Sequences of n-diagrams

Abstract

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc(), and the atomic diagram, D(), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

Computability-Theoretic Complexity of Countable Structures

“…The next result, which follows from the proofs in [32], gives a general sufficient condition for realizing arbitrary possible Turing degrees for the existential and atomic diagrams of isomorphic copies of a structure.…”

“…Theorem 5.7 (Harizanov, Knight and Morozov [32]). Using the notions above, assume that A is decidable, and that the relations ≤ n are c.e.…”

“…For every structure A, (D n (A)) n∈ is an -table. Harizanov, Knight and Morozov [32] investigated possible sequences of Turing degrees of n-diagrams for B A, (deg(D n (B))) n∈ .…”

“…In addition to linear orders, it would be worthwhile to investigate possible sequences of Turing degrees of n-diagrams for other natural mathematical structures. For models of completions of PA, see [32] for a conjecture about a characterization of these sequences.…”

Turing degrees of nonabelian groups

“…On the other hand, for an automorphically trivial structure, all isomorphic copies have the same Turing degree, and in the case of a finite language that degree must be 0 (see [13]). Harizanov, Knight and Morozov [12] showed that, while for every automorphically trivial structure M we have Richter [27] established the following general criterion for the existence of a structure whose isomorphism type has an arbitrary Turing degree. We will write A → B if A is embeddable in B.…”