Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra

In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of theorems in many finitely axiomatisable theories is nonrecursive, but the set of theorems for any finitely axiomatisable complete theory is recursive. Finitely presented groups might have an nonrecursive word problem, but finitely presented simple groups have a recursive word problem. In this article we introduce a topological framework based on closure spaces to show that many of these proofs can be obtained in a similar setting. We will show in particular that these statements can be generalized to cover arbitrary structures, with no finite or recursive presentation/axiomatization. This generalizes in particular work by Kuznetsov and others. Examples from first order logic and symbolic dynamics will be discussed at length

Satisfaction classes in nonstandard models of first-order arithmetic

A satisfaction class is a set of nonstandard sentences respecting Tarski's truth definition. We are mainly interested in full satisfaction classes, i.e., satisfaction classes which decides all nonstandard sentences. Kotlarski, Krajewski and Lachlan proved in 1981 that a countable model of PA admits a satisfaction class if and only if it is recursively saturated. A proof of this fact is presented in detail in such a way that it is adaptable to a language with function symbols. The idea that a satisfaction class can only see finitely deep in a formula is extended to terms. The definition gives rise to new notions of valuations of nonstandard terms; these are investigated. The notion of a free satisfaction class is introduced, it is a satisfaction class free of existential assumptions on nonstandard terms. It is well known that pathologies arise in some satisfaction classes. Ideas of how to remove those are presented in the last chapter. This is done mainly by adding inference rules to M-logic. The consistency of many of these extensions is left as an open question.Comment: Thesis for the degree of licentiate of philosophy, 74 pages, 4 figure

Unsolvability Cores in Classification Problems

Classification problems have been introduced by M. Ziegler as a generalization of promise problems. In this paper we are concerned with solvability and unsolvability questions with respect to a given set or language family, especially with cores of unsolvability. We generalize the results about unsolvability cores in promise problems to classification problems. Our main results are a characterization of unsolvability cores via cohesiveness and existence theorems for such cores in unsolvable classification problems. In contrast to promise problems we have to strengthen the conditions to assert the existence of such cores. In general unsolvable classification problems with more than two components exist, which possess no cores, even if the set family under consideration satisfies the assumptions which are necessary to prove the existence of cores in unsolvable promise problems. But, if one of the components is fixed we can use the results on unsolvability cores in promise problems, to assert the existence of such cores in general. In this case we speak of conditional classification problems and conditional cores. The existence of conditional cores can be related to complexity cores. Using this connection we can prove for language families, that conditional cores with recursive components exist, provided that this family admits an uniform solution for the word problem

A Second Step Towards Complexity-Theoretic Analogs of Rice's Theorem

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard. The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism.Comment: 14 pages. To appear in Theoretical Computer Scienc

On the isolated points in the space of groups

We investigate the isolated points in the space of finitely generated groups. We give a workable characterization of isolated groups and study their hereditary properties. Various examples of groups are shown to yield isolated groups. We also discuss a connection between isolated groups and solvability of the word problem.Comment: 30 pages, no figure. v2: minor changes, published version from March 200