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Embeddability into Relational Lattices Is Undecidable

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Relational and Algebraic Methods in Computer Science (RAMICS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10226))

Abstract

The natural join and the inner union operations combine relations of a database. Tropashko and Spight 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. Litak et al. proposed an axiomatization of relational lattices over the signature that extends the pure lattice signature with a constant and argued that the quasiequational theory of relational lattices over this extended signature is undecidable.

We prove in this paper that embeddability is undecidable for relational lattices. More precisely, it is undecidable whether a finite subdirectly-irreducible lattice can be embedded into a relational lattice. Our proof is a reduction from the coverability problem of a multimodal frame by a universal product frame and, indirectly, from the representability problem for relation algebras.

As corollaries we obtain the following results: the quasiequational theory of relational lattices over the pure lattice signature is undecidable and has no finite base; there is a quasiequation over the pure lattice signature which holds in all the finite relational lattices but fails in an infinite relational lattice.

Extended abstract, see [21] for a full version of this paper.

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Notes

  1. 1.

    In [13] such a lattice is called full relational lattice. The wording “class of relational lattices” is used there for the class of lattices that have an embedding into some lattice of the form \(\mathsf {R}(D,A)\).

  2. 2.

    As P(A) is not totally ordered, we avoid calling a morphism “non expanding map” as it is often done in the literature.

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Authors and Affiliations

  1. LIF, CNRS UMR 7279, Aix-Marseille Université, Marseille, France

    Luigi Santocanale

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  1. Luigi Santocanale
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Correspondence to Luigi Santocanale .

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Editors and Affiliations

  1. Data61, CSIRO, Sydney, New South Wales, Australia

    Peter Höfner

  2. CNRS, Lyon, France

    Damien Pous

  3. University of Sheffield, Sheffield, United Kingdom

    Georg Struth

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Santocanale, L. (2017). Embeddability into Relational Lattices Is Undecidable. In: Höfner, P., Pous, D., Struth, G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science(), vol 10226. Springer, Cham. https://doi.org/10.1007/978-3-319-57418-9_16

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  • DOI: https://doi.org/10.1007/978-3-319-57418-9_16

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-57418-9

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