Sensitivity and block sensitivity of nested canalyzing function

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.

💡 Research Summary

The paper investigates two fundamental complexity measures—sensitivity and block sensitivity—for the class of nested canalyzing functions (NCFs), and it also characterizes the subclass that is simultaneously monotone and nested canalyzing (MNCF).
Using the most recent structural description of NCFs, each function is represented by a tuple ((\sigma, a, b, c)) where (\sigma) is a permutation of the input variables, (a_i) are the canalyzing input values, (b_i) the corresponding output values, and (c) the final output when none of the canalyzing conditions are met. This representation makes explicit the hierarchical decision process: the first variable in the order that matches its canalyzing value forces the output, otherwise the evaluation proceeds to the next variable.
The authors first derive an exact formula for the sensitivity (s(f)) of any NCF. By analyzing how many input bits can be flipped to change the output at each decision level, they prove that for an (n)-variable NCF the sensitivity always lies in the interval (\big