76 research outputs found
Degree spectra of relations on structures of finite computable dimension
AbstractWe show that for every computably enumerable (c.e.) degree a>0 there is an intrinsically c.e. relation on the domain of a computable structure of computable dimension 2 whose degree spectrum is {0,a}, thus answering a question of Goncharov and Khoussainov (Dokl. Math. 55 (1997) 55–57). We also show that this theorem remains true with α-c.e. in place of c.e. for any α∈ω∪{ω}. A modification of the proof of this result similar to what was done in Hirschfeldt (J. Symbolic Logic, to appear) shows that for any α∈ω∪{ω} and any α-c.e. degrees a0,…,an there is an intrinsically α-c.e. relation on the domain of a computable structure of computable dimension n+1 whose degree spectrum is {a0,…,an}. These results also hold for m-degree spectra of relations
Degree spectra and computable dimensions in algebraic structures
AbstractWhenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the given class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings (with zero-divisors); integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories
Counting the changes of random Δ20 sets
We study the number of changes of the initial segment Zs ↾n for computable approximations of a Martin-Löf random Δ02Δ20 set Z. We establish connections between this number of changes and various notions of computability theoretic lowness, as well as the fundamental thesis that, among random sets, randomness is antithetical to computational power. We introduce a new randomness notion, called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↾n changes more than c2n times. We establish various connections with ω-c.e. tracing and omega;-c.e. jump domination, a new lowness property. We also examine some relationships to randomness theoretic notions of highness, and give applications to the study of (weak) Demuth cuppability.Fil: Figueira, Santiago. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas; ArgentinaFil: Hirschfeldt, Denis R.. University of Chicago; Estados UnidosFil: Miller, Joseph S.. University of Wisconsin; Estados UnidosFil: Ng, Keng Meng. Nanyang Technological University; SingapurFil: Nies, André. The University Of Auckland; Nueva Zeland
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