Sensitivity, Affine Transforms and Quantum Communication Complexity
In this paper, we study the Boolean function parameters sensitivity (), block sensitivity (), and alternation () under specially designed affine transforms and show several applications. For a function , and for and , the result of the transformation is defined as .
As a warm up, we study alternation under linear shifts (when is restricted to be the identity matrix) called the shift invariant alternation (the smallest alternation that can be achieved for the Boolean function by shifts, denoted by ). By a result of Lin and Zhang [LZ17], it follows that . Thus, to settle the Sensitivity Conjecture (), it suffices to argue that . However, we exhibit an explicit family of Boolean functions for which is .
Going further, we use an affine transform , such that the corresponding function satisfies , to prove that for , the bounded error quantum communication complexity of with prior entanglement, is . Our proof builds on ideas from Sherstov [She10] where we use specific properties of the above affine transformation. Using this, we show the following.
For a fixed prime and an , , any Boolean function that depends on all its inputs with must satisfy . Here, denotes the degree of the multilinear polynomial over which agrees with on Boolean inputs.
For Boolean function such that there exists primes and with for , the deterministic communication complexity - and are polynomially related. In particular, this holds when . Thus, for this class of functions, this answers an open question (see [BdW01]) about the relation between the two measures.
Restricting back to the linear setting, we construct linear transformation , such that the corresponding function satisfies, . Using this new relation, we exhibit Boolean functions (other than the parity function) such that is where is the number of non-zero coefficients in the Fourier representation of .
For a Boolean function , sensitivity of on , is the maximum number of indices , such that where with exactly the bit as . The sensitivity of (denoted by ) is the maximum sensitivity of over all inputs. A related parameter is the block sensitivity of (denoted by ), where we allow disjoint blocks of indices to be flipped instead of a single bit. Another parameter is the deterministic decision tree complexity (denoted by ) which is the depth of an optimal decision tree computing the function . The certificate complexity of (denoted by ) is the non-deterministic variant of the decision tree complexity. The parameter was originally studied by Cook et al. [CDR86] in connection with the CREW-PRAM model of computation. Subsequently, Nisan and Szegedy [NS92] (see also [Nis91]) introduced the parameters and and conjectured that for any function , - known as the Sensitivity Conjecture. Later developments, which revealed several connections between sensitivity, block sensitivity and the other Boolean function parameters, demonstrated the fundamental nature of the conjecture (see [HKP11] for a survey and several equivalent formulations of the conjecture). The best known upper bound for in terms of is due to He et al. [HLS17] improving a result of Ambainis et al. [And16].
Shi and Zhang [ZS10] studied the parity complexity variants of and and observed that such variants have the property that they are invariant under arbitrary invertible linear transforms (over ). They also showed existence of Boolean functions where under all invertible linear transforms of the function, the decision tree depth is linear while their parity variant of decision tree complexity is at most logarithmic in the input length.
Our Results : While the existing studies focus on understanding the Boolean function parameters under the effect of arbitrary invertible affine transforms, in this work, we study the relationship between the above parameters of Boolean functions , under specific affine transformations over . More precisely, we explore the relationship of the above parameters for the function and , where is defined as for specific and . We show the following results, and their corresponding applications, which we explain along with the context in which they are relevant.
Alternation under shifts : We study the parameters when the transformation is very structured - namely the matrix is the identity matrix and is a linear shift. More precisely, we study where is the shift. Observe that all the parameters mentioned above are invariant under shifts. A Boolean function parameter which is neither shift invariant nor invariant under invertible linear transforms is the alternation, a measure of non-monotonicity of Boolean function (see Section 2 for a formal definition). To see this for the case of shifts, if we take as the majority function on bits, then there exists shifts where while .
A recent result related to Sensitivity Conjecture by Lin and Zhang [LZ17] shows that . This bound for , implies that to settle the Sensitivity Conjecture, it suffices to show that is upper bounded by for all Boolean functions . However, the authors [DS18] ruled this out, by exhibiting a family of functions where is at least .
Observing that the parameters are invariant under shifts, we define a new quantity shift-invariant alternation, which is the minimum alternation of any function obtained from by shifting by a vector (see creftype 3.1). By the aforementioned bound on of Lin and Zhang [LZ17], it is easy to observe that . We also show that there exists a family of Boolean functions with (See Proposition 3.5).
It is conceivable that is much smaller compared to for a Boolean function and hence that can potentially be upper bounded by thereby settling the Sensitivity Conjecture. However, we rule this out by showing the following stronger gap, about the same family of functions demonstrated in [DS18] (see also [GSW16]).
There exists an explicit family of Boolean functions for which is
Block Sensitivity under Affine Transformations : We now generalize our theme of study to the affine transforms over . In particular, we explore how to design affine transformations in such a way that block sensitivity of the original function () is upper bounded by the sensitivity of the new function ().
For any and , there exists an affine transform such that for ,
where are not necessarily distinct.
The above transformation is used in Nisan and Szegedy (see Lemma 7 of [NS92]) to show that . Here, is the degree of the multilinear polynomial over reals that agrees with on Boolean inputs. We show another application of creftype 1.2 in the context of quantum communication complexity, a model for which was introduced by Yao [Yao93]. In this model, two parties Alice and Bob have to compute a function , where Alice is given an and Bob is given a . Both the parties have to come up with a quantum protocol where they communicate qubits via a quantum channel and compute while minimizing the number of qubits exchanged (which is the cost of the quantum protocol) in the process. In this model, we allow protocols to have prior entanglement. We define as the minimum cost quantum protocol computing with prior entanglement. For more details on this model, see [Raz03]. The corresponding analog in the classical setting is the bounded error randomized communication model where the parties communicate with bits and share an unbiased random source. We define as the minimum cost randomized protocol computing with error at most . It can be shown that .
One of the fundamental goals in quantum communication complexity is to see if there are functions where their randomized communication complexity is significantly larger than their quantum communication complexity. It has been the conjectured by Shi and Zhu [SZ09] that this is not the case in general (which they called as the Log-Equivalence Conjecture). In this work, we are interested in the case when is of the form where and is the string obtained by bitwise AND of and .
For , let be defined as . Is it true that for any such , ?
Since , answering the above question in positive would show that the classical randomized communication model is as powerful as the quantum communication model for the class of functions . This question for such restricted has also been proposed by Klauck [Kla07] as a first step towards answering the general question (see also [BdW01]). In this direction, Razborov [Raz03] showed that for the special case when is symmetric, satisfy . In the process, Razborov developed powerful techniques to obtain lower bounds on which were subsequently generalized by Sherstov [She08], Shi and Zhu [SZ09]. Subsequently, in a slightly different direction, Sherstov [She10] showed that instead of computing alone, if we consider to be the problem of computing both of and , then for all Boolean functions where and . Using creftype 1.2, we build on the ideas of Sherstov [She10] and obtain a lower bound for where .
Let and , then,
In this context, we make an important comparison with a result of Sherstov [She10]. He proved that for
, where is the input on which is maximum, (Corollary 4.5 of [She10]). Notice that and differ by a linear shift
of with .
Using the above result, for a prime , we show that if has small degree when expressed as a polynomial over (denoted by ), the quantum communication complexity of is large.
Fix a prime . Let where depends on all the variables. Let . For any such that , we have
1.1 Quantum communication lower bound from block sensitivity
Sherstov in [She10] showed the following lower bound on quantum communication cost of an affine shift of a Boolean function in terms of its block sensitivity.
Corollary 1.6 (Corollary 4.5 of [She10]).
Let be given. Then for some , the matrix obeys
In this section, we elaborate on why one cannot set for all Boolean functions and obtain Theorem 1.4. The above corollary crucially uses two results. The first one is Lemma 3.3 of [She10] which shows that there exists a Boolean function such that which is similar in spirit to creftype 1.2. The second one is Theorem 4.2 of [She10] which shows a lower bound for in terms of sensitivity of (where ). We reproduce the respective statements of both below.
Lemma 1.7 (Lemma 3.3 of [She10]).
Let . Then there exists a such that and for some
The function is defined as follows.
Let be the input on which is maximum and . Let be the sensitive blocks on . Define and . Let be the indices such that both and are both non-empty.
We observe that the above result of Sherstov (Lemma 3.3 of [She10]) can be seen as applying a suitable linear transform to the Boolean function to bound the block sensitivity of which is similar in spirit to creftype 1.2.
More precisely, the obtained in Lemma 3.3 of [She10] can be described as where is defined as, for ,
By definition as above, Sherstov showed that .
Theorem 1.9 (Theorem 4.2 of [She10]).
For a Boolean function with , if there exists an such that for and , then .
To use the above result, one way is to start with a function for which sensitivity is large at . To achieve, consider the shifted function where is the same input on which block sensitivity is maximized as before. This is because, by the choice of , will have maximum block sensitivity at which upon applying Lemma 3.3 of [She10] ensures that the function obtained has a large (i.e. sensitivity) at . This is exactly what is achieved in the proof of Corollary 4.5 of [She10].
Hence the choice is is tied up with the block sensitivity of function .
Observe that, though Theorem 1.4 does not answer creftype 1.3 in positive for all functions, we could show a class of Boolean function for which and are polynomially related. More specifically, we show this for the set of all Boolean functions such that there exists two distinct primes with and are sufficiently far apart (Theorem 1.10).
Let with . Fix . If there exists distinct primes , such that , then .
By the result of Gopalan et al. (Theorem 1.2, [GLS09]), any Boolean function with must have thereby satisfying the condition of Theorem 1.10. Hence for all such functions, Theorem 1.10 answers creftype 1.3 in positive. Observe that the same can also be derived from Theorem 1.5.
Alternation under Linear Transforms : We now restrict our study to linear transforms. Again, the aim is to design special linear transforms which transforms the parameters of interest for us. In particular, in this case, we show linear transforms for which we can upper bound the alternation of the original function in terms of the sensitivity of the resulting function. More precisely, we prove the following lemma:
For any , there exists an invertible linear transform such that for ,
We show an application of the above result in the context of sensitivity.
Nisan and Szegedy [NS92] showed that for any Boolean function , . However, the situation is quite different for - noticing that
for being parity on variables, and - the gap can even
Though parity may appear as a corner case, there are other functions like the Boolean inner product function
There exists a family of functions such that
In this section, we define the notations used. Define . For , define to be the indicator vector of the set . For , we denote (resp. ) as the string obtained by bitwise AND (resp. XOR) of and . We use to denote the bit of .
We now define the Boolean function parameters we use. Let and , we define, 1) the sensitivity of on as , 2) the block sensitivity of on , to be the maximum number of disjoint blocks such that and 3) the certificate complexity of on , to be the size of the smallest set such that fixing according to on the location indexed by causes the function to become constant. For , we define and are respectively called the sensitivity, the block sensitivity and the certificate complexity of . By definition, the three parameters are shift invariant, by which we mean , for where . Also, it can be shown that .
For , define if , . We define a chain on as such that for all , and . We define alternation of for a chain , denoted as the number of times the value of changes in the chain. We define alternation of a function as .
Every Boolean function can be expressed uniquely as a multilinear polynomial in over any field such that . Fix a prime . We denote (resp. ) to be the degree of the multilinear polynomial computing over reals (resp. ). We define as the depth of an optimal decision tree computing . It is known that for all Boolean functions , .
Sparsity of a Boolean function (denoted by ) is the number of non-zero Fourier coefficients in the Fourier representation of . For more details on this parameter, see [O’D14]. For more details on and other related parameters, see the survey by Buhrman, de Wolf [BdW02] and Hatami et al. [HKP11].
We consider the two party classical communication model. Given a function , Alice is given an and Bob is given . They can communicate with each other and their aim is to compute while communicating minimum number of bits. We call the procedure employed by Alice and Bob to computing as the protocol. We define as the minimum cost of a deterministic protocol computing . For functions of the form , it is known that [MO09]. For more details on communication complexity of Boolean functions, refer [KN06].
3 Warm up : Alternation under Shifts
In this section, as a warm-up, we study sensitivity and alternation under linear shifts (when the matrix is the identity matrix). We introduce a parameter, shift-invariant alternation (). We then show the existence of Boolean functions whose shift-invariant alternation is exponential in its sensitivity (see Proposition 1.1) thereby ruling out the possibility that can be upper bounded by a polynomial in for all Boolean functions .
Recall from the introduction that the parameters and are shift invariant while is not. We define a variant of alternation which is invariant under shifts.
Definition 3.1 (Shift-invariant Alternation).
For , the shift-invariant alternation (denoted by ) is defined as .
A family of functions with : We now exhibit a family of functions where for all , thereby ruling out the possibility that can be upper bounded by a polynomial in . The family is the same class of Boolean functions for which alternation is at least exponential in sensitivity due to [DS18].
Consider the family defined as follows.
The Boolean function is computed by a decision tree which is a full binary tree of depth with leaves. A leaf node is labeled as (resp. ) if it is the left (resp. right) child of its parent. All of the nodes (except the leaves) are labeled by a distinct variable.
We remark that Gopalan et al. [GSW16] demonstrates an exponential lower bound on tree sensitivity (introduced by them as a generalization of the
parameter sensitivity) in terms of decision tree depth for the same family of functions in creftype 3.2. We remark that, in general, lower bound on tree sensitivity need not implies a lower bound on alternation. For instance, if we consider the Majority function
The authors [DS18] have shown that for any , there exists of a chain of large alternation in . However, this is not sufficient to argue existence of a chain of large alternation under every linear shift. We now proceed to prove an exponential lower bound on in terms of for all .
For , .
Proof is by induction on . For , is a function on variables and it can be verified that for all , . Now consider an computed by a decision tree with the variable as its root. Let and be the left and right subtrees of in . Note that and depends on variables and belongs to by construction. Hence by induction, for all , and is at least . For , consider any where and . Since and are variable disjoint, completing the induction. ∎
A family of functions with : Lin and Zhang [LZ17] showed that for any Boolean function ,
This immediately gives the following proposition.
For any ,
We now exhibit a family of functions for which is at least .
Before proceeding, we show a tight composition result for alternation of Boolean functions when composed with (which is the bit Boolean OR function).
For functions where each , define the function as where for each , is input to the function .
Consider Boolean functions where each satisfy, . Then,
Let and be a chain in for which is maximized. Without loss of generality, let all the functions be non-constant. Let be the chain in obtained by restricting to variables of . Observe that if changes it value, it must be that at least one of the ’s have changed their evaluation along the chain . Since the functions are variable disjoint, such a change must be witnessed in the chain for some . Hence
To show that , we exhibit a chain in of alternation . Let be a chain in for which achieves maximum alternation. We construct a chain by “gluing” together these chains. More precisely, let by the chain such that for all , when restricted to the variables , we get a chain given by,
By construction of , since for all , at any input of the chain , there is exactly one that causes to alternate. Hence,
There exists a family of Boolean functions for which
We consider the Rubinstein’s function [Rub95] where the input is treated as matrix which evaluates to iff there is a row with two consecutive ones starting at the odd position and rest of the entries being zero. Alternatively, we can view as with where iff there are two consecutive ones starting at the odd position with rest of the entries as zero in . It can be verified that . Since , applying creftype 3.4 with for all , we get that . It is known that while [Rub95], thereby showing that . ∎
We remark that the above bound is stronger than what is needed in the context because, .
4 Affine Transforms : Lower Bounds on Quantum Communication Complexity
In this section, we study the affine transformation in its full generality applied to block sensitivity and sensitivity, and use it to prove Theorem 1.5 and Theorem 1.10 from the introduction. We achieve this using affine transforms as our tool (Section 4.1), by which we derive a new lower bound for in terms of (Section 4.2). Using this and a lower bound on (Proposition 4.3), we show that for any Boolean function , and any prime , . This immediately implies that if there is a such that is constant, then thereby answering creftype 1.3 in positive for such functions. We relax this requirement and show that if there exists distinct primes and for which and are not very close, then (Theorem 1.10).
4.1 Upper Bound for Block Sensitivity via Affine Transforms
In this section, we describe our main tool. Given an and any , we exhibit an affine transform such that for , .
Before describing the affine transform, we note that a linear transform is already known to achieve a weaker bound of due to Sherstov [She10].
Proposition 4.1 (Lemma 3.3 of [She10]).
For any , there exists a linear transform such that for , .
Now we describe an affine transform which improves the bound on in the above proposition to linear in . This affine transform has already been used in Nisan and Szegedy (see Lemma 7 of [NS92]) to show that . Since the exact form of is relevant in the subsequent arguments, we explicitly prove it here bringing out the structure of the affine transform that we require.
For any and , there exists an affine transform such that for ,
where are not necessarily distinct.