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arXiv:1810.02393

On Block Sensitivity and Fractional Block Sensitivity

Andris Ambainis, Krišjānis Prūsis, Jevgēnijs Vihrovs

Published Oct 4, 2018• 0 citations
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Abstract

We investigate the relation between the block sensitivity $\text{bs}(f)$ and fractional block sensitivity $\text{fbs}(f)$ complexity measures of Boolean functions. While it is known that $\text{fbs}(f) = O(\text{bs}(f)^2)$, the best known separation achieves $\text{fbs}(f) = \left(\frac{1}{3\sqrt2} +o(1)\right) \text{bs(f)}^{3/2}$. We improve the constant factor and show a family of functions that give $\text{fbs}(f) = \left(\frac{1}{\sqrt6}-o(1)\right) \text{bs}(f)^{3/2}.$

DOI
10.48550/arxiv.1810.02393

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In this paper we study the fractional block sensitivity of Boolean functions. Recently, Tal [Tal13] and Gilmer, Saks, and Srinivasan [GSS13] independently introduced this complexity measure, denoted by fbs(f), and showed that it is equal (up to a constant factor) to the randomized certificate complexity, denoted by RC(f), which was introduced by Aaronson [Aar03]. In this paper, we relate the fractional block sensitivity to other complexity measures such as sensitivity s(f) and approximate degree deg(f). As a consequence we obtain the following results: 1. We show that deg(f) = Ω( √ RC(f)), solving an open question posed by Aaronson [Aar03]. This also implies that deg(f) = Ω(QC(f)), where QC(f) is the quantum certificate complexity of f. As both deg(f) and QC(f) serve as lower bounds for the bounded error quantum query complexity, this shows that deg(f) is always a tighter lower bound compared to QC(f). 2. (a) We show that every transitive function on n variables must have RC(f) = Ω(n), QC(f) = Ω(n) and deg(f) = Ω(n), and note that all these bounds are tight. This is a strengthening of the previous lower bounds given by [SYZ04] and [Sun07]. (b) We show that Chakraborty’s [Cha11] example of a transitive function with s(f) = O(n) is optimal unless there is better than quadratic separation between the block sensitivity and the sensitivity. 3. Using fractional block sensitivity, we show that the zero error randomized decision tree complexity, R0(f), is upper bounded by O(R2(f) 2 · logR2(f)) where R2(f) is the two-sided bounded error randomized decision tree complexity of f . This improves the previous best relation between these two complexity measures given by Midrijanis [Mid05] of R0(f) = O(R2(f) 2 · log n) (where n is the number of variables). 4. We show that the (non-negative weight) adversary methods to lower bound the bounded error quantum query complexity of f can not give better bounds than √ RC(f)RC(f). This refines the earlier bound of √ C0(f)C1(f) by Spalek and Szegedy [SS06] and strengthens the so called certificate complexity barrier to its randomized analogue. ∗Centre for Quantum Technologies, Singapore Email: kulraghav@gmail.com †The Weizmann Institute of Science, Rehovot, Israel. Email: avishay.tal@weizmann.ac.il. Research supported by an Adams Fellowship of the Israel Academy of Sciences and Humanities, by an Israel Science Foundation grant and by the Israeli Centers of Research Excellence (I-CORE) program. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Revision 1 of Report No. 168 (2013)
The spectral sensitivity λ(f)λ(f)λ(f) of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is bs(f)=O(λ(f)2)bs(f)=O(λ(f)^{2})bs(f)=O(λ(f)2)? We show that it is not. We construct a total Boolean function on 201758420175842017584 variables with bs(f)≥14011bs(f)\ge 14011bs(f)≥14011 and λ(f)≤89.0162λ(f)\le 89.0162λ(f)≤89.0162, so that bs(f)≥λ(f)2.127bs(f)\geλ(f)^{2.127}bs(f)≥λ(f)2.127, and hence by composition a family with λ(fn)→∞λ(f_n)\to\inftyλ(fn​)→∞ and bs(fn)=Ω(λ(fn)2.127)bs(f_n)=Ω(λ(f_n)^{2.127})bs(fn​)=Ω(λ(fn​)2.127). The function is the indicator of a union of kkk subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on 125512551255 variables reaches an exponent near 2.202.202.20, and exhibit a member on 303030 variables whose exponent already exceeds 222 and whose spectral sensitivity can be computed exactly.
We provide new query complexity separations against sensitivity for total Boolean functions: a power 3 separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power 2.22 separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity. We also show that one-sided unambiguous certificate complexity is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between block sensitivity and sensitivity. Along the way, we give a power 1.22 separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power 1.128 separation due to Goos [FOCS 2015]. As a consequence, we obtain an improved lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.
For any Boolean functions f and g, the question whether R(f∘g)=Θ~(R(f)⋅R(g))\text {R}(f\circ g) = \tilde{\Theta }(\text {R}(f) \cdot \text {R}(g))R(f∘g)=Θ~(R(f)⋅R(g)) is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg~(f∘g)=Θ~(deg~(f)⋅deg~(g))\widetilde{\text {deg}}(f\circ g) = \tilde{\Theta }(\widetilde{\text {deg}}(f)\cdot \widetilde{\text {deg}}(g))deg​(f∘g)=Θ~(deg​(f)⋅deg​(g)) . These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R\text {R}R and deg~\widetilde{\text {deg}}deg​ compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f∘g)=Ω(noisyR(f)⋅R(g))\text {R}(f \circ g) = \Omega (\text {noisyR}(f)\cdot \text {R}(g))R(f∘g)=Ω(noisyR(f)⋅R(g)) . This implies that composition holds iff noisyR(f)=Θ~(R(f))\text {noisyR}(f) = \tilde{\Theta }(\text {R}(f))noisyR(f)=Θ~(R(f)) . We show two results: On the other hand, we prove that deg~(f∘g)=Ω~(bs(f)⋅deg~(g)), \widetilde{\text {deg}}(f\circ g) = \widetilde{\Omega }(\sqrt{\text {bs}(f)} \cdot \widetilde{\text {deg}}(g)), deg​(f∘g)=Ω(bs(f)​⋅deg​(g)), where bs(f)\text {bs}(f)bs(f) is the block sensitivity of f. This implies that deg~\widetilde{\text {deg}}deg​ composes when deg~(f)\widetilde{\text {deg}}(f)deg​(f) is asymptotically equal to bs(f)\sqrt{\text {bs}(f)}bs(f)​ . It is already known that both R\text {R}R and deg~\widetilde{\text {deg}}deg​ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function.
The randomized query complexity 𝖱(f) of a boolean function f: {0,1}ⁿ → {0,1} is famously characterized (via Yao’s minimax) by the least number of queries needed to distinguish a distribution 𝒟₀ over 0-inputs from a distribution 𝒟₁ over 1-inputs, maximized over all pairs (𝒟₀,𝒟₁). We ask: Does this task become easier if we allow query access to infinitely many samples from either 𝒟₀ or 𝒟₁? We show the answer is no: There exists a hard pair (𝒟₀,𝒟₁) such that distinguishing 𝒟₀^∞ from 𝒟₁^∞ requires Θ(𝖱(f)) many queries. As an application, we show that for any composed function f∘g we have 𝖱(f∘g) ≥ Ω(fbs(f)𝖱(g)) where fbs denotes fractional block sensitivity.