[Paper Review] Sensitivity and block sensitivity of nested canalyzing function

Yuan Li, John O. Adeyeye|arXiv (Cornell University)|Sep 7, 2012

Sparse and Compressive Sensing Techniques7 references3 citations

TL;DR

This paper provides exact formulas for the sensitivity and block sensitivity of nested canalyzing functions (NCFs), proving both are equal and bounded between $\frac{n+1}{2}$ and $n$, with tight bounds. It further characterizes all functions that are both monotone and nested canalyzing, deriving their exact count using multinomial coefficients.

ABSTRACT

Based on a recent characterization of nested canalyzing function (NCF), we obtain the formula of the sensitivity of any NCF. Hence we find that any sensitivity of NCF is between $\frac{n+1}{2}$ and $n$. Both lower and upper bounds are tight. We prove that the block sensitivity, hence the $l$-block sensitivity, is same to the sensitivity. It is well known that monotone function also has this property. We eventually find all the functions which are both monotone and nested canalyzing (MNCF). The cardinality of all the MNCF is also provided.

Motivation & Objective

• To derive an exact formula for the sensitivity of any nested canalyzing function (NCF) using its unique algebraic normal form and layer number.
• To prove that block sensitivity equals sensitivity for all NCFs, extending a property previously known only for monotone functions.
• To characterize the set of Boolean functions that are both monotone and nested canalyzing (MNCFs), and to compute their exact cardinality.
• To establish tight bounds on sensitivity of NCFs, showing it ranges from $\frac{n+1}{2}$ to $n$, with both bounds achievable.
• To provide a complete combinatorial characterization of MNCFs using layer structure and variable profiles, leveraging multinomial coefficients.

Proposed method

• Utilizes the unique algebraic normal form (ANF) of NCFs as characterized in prior work, expressed as $ f = M_1(M_2(\cdots(M_r \oplus 1)\oplus 1)\cdots)\oplus b $, where $ M_i $ are products of variables or their complements.
• Introduces the concept of 'layer number' $ r $, which defines the hierarchical structure of NCFs and determines sensitivity behavior.
• Analyzes sensitivity at each input by tracking how flipping individual bits affects the output through the nested structure of $ M_i $, particularly focusing on the first zero bit in the chain.
• Proves block sensitivity equals sensitivity by showing that any minimal block changing the function value must consist of a single bit, due to the hierarchical dependency in the NCF structure.
• Applies combinatorial techniques to count MNCFs by fixing the layer structure $[k_1, \dots, k_r]$ with $k_i \geq 1$ for $i < r$ and $k_r \geq 2$, and assigning consistent polarity values to maintain monotonicity.
• Uses multinomial coefficients $ \binom{n}{k_1, \dots, k_r} $ to count valid variable assignments per profile, multiplied by 4 for choices of polarity and output value.

Experimental results

Research questions

• RQ1 What is the exact sensitivity of any nested canalyzing function, and what are its tight bounds?
• RQ2 Is block sensitivity equal to sensitivity for all nested canalyzing functions, as it is for monotone functions?
• RQ3 Which Boolean functions are both monotone and nested canalyzing, and how many such functions exist?
• RQ4 How does the layer number and variable profile of an NCF influence its sensitivity and monotonicity?
• RQ5 Can the complete set of monotone nested canalyzing functions be enumerated, and what is their exact cardinality?

Key findings

• The sensitivity of any NCF is bounded between $\frac{n+1}{2}$ and $n$, and both bounds are tight, with the lower bound achieved when the layer number $ r = n $.
• Block sensitivity equals sensitivity for all NCFs, meaning $ bs(f) = s(f) $, due to the hierarchical structure forcing minimal blocks to be single-bit.
• The sensitivity of an NCF depends only on the layer number $ r $: for $ r = 1 $, sensitivity is $ n $; for $ r > 1 $, it is determined by the first layer where the value flips.
• All functions that are both monotone and nested canalyzing (MNCFs) are characterized by alternating polarities across layers: $ M_i = \prod (x_j \oplus a) $, $ M_{i+1} = \prod (x_j \oplus \overline{a}) $ for odd $ i $.
• The total number of MNCFs is $ 4 \sum_{\substack{k_1+\cdots+k_r=n \\ k_i\geq1, i<r, k_r\geq2}} \binom{n}{k_1,\dots,k_r} $, derived from multinomial coefficients over valid layer profiles.
• The cardinality formula accounts for 4 choices: two for the base polarity $ a \in \{0,1\} $, and two for the output value $ b \in \{0,1\} $, with variable assignments constrained by layer structure.