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Representability is not decidable
for finite relation algebras


Authors: Robin Hirsch and Ian Hodkinson
Journal: Trans. Amer. Math. Soc. 353 (2001), 1403-1425
MSC (1991): Primary 03G15; Secondary 03G05, 06E25, 03D35
DOI: https://doi.org/10.1090/S0002-9947-99-02264-3
Published electronically: April 23, 1999
MathSciNet review: 1806735
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove that there is no algorithm that decides whether a finite relation algebra is representable.

Representability of a finite relation algebra $\mathcal A$ is determined by playing a certain two player game $G({\mathcal A})$ over `atomic $\mathcal A$-networks'. It can be shown that the second player in this game has a winning strategy if and only if $\mathcal A$ is representable.

Let $\tau$ be a finite set of square tiles, where each edge of each tile has a colour. Suppose $\tau$ includes a special tile whose four edges are all the same colour, a colour not used by any other tile. The tiling problem we use is this: is it the case that for each tile $T \in \tau$ there is a tiling of the plane ${\mathbb Z}\times {\mathbb Z}$ using only tiles from $\tau$ in which edge colours of adjacent tiles match and with $T$ placed at $(0,0)$? It is not hard to show that this problem is undecidable.

From an instance of this tiling problem $\tau$, we construct a finite relation algebra $RA(\tau)$ and show that the second player has a winning strategy in $G(RA(\tau))$ if and only if $\tau$ is a yes-instance. This reduces the tiling problem to the representation problem and proves the latter's undecidability.


References [Enhancements On Off] (What's this?)

  • [AMN91] H. Andréka, J. D. Monk, and I. Németi. Algebraic Logic. Colloq. Math. Soc. J. Bolyai. North-Holland, Amsterdam, 1991. Conference Proceedings, Budapest, 1988. MR 92m:03003
  • [Ber66] R. Berger. The undecidability of the domino problem. Memoirs of the American Mathematical Society, vol. 66, 1966. MR 36:49
  • [HH97a] R. Hirsch and I. Hodkinson. Complete representations in algebraic logic. Journal of Symbolic Logic, 1997. 62(3):816-847, 1997. MR 98m:03123
  • [HH97b] R. Hirsch and I. Hodkinson. Step by step - building representations in algebraic logic. Journal of Symbolic Logic. 62(1):225-279, 1997. MR 98g:03145
  • [Hir95] R. Hirsch. Completely representable relation algebras. Bulletin of the Interest Group in Propositional and Predicate Logics, 3(1):77-91, 1995. MR 96d:03082
  • [HMT85] L. Henkin, J. D. Monk, and A. Tarski. Cylindric Algebras. Part II. North-Holland, 1985. MR 86m:03095b
  • [Hod97] I. Hodkinson. Atom structures of cylindric algebras and relation algebras. Ann. Pure Appl. Logic, 89(2-3):117-148, 1997. CMP 98:06
  • [JT48] B. Jónsson and A. Tarski. Representation problems for relation algebras. Bulletin of the American Mathematical Society, 54, pp. 80 and 1192, 1948.
  • [JT52] B. Jónsson and A. Tarski. Boolean algebras with operators. II. American Journal of Mathematics, 74:127 - 162, 1952. MR 13:524g
  • [Lyn50] R. Lyndon. The representation of relational algebras. Annals of Mathematics, 51(3):707-729, 1950. MR 12:237a
  • [Lyn56] R. Lyndon. The representation of relation algebras, II. Annals of Mathematics, 63(2):294-307, 1956. MR 18:106f
  • [Ma82] R. Maddux. Some varieties containing relation algebras. Transactions of the American Mathematical Society, 272(2):501-526, 1982. MR 84a:03079
  • [Ma91a] R. Maddux. Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interconnections. In [AMN91], pages 361-392. MR 93c:03082
  • [Ma91b] R. Maddux. The origin of relation algebras in the development and axiomatization of the calculus of relations. Studia Logica, 5(3-4):421-456, 1991. MR 93c:03082
  • [Ma94] R. Maddux. A perspective on the theory of relation algebras. Algebra Universalis, 31:456-465, 1994. MR 95c:03143
  • [Mon64] J. D. Monk. On representable relation algebras. Michigan Mathematics Journal, 11:207-210, 1964. MR 30:3016
  • [Nem96] I. Németi. A fine-structure analysis of first-order logic. In M. Marx, L. Pólos, and M. Masuch, editors, Arrow Logic and Multi-Modal Logic, Studies in Logic, Language and Information, pages 221-247. CSLI Publications & FoLLI, 1996. CMP 97:09
  • [TG87] A. Tarski and S. R. Givant. A Formalization of Set Theory Without Variables. Number 41 in Colloquium Publications in Mathematics. American Mathematical Society, Providence, Rhode Island, 1987. MR 89g:03012

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Additional Information

Robin Hirsch
Affiliation: Department of Computer Science, University College, Gower Street, London WC1E 6BT, U.K.
Email: r.hirsch@cs.ucl.ac.uk

Ian Hodkinson
Affiliation: Department of Computing, Imperial College, Queen’s Gate, London SW7 2BZ, U.K.
Email: imh@doc.ic.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-99-02264-3
Keywords: Relation algebra, representation, undecidability, tiling problem, games, algebraic logic
Received by editor(s): April 2, 1997
Received by editor(s) in revised form: November 11, 1997
Published electronically: April 23, 1999
Additional Notes: Research of the second author was partially supported by UK EPSRC grant GR/K54946. Many thanks to Peter Jipsen, Maarten Marx, Szabolcs Mikulás, Mark Reynolds, Yde Venema and the referee for useful comments and for pointing out serious errors in early drafts of this paper.
Article copyright: © Copyright 1999 American Mathematical Society

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