Chapter 1 Pure computable model theory

Harizanov, Valentina S.

doi:10.1016/s0049-237x(98)80002-5

Cited by 56 publications

(46 citation statements)

References 135 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, any computable tree with no dead ends can be coded into a complete decidable theory. For more details, see Harizanov [5].…”

Section: Theorem (Csima) Let T Be a Complete Decidable Theory Let Smentioning

confidence: 99%

Computable trees, prime models, and relative decidability

Hirschfeldt

2005

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We show that for every computable tree T with no dead ends and all paths computable, and every D > T ∅, there is a D-computable listing of the isolated paths of T . It follows that for every complete decidable theory T such that all the types of T are computable and every D > T ∅, there is a D-decidable prime model of T . This result extends a theorem of Csima and yields a stronger version of the theorem, due independently to Slaman and Wehner, that there is a structure with presentations of every nonzero degree but no computable presentation.

show abstract

“…On the other hand, any computable tree with no dead ends can be coded into a complete decidable theory. For more details, see Harizanov [5].…”

Section: Theorem (Csima) Let T Be a Complete Decidable Theory Let Smentioning

confidence: 99%

Computable trees, prime models, and relative decidability

Hirschfeldt

2005

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…However, since the adjacency relation is Π 0 1 -definable, its degree spectrum clearly cannot contain any degrees that are not c.e. Many results are known about the relationship between definability of a relation and upper bounds on its degree spectrum; we refer the reader to [9] for details.…”

Section: Lemma 12 If S Is Spectrally Universal For a Theory T Thenmentioning

confidence: 99%

Spectra of structures and relations

Harizanov

Miller²

2007

J. symb. log.

Self Cite

View full text Add to dashboard Cite

We consider embeddings of structures which preserve spectra: if g : M → S with S computable, then M should have the same Turing degree spectrum (as a structure) that g(M) has (as a relation on S). We show that the computable dense linear order L is universal for all countable linear orders under this notion of embedding, and we establish a similar result for the computable random graph G. Such structures are said to be spectrally universal. We use our results to answer a question of Goncharov, and also to characterize the possible spectra of structures as precisely the spectra of unary relations on G. Finally, we consider the extent to which all spectra of unary relations on the structure L may be realized by such embeddings, offering partial results and building the first known example of a structure whose spectrum contains precisely those degrees c with c ≥ T 0 .

show abstract

“…For computable model theory, see [8], [13], [12], and [24]. In the sections that follow we will use some facts from algebra and number theory.…”

Section: Dgsp(a) = {Deg(b) : B ∼ = A}mentioning

confidence: 99%

Turing degrees of isomorphism types of algebraic objects

Calvert¹,

Harizanov²,

Shlapentokh³

2007

Journal of the London Mathematical Society

View full text Add to dashboard Cite

Abstract. The Turing degree spectrum of a countable structure A is the set of all Turing degrees of isomorphic copies of A. The Turing degree of the isomorphism type of A, if it exists, is the least Turing degree in its degree spectrum. We show there are countable fields, rings, and torsion-free abelian groups of arbitrary rank, whose isomorphism types have arbitrary Turing degrees. We also show that there are structures in each of these classes whose isomorphism types do not have Turing degrees.

show abstract

Chapter 1 Pure computable model theory

Cited by 56 publications

References 135 publications

Computable trees, prime models, and relative decidability

Computable trees, prime models, and relative decidability

Spectra of structures and relations

Turing degrees of isomorphism types of algebraic objects

Contact Info

Product

Resources

About