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Complexity of equational theory of relational algebras with standard projection elements

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Abstract

The class \(\mathsf{TPA}\) of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of \(\mathsf{TPA}\) nor the first order theory of \(\mathsf{TPA}\) are decidable. Moreover, we show that the set of all equations valid in \(\mathsf{TPA}\) is exactly on the \(\Pi ^1_1\) level. We consider the class \(\mathsf{TPA}^-\) of the relation algebra reducts of \(\mathsf{TPA}\)’s, as well. We prove that the equational theory of \(\mathsf{TPA}^-\) is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work.

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Acknowledgments

Research supported by Hungarian National Foundation for Scientific Research Grant Numbers T030314, T034861, and T035192.

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Authors and Affiliations

  1. School of Computer Science and Information Systems, Birkbeck College, University of London, London, UK

    Szabolcs Mikulás

  2. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    Ildikó Sain

  3. Department of Algebra, Budapest University of Technology and Economics, Budapest, Hungary

    András Simon

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  1. Szabolcs Mikulás
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  2. Ildikó Sain
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Correspondence to Ildikó Sain.

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Mikulás, S., Sain, I. & Simon, A. Complexity of equational theory of relational algebras with standard projection elements. Synthese 192, 2159–2182 (2015). https://doi.org/10.1007/s11229-015-0689-1

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