Self-Correlation and Maximum Independence in Finite Relations

TL;DR

This paper explores the relationship between independence and self-correlation in finite relations, introducing a fixed point approach to identify maximum independent partitions based on self-correlated subsets.

Contribution

It establishes a novel connection between independence and self-correlation, and presents a fixed point method to compute maximum independent partitions in finite relations.

Findings

Maximum independent partition is the least fixed point of a specific inflationary transformer.

The fixed point can be obtained as the limit of a standard approximation sequence.

The approach generalizes concepts from Kleene's fixed point theorem to relation analysis.

Abstract

We consider relations with no order on their attributes as in Database Theory. An independent partition of the set of attributes S of a finite relation R is any partition X of S such that the join of the projections of R over the elements of X yields R. Identifying independent partitions has many applications and corresponds conceptually to revealing orthogonality between sets of dimensions in multidimensional point spaces. A subset of S is termed self-correlated if there is a value of each of its attributes such that no tuple of R contains all those values. This paper uncovers a connection between independence and self-correlation, showing that the maximum independent partition is the least fixed point of a certain inflationary transformer alpha that operates on the finite lattice of partitions of S. alpha is defined via the minimal self-correlated subsets of S. We use some additional properties of alpha to show the said fixed point is still the limit of the standard approximation sequence, just as in Kleene's well-known fixed point theorem for continuous functions.