# Combinatorial and Algorithmic Properties of One Matrix Structure at Monotone Boolean Functions

## TL;DR

This paper introduces a new matrix structure for monotone Boolean functions, explores its properties, and develops efficient algorithms for generating and identifying such functions using minimal membership queries.

## Contribution

It defines a novel matrix structure for monotone Boolean functions and presents three algorithms leveraging its properties for function generation and identification.

## Key findings

- Algorithms generate all monotone Boolean functions in lexicographic order.
- The identification algorithm uses at most n membership queries and runs in Θ(n) time.
- Experimental results show the algorithm's query complexity is proportional to the size of minimal true and false vector sets.

## Abstract

One matrix structure in the area of monotone Boolean functions is defined here. Some of its combinatorial, algebraic and algorithmic properties are derived. On the base of these properties, three algorithms are built. First of them generates all monotone Boolean functions of $n$ variables in lexicographic order. The second one determines the first (resp. the last) lexicographically minimal true (resp. maximal false) vector of an unknown monotone function $f$ of $n$ variables. The algorithm uses at most $n$ membership queries and its running time is $\Theta(n)$. It serves the third algorithm, which identifies an unknown monotone Boolean function $f$ of $n$ variables by using membership queries only. The experimental results show that for $1\leq n\leq 6$, the algorithm determines $f$ by using at most $m.n$ queries, where $m$ is the combined size of the sets of minimal true and maximal false vectors of $f$.