Representability is not decidable for finite relation algebras (1999) [12 citations — 7 self]
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@ARTICLE{Hirsch99representabilityis,author = {Robin Hirsch and Ian Hodkinson},
title = {Representability is not decidable for finite relation algebras},
journal = {Trans. Amer. Math. Soc},
year = {1999},
volume = {353}
}
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Abstract:
Abstract. We prove that there is no algorithm that decides whether a finite relation algebra is representable. Representability of a finite relation algebra A is determined by playing a certain two player game G(A) over `atomic A-networks'. It can be shown that the second player in this game has a winning strategy if and only if A is representable. Let be a finite set of square tiles, where each edge of each tile has a colour. Suppose 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 2 there is a tiling of the plane Z\Theta Zusing only tiles from 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 we construct a finite relation algebra RA() and show that the second player has a winning strategy in G(RA()) if and only if is a yesinstance. This reduces the tiling problem to the representation problem and proves the latter's undecidability. 1.
