## Representability is not decidable for finite relation algebras (1999)

Venue: | Trans. Amer. Math. Soc |

Citations: | 15 - 6 self |

### BibTeX

@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}

}

### Years of Citing Articles

### OpenURL

### 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 τ be a finite set of square tiles, where each edge of each tile has a colour. Suppose τ 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 ∈ τ there is a tiling of the plane Z × Z using only tiles from τ 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 τ, we construct a finite relation algebra RA(τ) and show that the second player has a winning strategy in G(RA(τ)) if and only if τ is a yes-instance. This reduces the tiling problem to the representation problem and proves the latter’s undecidability. 1.

### Citations

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Citation Context ... f(x,y)). We call such an f a tiling. If there is no such tiling, then we have a no-instance. The tiling problem (given an instance, is it a yes-instance or a no-instance?) is known to be undecidable =-=[Ber66]-=-. It is not hard to show from this that the following problem is also undecidable. Given a finite set of tiles {T0,... ,Tk−1} as above, is it the case that for each i<kthere is a tiling f i of the pla... |

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Citation Context ...s first axiomatization and produced a rather complex but correct axiomatization in [Lyn56]. Three separate axiomatizations of the closely related class of representable cylindric algebras appeared in =-=[HMT85]-=-, and alternative axiomatizations of representable relation and cylindric algebras appeared in [HH97b]. 1Strictly, we have to consider the isomorphism type of members of these classes, so that we are ... |

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Citation Context ...being Tarski’s axiomatization of the relation algebras [JT52, Definition 4.1]. This axiomatization turned out not to be complete [Lyn50, Mon64] and Lyndon proposed a stronger, infinite axiomatization =-=[Lyn50]-=- which we will refer to here as the Lyndon conditions. It turned out that the Lyndon conditions were not sound over RRA: there are representable relation algebras that fail some of the Lyndon conditio... |

49 |
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Citation Context ...A is representable if and only if ∃ has a winning strategy in G(A). Proof. See, for example, [HH97b, Theorem 9] or [HH97a, Proposition 13]. The idea is essentially in [Lyn50] and is well known (e.g., =-=[Ma82]-=-). 3. The tiling problem An instance τ of the tiling problem is a finite set of square tiles τ ={T0,... ,Tk−1}. Each tile has a colour on each of its four edges: the four colours on the tile Ti are To... |

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Citation Context ...ated. It is interesting to consider alternatives to this approach. For example, it has been known for some time that the representable relation algebras cannot be defined by a finite number of axioms =-=[Mon64]-=-, and this alone suggests that for finite relation algebras the representability problem is undecidable. However, finite axiomatizability and decidability are not the same. If a class is finitely axio... |

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Citation Context ...rate axiomatizations of the closely related class of representable cylindric algebras appeared in [HMT85], and alternative axiomatizations of representable relation and cylindric algebras appeared in =-=[HH97b]-=-. 1Strictly, we have to consider the isomorphism type of members of these classes, so that we are dealing with sets and not classes.sREPRESENTABILITY IS NOT DECIDABLE 1405 However, the Lyndon conditio... |

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Citation Context ...table relation algebras among the finite relation algebras; and in this paper, where we deal only with finite relation algebras, we will use a variant of these conditions to test representability. In =-=[HH97a]-=- the Lyndon conditions are expressed in terms of a winning strategy for the second, ‘existential player’ in a certain two-player game, played over a relation algebra. 2 Here we define a variant of the... |

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Citation Context ...at undecidability results for relation algebra should be obtainable by this result. However we have not been able to obtain the result of the present paper in that way. Our construction originated in =-=[Hir95]-=- and has been used in different forms in [HH97a, Hod97]. We assume some familiarity with relation algebras. The uninitiated might try [JT48, JT52, Ma91b, Ma91a], for example. 2. Representability and G... |

3 | A perspective on the theory of relation algebras - Maddux - 1994 |

2 | Completely representable relation algebras
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Citation Context ...t undecidability results for relation algebra should be obtainable by this result. However, we have not been able to obtain the result of the present paper in that way. Our construction originated in =-=[Hir95]-=- and has been used in different forms in [HH97a, Hod97]. We assume some familiarity with relation algebras. The uninitiated might try [JT48, JT52, Ma91b, Ma91a], for example. 2. Representability and g... |

2 |
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Citation Context ... He further shows that the equational theory of this class is decidable. One of the main motivations for Tarski's study of relation algebras was to define an alternative foundation for set theory. In =-=[TG87]-=- it is shown that relation algebra can act as a vehicle for set theory and hence all of mathematics. It would seem then, that undecidability results for relation algebra should be obtainable by this r... |

1 |
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Citation Context ...: there are representable relation algebras that fail some of the Lyndon conditions. Lyndon explained the error in his first axiomatization and produced a rather complex but correct axiomatization in =-=[Lyn56]-=-. Three separate axiomatizations of the closely related class of representable cylindric algebras appeared in [HMT85], and alternative axiomatizations of representable relation and cylindric algebras ... |