Logical reduction of relations: from relational databases to Peirce's reduction thesis

TL;DR

This paper explores the logical reduction of relations in databases using algebraic frameworks, introducing new concepts like ternarity to analyze relation complexity and refining Peirce's reduction thesis across finite and infinite domains.

Contribution

It unifies various algebraic approaches to relation reduction and introduces the concept of ternarity to measure relation complexity, refining classical reduction results.

Findings

Refined Peirce's reduction thesis for finite and infinite domains.

Introduced ternarity as a new measure of relation complexity.

Systematic study of irreducible relations and reduction methods.

Abstract

We study logical reduction (factorization) of relations into relations of lower arity by Boolean or relative products that come from applying conjunctions and existential quantifiers to predicates, i.e. by primitive positive formulas of predicate calculus. Our algebraic framework unifies natural joins and data dependencies of database theory and relational algebra of clone theory with the bond algebra of C.S. Peirce. We also offer new constructions of reductions, systematically study irreducible relations and reductions to them, and introduce a new characteristic of relations, ternarity, that measures their `complexity of relating' and allows to refine reduction results. In particular, we refine Peirce's controversial reduction thesis, and show that reducibility behavior is dramatically different on finite and infinite domains.