Boolean Nearest Neighbor Language in the Knowledge Compilation Map

TL;DR

This paper investigates the Boolean Nearest Neighbor (BNN) representation of Boolean functions, analyzing its placement in the Knowledge Compilation Map and comparing its succinctness and computational complexity to other standard languages.

Contribution

It establishes the position of BNN in the KCM, compares its succinctness with other languages, and analyzes the complexity of queries and transformations for BNN.

Findings

BNN's position in the Knowledge Compilation Map is determined.

Comparative analysis of BNN's succinctness with standard languages.

Complexity results for queries and transformations on BNN representations.

Abstract

The Boolean Nearest Neighbor (BNN) representation of Boolean functions was recently introduced by Hajnal, Liu and Turan. A BNN representation of $f$ is a pair $(P,N)$ of sets of Boolean vectors (called positive and negative prototypes) where $f(x)=1$ for every positive prototype $x \in P$, $f(x)=0$ for all every negative prototype $x \in N$, and the value $f(x)$ for $x \not\in P \cup N$ is determined by the type of the closest prototype. The main aim of this paper is to determine the position of the BNN language in the Knowledge Compilation Map (KCM). To this end, we derive results which compare the succinctness of the BNN language to several standard languages from KCM, and determine the complexity status of most standard queries and transformations for BNN inputs.