Decision Trees with Hypotheses for Recognition of Monotone Boolean Functions and for Sorting

TL;DR

This paper explores decision trees that incorporate both attribute-based and hypothesis-based queries for recognizing monotone Boolean functions and sorting, comparing their complexities and analyzing entropy-based construction methods.

Contribution

It introduces a framework for decision trees with hypotheses, compares their optimal complexities for specific problems, and evaluates entropy-based algorithms for decision rule length.

Findings

Decision trees with hypotheses can reduce complexity in recognition and sorting tasks.

Entropy-based greedy algorithms produce concise decision rules.

Optimal decision trees vary significantly with problem size and query type.

Abstract

In this paper, we consider decision trees that use both queries based on one attribute each and queries based on hypotheses about values of all attributes. Such decision trees are similar to ones studied in exact learning, where not only membership but also equivalence queries are allowed. We investigate the problem of recognition of monotone Boolean functions with $n$ variables, $n=2, \ldots, 4$, and the problem of sorting $n$ pairwise different elements from linearly ordered set, $n=3, \ldots, 6$. For each of these problems, we compare the complexity of different types of optimal (relative to the depth or the number of realizable nodes) decision trees with hypotheses. We also study the complexity of decision trees constructed by entropy-based greedy algorithm and analyze the length of decision rules derived from these trees.