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- W1753706763 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 âatomic $mathcal A$-networksâ. 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 latterâs undecidability." @default.
- W1753706763 created "2016-06-24" @default.
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- W1753706763 date "1999-04-23" @default.
- W1753706763 modified "2023-09-23" @default.
- W1753706763 title "Representability is not decidable for finite relation algebras" @default.
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- W1753706763 doi "https://doi.org/10.1090/s0002-9947-99-02264-3" @default.
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