7 research outputs found
The equational theory of the natural join and inner union is decidable
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 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.Comment: arXiv admin note: text overlap with arXiv:1607.0298
Relational lattices via duality
The natural join and the inner union combine in different ways tables of a
relational database. Tropashko [18] observed that these two operations are the
meet and join in a class of lattices-called the relational lattices- and
proposed lattice theory as an alternative algebraic approach to databases.
Aiming at query optimization, Litak et al. [12] initiated the study of the
equational theory of these lattices. We carry on with this project, making use
of the duality theory developed in [16]. The contributions of this paper are as
follows. Let A be a set of column's names and D be a set of cell values; we
characterize the dual space of the relational lattice R(D, A) by means of a
generalized ultrametric space, whose elements are the functions from A to D,
with the P (A)-valued distance being the Hamming one but lifted to subsets of
A. We use the dual space to present an equational axiomatization of these
lattices that reflects the combinatorial properties of these generalized
ultrametric spaces: symmetry and pairwise completeness. Finally, we argue that
these equations correspond to combinatorial properties of the dual spaces of
lattices, in a technical sense analogous of correspondence theory in modal
logic. In particular, this leads to an exact characterization of the finite
lattices satisfying these equations.Comment: Coalgebraic Methods in Computer Science 2016, Apr 2016, Eindhoven,
Netherland
Relational lattices: from databases to Universal Algebra
Relational lattices are obtained by interpreting lattice connectives as natural join and inner union between database relations. Our study of their equational theory reveals that the variety generated by relational lattices has not been discussed in the existing literature. Furthermore, we show that addition of just the header constant to the lattice signature leads to undecidability of the quasiequational theory. Nevertheless, we also demonstrate that relational lattices are not as intangible as one may fear: for example, they do form a pseudoelementary class. We also apply the tools of Formal Concept Analysis and investigate the structure of relational lattices via their standard contexts. Furthermore, we show that the addition of typing rules and singleton constants allows a direct comparison with the monotonic relational expressions of Sagiv and Yannakakis