Self-Correlation and Maximum Independence in Finite Relations

Dilian Gurov
(KTH Royal Institute of Technology, Stockholm, Sweden)
Minko Markov
(St. Kliment Ohridski University of Sofia, Sofia, Bulgaria)

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.

In Ralph Matthes and Matteo Mio: Proceedings Tenth International Workshop on Fixed Points in Computer Science (FICS 2015), Berlin, Germany, September 11-12, 2015, Electronic Proceedings in Theoretical Computer Science 191, pp. 60–74.
Published: 9th September 2015.

ArXived at: http://dx.doi.org/10.4204/EPTCS.191.7 bibtex PDF
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