Complexity classes of partial recursive functions

This paper studies possible extensions of the concept of complexity class of recursive functions to partial recursive functions. Many of the well-known results for total complexity classes are shown to have corresponding, though not exactly identical, statements for partial classes. In particular, with two important exceptions, all results on the presentation and decision problems of membership for the two most reasonable definitions of partial classes are the same as for total classes. The exceptions concern presentations of the complements and maximum difficulty for decision problems of the more restricted form of partial classes.The last section of this paper shows that it is not possible to have an “intersection theorem,” corresponding to the union theorem of McCreight and Meyer, either for complexity classes or complexity index sets

Isomorphism versus commensurability for a class of finitely presented groups

We construct a class of finitely presented groups where the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are first examples of such a contrastive complexity behaviour with respect to the isomorphism problem

On the difficulty of presenting finitely presentable groups

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal