R Hirsch and I Hodkinson. Representability is not decidable for nite relation algebras. Trans. Amer. Math. Soc., 1999? To appear.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....necessarily in nite [19] equational axiomatization of RRA, such as the one by Lyndon [9] So the set of nite nonrepresentable algebras is recursively enumerable. Is it recursive This natural question was open for more than 20 years before being answered in the negative by Hirsch and Hodkinson [5]. Long before these results were obtained, Tarski remarked to me that it would be a challenging task to develop a structure theory for nite relation algebras. He was right. ....

Robin Hirsch and Ian Hodkinson, Representability is not decidable for nite relation algebras, Transactions of the American Mathematical Society 353 (2001), 1403-1425. 10 ROGER D. MADDUX

....an n ary relation on the atoms of A and check that it is an n dimensional basis in the sense of de nition 6 below; this algorithm accepts A if and only if A 2 B n , if and only if A 2 B n (because A is nite, so that A = A ) if and only if A 2 RA n , as required. It was shown in [HH99c] that the corresponding problem for RRA is undecidable, so this is an advantage of using RA n instead. 1.9 Complete relativised representations Now that we have a notion of representation, we can look for a direct n dimensional analogue of CRA. Earlier, we said that this was B n , but this ....

....relative to RA n if no RA l (n l ) is. In a similar way, we obtain the result discussed earlier: Theorem 2 For any n with 5 n , CRA n is not elementary. Our construction in the proofs of these two results originated in [Hir95] and has been used in di erent forms in [HH97b, Hod97, HH99c] The version presented here is in a fairly general form (theorem 39) and may be useful in other applications. We will discuss it in more detail later. Games are used throughout, here. It may be worth repeating here that for n = 3; 4, CRA n is elementary: we have CRA n = B n = fA 2 RA n : A is ....

R Hirsch and I Hodkinson. Representability is not decidable for nite relation algebras. Trans. Amer. Math. Soc., 1999? To appear.

....and all the logics of the form L S5 2 are undecidable and non nitely axiomatizable, if K L S5. Thus K 3 is a natural example of an undecidable but recursively enumerable logic which has the nite model property. In the proofs we will use the following result of Hirsch Hodkinson [8]: It is undecidable whether a nite simple relation algebra is representable. 1 For any natural number n 3 and any nite simple relation algebra A (see Section 3 for de nitions) we de ne (in a recursive way) a nite n frame FA;n and a 3 modal formula A , and prove the following lemmas. ....

....2, since if L were nitely axiomatizable then there would be a recursive test for nite frames being frames for L. We prove Lemmas 5, 6 and Theorem 4 in Section 3. Note that if L is recursively enumerable and nite product frames for L are also recursively enumerable (such as, e.g. for 1 In [8] this statement is not claimed for nite simple relation algebras, but for nite relation algebras in general only. However, this implies the result also for nite simple relation algebras, by taking subdirect decompositions. Or, in another way: the relation algebras constructed in the proof of ....

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R. Hirsch and I. Hodkinson. Representability is not decidable for nite relation algebras. Trans. Amer. Math. Soc., 353:1403-1425, 2001.

....and all the logics of the form L S5 2 are undecidable and non nitely axiomatizable, if K L S5. This way K 3 is a natural example of an undecidable but recursively enumerable logic which has the nite model property. In the proofs we will use the following result of Hirsch Hodkinson [7]: It is undecidable whether a nite simple relation algebra is representable. 1 (1) Here we give the proofs for the case n = 3 only, but they can be easily modi ed for any n 3. For any nite simple relation algebra A (see Section 3 for de nitions) we de ne (in a recursive way) a nite ....

....de ne (in a recursive way) a nite 3 modal frame FA and a 3 modal formula A , and prove the following lemmas. Lemma 5. Let L be any set of 3 modal formulas with K 3 L S5 3 . Then the following are equivalent: i) FA is a frame for L. ii) The formula : A does not belong to L. 1 In [7] this statement is not claimed for nite simple relation algebras, but for relation algebras in general only. However, the relation algebras contructed in the proof therein are clearly simple, thus the proof works for simple relation algebras as well. ON MODAL LOGICS BETWEEN K 3 AND S5 3 3 ....

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R. Hirsch and I. Hodkinson. Representability is not decidable for nite relation algebras. Trans. Amer. Math. Soc., 2000. To appear.

....n 2 ary relation on the atoms of A and check that it is an n dimensional basis in the sense of de nition 6 below; this algorithm accepts A if and only if A 2 B n , if and only if A 2 B n (because A is nite, so that A = A ) if and only if A 2 RA n , as required. It was shown in [HH99c] that the corresponding problem for RRA is undecidable, so this is an advantage of using RA n instead. 1.9 Complete relativised representations Now that we have a notion of representation, we can look for a direct n dimensional analogue of CRA. Earlier, we said that this was B n , but this ....

....relative to RA n if no RA l (n l ) is. In a similar way, we obtain the result discussed earlier: Theorem 2 For any n with 5 n , CRA n is not elementary. Our construction in the proofs of these two results originated in [Hir95] and has been used in di erent forms in [HH97b, Hod97, HH99c] The version presented here is in a fairly general form (theorem 39) and may be useful in other applications. We will discuss it in more detail later. Games are used throughout, here. 6 It may be worth repeating here that for n = 3; 4, CRA n is elementary: we have CRA n = B n = fA 2 RA n : A ....

R Hirsch and I Hodkinson. Representability is not decidable for nite relation algebras. Trans. Amer. Math. Soc., 1999? To appear.

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