Identification of a monotone Boolean function with $k$ "reasons" as a combinatorial search problem

TL;DR

This work studies the query complexity of identifying a monotone Boolean function with $k$ minimal reasons, equivalently the identification of a $k$-member upfamily or a $k$-element antichain in $2^{[n]}$, under adaptive and non-adaptive models. It develops a combinatorial-search framework linking upfamilies, minimal covers, and antichains, and provides tight non-adaptive bounds $h(n,k)$ with explicit formulas for all regimes, as well as adaptive results that connect to the numbers of minimal covers via $g(n,m)$. For fixed $k\,ge 3$, the adaptive case yields a precise asymptotic $f(n,k)=(1+o(1))(n/(k-1))^{k-1}$, while the special case $k=2$ has the tight bound $f(n,2)=2n$ for large $n$. The approach combines explicit non-adaptive constructions, adversarial lower bounds, and the Gainanov algorithm to iteratively identify elements of the hidden $k$-set family, offering a near-complete characterization of adaptive and non-adaptive query complexities in this monotone-function identification setting.

Abstract

We study the number of queries needed to identify a monotone Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$. A query consists of a 0-1-sequence, and the answer is the value of $f$ on that sequence. It is well-known that the number of queries needed is $\binom{n}{\lfloor n/2\rfloor}+\binom{n}{\lfloor n/2\rfloor+1}$ in general. Here we study a variant where $f$ has $k$ ``reasons'' to be 1, i.e., its disjunctive normal form has $k$ conjunctions if the redundant conjunctions are deleted. This problem is equivalent to identifying an upfamily in $2^{[n]}$ that has exactly $k$ minimal members. We find the asymptotics on the number of queries needed for fixed $k$. We also study the non-adaptive version of the problem, where the queries are asked at the same time, and determine the exact number of queries for most values of $k$ and $n$.

Theorems & Definitions (14)

• Proposition 1.1
• proof
• Theorem 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 1.5
• Proposition 1.6
• proof : Proof of Theorem \ref{['nona']}
• proof : Proof of Theorem \ref{['adap']}
• proof : Proof of Proposition \ref{['fixadap']}