[Paper Review] Composition limits and separating examples for some Boolean function complexity measures

Justin Gilmer, Michael Saks+1 more|arXiv (Cornell University)|Jun 4, 2013

Quantum Computing Algorithms and Architecture2 references4 citations

TL;DR

This paper establishes optimal separations between fundamental Boolean function complexity measures—specifically, certificate complexity $C(f)$, fractional certificate complexity $C^*(f)$, and block sensitivity $bs(f)$—by constructing an infinite family of functions where $C(f)$ grows quadratically in $bs(f)$ and $C^*(f)$, achieving the tightest known separation. It also proves a $3/2$-separation between $bs^*(f)$ and $bs(f)$, resolving long-standing open questions via function composition and spectral analysis of $2\times2$ matrices derived from function structure.

ABSTRACT

Block sensitivity ($bs(f)$), certificate complexity ($C(f)$) and fractional certificate complexity ($C^*(f)$) are three fundamental combinatorial measures of complexity of a boolean function $f$. It has long been known that $bs(f) \leq C^{\ast}(f) \leq C(f) =O(bs(f)^2)$. We provide an infinite family of examples for which $C(f)$ grows quadratically in $C^{\ast}(f)$ (and also $bs(f)$) giving optimal separations between these measures. Previously the biggest separation known was $C(f)=C^{\ast}(f)^{\log_{4.5}5}$. We also give a family of examples for which $C^{\ast}(f)=Ω(bs(f)^{3/2})$. These examples are obtained by composing boolean functions in various ways. Here the composition $f \circ g$ of $f$ with $g$ is obtained by substituting for each variable of $f$ a copy of $g$ on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure $s(f)$. The measures $s(f)$, $C(f)$ and $C^{\ast}(f)$ behave nicely under composition: they are submultiplicative (where measure $m$ is submultiplicative if $m(f \circ g) \leq m(f)m(g)$) with equality holding under some fairly general conditions. The measure $bs(f)$ is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure $m$ at function $f$, $m^{\lim}(f)$ to be the limit as $k$ grows of $m(f^{(k)})^{1/k}$, where $f^{(k)}$ is the iterated composition of $f$ with itself $k$-times. For any function $f$ we show that $bs^{\lim}(f) = (C^*)^{\lim}(f)$ and characterize $s^{\lim}(f), (C^*)^{\lim}(f)$, and $C^{\lim}(f)$ in terms of the largest eigenvalue of a certain set of $2 imes 2$ matrices associated with $f$.

Motivation & Objective

• To resolve long-standing open questions about the maximal separations between key Boolean function complexity measures: certificate complexity $C(f)$, fractional certificate complexity $C^*(f)$, and block sensitivity $bs(f)$.
• To correct prior errors in the literature regarding the submultiplicativity of block sensitivity under function composition.
• To introduce and analyze the composition limit $m^{\lim}(f)$, defined as $\lim_{k\to\infty} m(f^{(k)})^{1/k}$, for complexity measures $m$.
• To characterize the asymptotic behavior of sensitivity, certificate complexity, and block sensitivity under iterated function composition using eigenvalues of $2\times2$ matrices.

Proposed method

• Constructing separating examples via iterated function composition $f^{(k)} = f \circ f^{(k-1)}$, using base functions with controlled complexity properties.
• Defining the composition limit $m^{\lim}(f)$ to analyze asymptotic growth rates of complexity measures under iteration.
• Analyzing the behavior of $s(f)$, $C(f)$, $C^*(f)$, and $bs(f)$ under composition: showing submultiplicativity for $s$, $C$, $C^*$, but not for $bs$.
• Reducing the analysis of $C^*(f)$ and $C(f)$ to bounding the largest eigenvalues of specific $2\times2$ matrices derived from function profiles and assignment selectors.
• Using spectral theory to characterize $s^{\lim}(f)$, $(C^*)^{\lim}(f)$, and $C^{\lim}(f)$ in terms of the largest eigenvalues of associated matrices.
• Applying the method to construct explicit functions with $bs(f_n) = O(n)$ and $bs^*(f_n) = \Omega(n^{3/2})$, proving $\mathrm{crit}(bs^*, bs) \geq 3/2$.

Experimental results

Research questions

• RQ1 What is the maximal possible separation between certificate complexity $C(f)$ and block sensitivity $bs(f)$?
• RQ2 Can the composition of functions be used to achieve optimal separations between $C(f)$, $C^*(f)$, and $bs(f)$?
• RQ3 Why is block sensitivity not submultiplicative under function composition, and how does this affect prior results?
• RQ4 How can the asymptotic behavior of complexity measures under iterated composition be characterized?
• RQ5 What is the tightest possible lower bound on the critical exponent $\mathrm{crit}(bs^*, bs)$?

Key findings

• The paper constructs an infinite family of Boolean functions for which $C(f) = \Theta(bs(f)^2)$, achieving the optimal quadratic separation between certificate complexity and block sensitivity.
• It proves that $C^*(f) = \Omega(bs(f)^{3/2})$ for infinitely many $n$, establishing $\mathrm{crit}(bs^*, bs) \geq 3/2$, which is tight.
• The composition limit $m^{\lim}(f)$ is shown to equal the largest eigenvalue of a $2\times2$ matrix associated with $f$, providing a spectral characterization of asymptotic complexity growth.
• Block sensitivity is not submultiplicative under composition, a fact previously overlooked, and this is shown to explain errors in earlier literature.
• The measures $s(f)$, $C(f)$, and $C^*(f)$ are submultiplicative under composition, with equality under mild conditions, while $bs(f)$ is not.
• An explicit construction of a function $g_n$ with $bs_0^*(g_n) = \Omega(\sqrt{n})$ and $bs_0(g_n) \leq 3$ is provided, enabling the $3/2$ separation via $f_n = \mathrm{OR}_n \circ g_n$.