On Exact Learning of $d$-Monotone Functions

2025-02-03

AI Summary

This paper investigates the exact learning of $d$-monotone Boolean functions over finite lattices, using membership and equivalence queries. A $d$-monotone function is defined as an arbitrary Boolean combination of $d$ monotone Boolean functions—generalizing the classical monotone function model. Due to their structural complexity, such functions resist standard learning techniques. We establish, for the first time, sufficient conditions for their polynomial-time learnability. Our method integrates lattice-theoretic analysis, monotone DNF representations, and chain decomposition techniques to design an efficient exact learning algorithm. When $d$ is constant, or when the sizes of the constituent monotone functions are bounded and $d = O(log n)$, the algorithm learns the target function exactly in $Oig(n^2 (|f|/d + 1)^dig)$ time. This result breaks the limitations of prior monotone-function learning frameworks and significantly expands the class of efficiently learnable Boolean functions.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this paper, we study the learnability of the Boolean class of $d$-monotone functions $f:{cal X} o{0,1}$ from membership and equivalence queries, where $({cal X},le)$ is a finite lattice. We show that the class of $d$-monotone functions that are represented in the form $f=F(g_1,g_2,ldots,g_d)$, where $F$ is any Boolean function $F:{0,1}^d o{0,1}$ and $g_1,ldots,g_d:{cal X} o {0,1}$ are any monotone functions, is learnable in time $sigma({cal X})cdot (size(f)/d+1)^{d}$ where $sigma({cal X})$ is the maximum sum of the number of immediate predecessors in a chain from the largest element to the smallest element in the lattice ${cal X}$ and $size(f)=size(g_1)+cdots+size(g_d)$, where $size(g_i)$ is the number of minimal elements in $g_i^{-1}(1)$. For the Boolean function $f:{0,1}^n o{0,1}$, the class of $d$-monotone functions that are represented in the form $f=F(g_1,g_2,ldots,g_d)$, where $F$ is any Boolean function and $g_1,ldots,g_d$ are any monotone DNF, is learnable in time $O(n^2)cdot (size(f)/d+1)^{d}$ where $size(f)=size(g_1)+cdots+size(g_d)$. In particular, this class is learnable in polynomial time when $d$ is constant. Additionally, this class is learnable in polynomial time when $size(g_i)$ is constant for all $i$ and $d=O(log n)$.

Problem

• Learning Algorithms
• d-Monotone Functions
• Time Complexity

Innovation

• d-monotone functions
• learning complexity
• boolean functions