussed in Section 2 and Section 4.5 of this paper. The block sensitivity and sensitivity of AND-of-ORs function on k 2 variables is k. AND-of-ORs has full degree and hence its decision tree complexity is also k 2 . The certificate complexity of AND-of-ORs function is k. f ⋄ g(x 11 , . . . , x mn ) = f g(x 11 , . . . , x 1n ), . . . , g(x m1 , . . . , x mn ) .
Kushilevitz exhibited a function f that provides the largest gap in the exponent of a polynomial in deg(f ) that gives an upper bound on bs(f ). Never published by Kushilevitz, the function appears in footnote 1 of the Nisan-Wigderson paper [22]. It was discussed in Section 2 of this paper. h(z 1 , . . . , z 6 ) = i z i − ij z i z j + z 1 z 3 z 4 + z 1 z 2 z 5 + z 1 z 4 z 5 + z 2 z 3 z 4 + z 2 z 3 z 5 + z 1 z 2 z 6 + z 1 z 3 z 6 + z 2 z 4 z 6 + z 3 z 5 z 6 + z 4 z 5 z 6 .
Observe that the auxiliary function h on 6 variables in Kushilevitz's example has degree 3 and full sensitivity on the 0 input. Let the function f be obtained by composing h with itself k times. It is defined on n = 6 k variables and has full sensitivity, block sensitivity, decision tree complexity and ceritificate complexity. Degree of f is 3 k = n log 6 3 .