The equational theory of the natural join and inner union is decidable

TL;DR

This paper proves that the equational theory of relational lattices, which model natural join and inner union operations in databases, is decidable, providing an algorithm to determine term equality.

Contribution

It establishes the decidability of the equational theory of relational lattices, contrasting previous undecidability results for their quasiequational theory.

Findings

Decidability of the equational theory of relational lattices.

Existence of a triple exponential bound on counterexamples.

Algorithm for deciding term equality in relational lattices.

Abstract

The natural join and the inner union operations combine relations of a database. Tropashko and Spight [24] realized that these two operations are the meet and join operations in a class of lattices, known by now as the relational lattices. They proposed then lattice theory as an algebraic approach to the theory of databases, alternative to the relational algebra. Previous works [17, 22] proved that the quasiequational theory of these lattices-that is, the set of definite Horn sentences valid in all the relational lattices-is undecidable, even when the signature is restricted to the pure lattice signature. We prove here that the equational theory of relational lattices is decidable. That, is we provide an algorithm to decide if two lattice theoretic terms t, s are made equal under all intepretations in some relational lattice. We achieve this goal by showing that if an inclusion t $\le$ s fails in any of these lattices, then it fails in a relational lattice whose size is bound by a triple exponential function of the sizes of t and s.