Sensitivity, affine transforms and quantum communication complexity
Introduction
For a Boolean function , sensitivity of f on , is the maximum number of indices , such that where with exactly the ith bit as 1. The sensitivity of f (denoted by ) is the maximum sensitivity of f over all inputs. A related parameter is the block sensitivity of f (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 f. The certificate complexity of f (denoted by ) is the non-deterministic variant of the decision tree complexity. The parameter was originally studied by Cook et al. [1] in connection with the CREW-PRAM model of computation. Subsequently, Nisan and Szegedy [2] (see also [3]) 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 [4] for a survey and several equivalent formulations of the conjecture). This conjecture has recently been resolved in [5] by showing the following which implies that . Theorem 1.1 Sensitivity Theorem [5] For every Boolean function f, .
Shi and Zhang [6] 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 f, where g is defined as for specific and (where is M not necessarily invertible). 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 M is the identity matrix and is a linear shift. More precisely, we study where b 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 f as the majority function on n bits, then there exists shifts where while .
A result related to Sensitivity Conjecture by Lin and Zhang [7] 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 f. However, the authors [8] 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 g obtained from f upon shifting by a vector (Definition 3.1). By the aforementioned bound on of [7], it is easy to observe that . We also show that there exists a family of Boolean functions f with (Proposition 3.5).
It is conceivable that is much smaller compared to for a Boolean function f 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 [8] (see also [9]). Proposition 1.2 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 (f) is upper bounded by the sensitivity of the new function (g). We use to denote the number of sensitive blocks of f on the input a. Lemma 1.3 For any and , there exists an affine transform such that for , , and where are not necessarily distinct.
The above transformation is used in Nisan and Szegedy (see Lemma 7 of [10]) to show that . Here, is the degree of the multilinear polynomial over reals that agrees with f on Boolean inputs. We show another application of Lemma 1.3 in the context of quantum communication complexity, a model for which was introduced by Yao [11]. 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 f 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 F with prior entanglement. For more details on this model, see [12]. 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 F with error at most 1/3. 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 [13] that this is not the case in general (which they called 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 x and y. Question 1.4 For , let be defined as . Is it true that for any such F, ?
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 F has also been proposed by Klauck [14] as a first step towards answering the general question (see also [15]). In this direction, Razborov [12] showed that for the special case when f is symmetric, satisfy . In the process, Razborov developed powerful techniques to obtain lower bounds on which were subsequently generalized by Sherstov [16], Shi and Zhu [13]. Subsequently, in a slightly different direction, Sherstov [17] showed that instead of computing alone, if we consider F to be the problem of computing both of and , then for all Boolean functions f where and . Using Lemma 1.3, we build on the ideas of Sherstov [17] and obtain a lower bound for where . Theorem 1.5 Let and , then,
Using the above result, for a prime p, we show that if f has small degree when expressed as a polynomial over (denoted by ), the quantum communication complexity of F is large. Theorem 1.6 Fix a prime p. Let where f depends on all the variables. Let . For any such that , we have
Observe that, though Theorem 1.5 does not answer Question 1.4 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 f such that there exists two distinct primes with and are sufficiently far apart (Theorem 1.7). Theorem 1.7 Let with . Fix . If there exist distinct primes p, q such that , then .
Alternation under linear transforms: We now restrict our study to linear transforms. Again, in this context, the aim is to design special linear transforms for the parameters of interest. 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: Lemma 1.8 For any , there exists an invertible linear transform such that for ,
We show an application of the above result in the context of the parameter sensitivity. Nisan and Szegedy [10] showed that for any Boolean function f, . However, the situation is quite different for - noticing that for f being parity on n variables, and - the gap can even be unbounded. Though parity may appear as a corner case, there are other functions like the Boolean inner product function2 whose -degree is constant while sensitivity is thereby ruling out the possibility that . It is known that if f is not the parity on n variables (or its negation), [19], [20]. Hence, as a structural question about the two parameters, we ask: for f other than the parity function, is it true that .3 In fact, the Sensitivity Theorem (Theorem 1.1) by [5] implies that for every Boolean function f, . Hence, if we could answer our question in affirmative, it would imply that and are polynomially related. We use Lemma 1.8, which is in the theme of studying alternation and sensitivity in the context of linear transformations, to show that this is not the case, by exhibiting a family of functions where the gap is exponential.
Theorem 1.9 There exists a family of functions such that
This family of Boolean functions also rules out a potential approach to settle the XOR Log-Rank conjecture via the recently settled Sensitivity conjecture [5]. We elaborate on this approach and how our function family rules it out in Appendix 5.
Section snippets
Preliminaries
In this section, we define the notations used. Define . For , define to be the indicator vector of the set S. For , we denote (resp. ) as the string obtained by bitwise AND (resp. XOR) of x and y. We use to denote the ith bit of x.
We now define the Boolean function parameters we use. Let and , we define, 1) the sensitivity of f on a as , 2) the block sensitivity of f on a, to
Warm up: alternation under shifts
In this section, as a warm-up, we study sensitivity and alternation under linear shifts (when the matrix M 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 1.2) thereby ruling out the possibility that can be upper bounded by a polynomial in for all Boolean functions f.
Recall from the introduction that the
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 1.6 and Theorem 1.7 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 4.4), we show that for any Boolean function f, and any prime p, . This
Linear transforms: sensitivity versus sparsity
Continuing in the theme of affine transforms, in this section, we first establish an upper bound on alternation of a function in terms of sensitivity of the function after application of a suitable linear transform. Using this, we show the existence of a function whose sensitivity is asymptotically as large as square root of sparsity (see introduction for a motivation and discussion). Lemma 1.8 For any , there exists an invertible linear transform such that for ,
Conclusion and future directions
In this paper, we study the Boolean function complexity measures, namely sensitivity, block sensitivity, and alternation under affine transforms. We showed design of special transforms which achieves structurally revealing statements about the resulting function. We used their properties to show lower bounds on the bounded error quantum communication complexity of Boolean function whose -degree is small. We showed that classical and quantum communication complexity are polynomially related
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgement
The authors would like to thank the anonymous reviewers for their constructive comments to this paper, specifically for pointing out an error in the earlier version of Theorem 1.5 by giving examples. See the Remark 4.2 and the discussion after Theorem 1.5 of this paper.
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