 # On Block Sensitivity and Fractional Block Sensitivity

We investigate the relation between the block sensitivity bs(f) and fractional block sensitivity fbs(f) complexity measures of Boolean functions. While it is known that fbs(f) = O(bs(f)^2), the best known separation achieves fbs(f) = (1/3√(2) +o(1)) bs(f)^3/2. We improve the constant factor and show a family of functions that give fbs(f) = (1/√(6)-o(1)) bs(f)^3/2.

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## 1 Introduction

The query complexity of Boolean functions is one of the simplest models of computation. In this setting, the cost of the computation is the number of the input bits one needs to query to decide the value of the function on this input. One of the main challenges is to precisely relate the computational power of the decision tree complexity , randomized decision tree complexity and quantum decision tree complexity (see [ABDK16] for the currently known relations between various complexity measures).

Block sensitivity is a useful intermediate measure that has been used to show polynomial relations between the above measures. Fractional block sensitivity (aka fractional certificate complexity , randomized certificate complexity [Aar08]) is a recently introduced measure that is a relaxation of block sensitivity [Tal13]. It has been used to show a tight relation (up to logarithmic factors) between the zero-error randomized decision tree complexity and two-sided bounded error randomized decision tree complexity [KT16].

The relation between and has been only partially understood. On one hand, and this inequality is tight. On the other hand, it is known that but the best known separation gives [GSS16]. We show a family of functions that give a constant factor improvement, .

## 2 Definitions

Let be a Boolean function on variables. We denote the input to by a binary string , so that the -th variable is . For an index set , let be the input obtained from an input by flipping every bit , .

We briefly define the notions of sensitivity, certificate complexity and variations on them. For more information on them and their relations to other complexity measures (such as deterministic, probabilistic and quantum decision tree complexities), we refer the reader to the surveys by Buhrman and de Wolf [BdW02] and Hatami et al. [HKP11].

The sensitivity complexity of on an input is defined as

 s(f,x)=|{i∈[n]∣f(x)≠f(x{i})}|. (1)

The sensitivity of is defined as .

The block sensitivity of on an input is defined as the maximum number such that there are pairwise disjoint subsets of for which . We call each a block. The block sensitivity of is defined as .

The fractional block sensitivity of on an input

is the optimal value of the following linear program, where each sensitive block of

is assigned a real valued weight :

 max∑f(x)≠f(xB)wBsubject % to: ∀i∈[n]:∑B∋iwB≤1, ∀B:0≤wB≤1.

The fractional block sensitivity of is defined as .

A certificate of is a partial assignment of the input such that is constant on this restriction. We call the length of . If is always 0 on this restriction, the certificate is a 0-certificate. If is always 1, the certificate is a 1-certificate.

The certificate complexity of on an input is defined as the minimum length of a certificate that satisfies. The certificate complexity of is defined as .

The fractional certificate complexity of on an input is the optimal value of the following linear program, where each position is assigned a real valued weight :

 min∑i∈[n]visubject to: ∀B s.t. f(x)≠f(xB):∑i∈Bvi≥1, ∀i∈[n]:0≤vi≤1.

The fractional certificate complexity of is defined as .

For any of these measures , define . In that way, we define the measures , , , , , , , , , . In particular, .

One can show that [Tal13]. In fact, the linear programs of and are duals of each other. Therefore, .

## 3 Separation

The separation in [GSS16] composes a graph property Boolean function (namely, whether a given graph is a star graph) with the function. We build on these ideas and define a new graph property for the composition that gives a larger separation.

###### Theorem 1.

There exists a family of Boolean functions such that

 fbs(f)=(1√6−o(1))bs(f)3/2.
###### Proof.

Let be a multiple of 3. An input on variables encodes a graph on vertices. Let iff the vertices and are connected by an edge in .

We define an auxiliary function . Partition into three sets such that . Let iff:

• there is some vertex that is connected to every other vertex by an edge (a star graph);

• for any , no two vertices such that are connected by an edge.

Formally, iff satisfies one of the following 1-certificates : assigns 1 to every edge in , and assigns 0 to every edge in .

Now we calculate the values of .

• .

Consider an input describing a triangle graph between vertices . For this input . Let be an input obtained from by removing the edge and adding all the missing edges , for all . The corresponding graph is a star graph, therefore, . Let be the sensitive block that flips to . Similarly define and . None of the three blocks overlap, hence .

Now we prove that . Assume the contrary, that there exists an input with . Then has (at least) 4 non-overlapping sensitive blocks . Each satisfies one of the 1-certificates, each a different one. There are 4 such certificates, therefore at least two of them require a star at vertices belonging to the same . The corresponding certificates and both assign 1 at the edge . On the other hand, every other assigns 0 at . Therefore, of the 4 certificates corresponding to , two assign 1 to this edge and two assign 0 to this edge. Then, regardless of the value of , we would need to flip it in two of the blocks : a contradiction, since the blocks don’t overlap. Therefore no such exists.

• .

Examine any 1-certificate . Find three indices (this is possible, as ). Any input that satisfies has . On the other hand, any other 1-certificate requires at least two of the variables to be 0. Hence, the Hamming distance between and is at least two. Therefore, flipping any position of that is fixed in changes the value of the function as well. Thus, . As , we have

 bs(f,x)=C(f,x)=|Ci|=3(N/32)+2N3=N26+N6.
• .

Examine the all zeros input . Any sensitive block of this input flips the edges on a star from some vertex. Therefore, any position is flipped by exactly two of the sensitive blocks. The weights for each sensitive block then give a feasible solution for the fractional block sensitivity linear program. As there are sensitive blocks, .

To obtain the final function we use the following lemma:

###### Lemma 2 (Proposition 31 in [Gss16]).

Let be a non-constant Boolean function and

 f=OR(g(1),…,g(m)),

an composed with copies of . Then for complexity measures , we have

 M1(f) =M1(g) M0(f) =m⋅M0(g).

Let . Then . On the other hand, . Therefore, we have

 fbs(f)≥(N26+N6)⋅N6=bs(f)⋅(1√6−o(1))√bs(f)=(1√6−o(1))bs(f)3/2.

## References

• [Aar08] Scott Aaronson. Quantum certificate complexity. Journal of Computer and System Sciences, 74(3):313–322, 2008.
• [ABDK16] Scott Aaronson, Shalev Ben-David, and Robin Kothari. Separations in query complexity using cheat sheets. In

Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing

, STOC ’16, pages 863–876, New York, NY, USA, 2016. ACM.
• [BdW02] Harry Buhrman and Ronald de Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21–43, 2002.
• [GSS16] Justin Gilmer, Michael Saks, and Srikanth Srinivasan. Composition limits and separating examples for some Boolean function complexity measures. Combinatorica, 36(3):265–311, 2016.
• [HKP11] Pooya Hatami, Raghav Kulkarni, and Denis Pankratov. Variations on the Sensitivity Conjecture. Number 4 in Graduate Surveys. Theory of Computing Library, 2011.
• [KT16] Raghav Kulkarni and Avishay Tal. On fractional block sensitivity. Chicago Journal of Theoretical Computer Science, 8:1–16, 2016.
• [Tal13] Avishay Tal. Properties and applications of Boolean function composition. In Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, ITCS ’13, pages 441–454, New York, NY, USA, 2013. ACM.