DATALOG SIRUPs Uniform Boundedness is Undecidable

Lemma 2.19 If C(g) does not hold then for each constant C there exist a database D, , and a fact which can be proved, with the rule R, in the database D, but the proof requires more than C steps. Proof: We proceed in a similar way as in the proof of lemma 2.16, with the following diierences: (i) we no longer assume that the IDB input is empty. Instead, we require that there are the following CONFIG facts in the input: (ii) we require that for each x C Motorway(joker; joker; : : :joker; A px) holds. This ends the proof of Theorem 2.20 Uniform boundedness of single rule DATALOG programs is undecidable. 3 Acknowledgment This paper has been written while the author was enjoying the hospitality of the people of Laboratoire d'Informatique Fondamentale in Lille (France). It is much to little to merely write that I had perfect conditions for work there. I would like to thank Philippe Devienne and Jean-Christophe Routier for helpful discussion and Leszek Pacholski for his spiritual support. In this way, one could think, we secure that it is possible to start the computation of the Achilles-Turtle machine in each place, where any derivation step is made. But it is not enough to go in the footsteps of the proof of lemma 2.15. We require there, that the initial connguration is not only provable, what is really secured by the would-be rule above, but that it is provable in a bounded number of steps (in fact, just one step, in the previous section). We are to think of a new trick to assure that. 2.7 Single rule program: how to construct it The single recursive rule R is: Where the constant joker occurs p times in the "predicate" Motorway. We have added two additional arguments to the recursive predicate here. The rule asserts that if something can be derived then its second argument is run. So, if only the constants run and jump are not interpreted in the same way in the database, then the fact A 2) can not be proved by the program, and if it is prov-able then it is provable in 0 steps (is given as a part of the input). The fact CONFIG(run; jump; joker; joker; joker) does not require deep proofs: in fact if any proof at all is possible then the fact is given in the …