Degrees of presentability of structures. II

Abstract

Keywords: admissible set, semilattice of degrees of Σ-definability.We show that the property of being locally constructivizable is inherited under Muchnik reducibility, which is weakest among the effective reducibilities considered over countable structures. It is stated that local constructivizability of level higher than 1 is inherited under Σ-reducibility but is not inherited under Medvedev reducibility. An example of a structure M and a relation P ⊆ M is constructed for which (M, P ) ≡ M but (M, P ) ≡ Σ M. Also, we point out a class of structures which are effectively defined by a family of their local theories.This paper is a continuation of [1, 2] and uses the same notation. PROPERTIES OF STRUCTURES INHERITED UNDER EFFECTIVE REDUCIBILITIESBelow we show that the condition of a structure M having a degree specified in [1, Thm. 7] is essential. To do this, we state necessary conditions for effective reducibilities between structures, in particular, conditions that are necessary for being Σ-definable.Structure M is said to be locally constructivizable [3] if Th ∃ (M,m) is computably enumerable (c.e.) for anym ∈ M <ω . As noted in [3], the local constructivizability of M is equivalent to the fact that for every tuplem ∈ M <ω , there exist a constructivizable structure N and a tuplen ∈ N <ω such that Th ∃ (M,m) = Th ∃ (N,n). For structures M and N, by writing M ∃ N we mean that for every tuplē m ∈ M <ω , there is a tuplen ∈ N <ω for which Th ∃ (M,m) e Th ∃ (N,n). In particular, if M is locally constructivizable then M ∃ N for any structure N.That a structure M is locally constructivizable if so is N with M Σ N was first mentioned in [3]. A direct generalization of this fact is the following: if M Σ N then M ∃ N. In order to state other necessary conditions of Σ-definability (which are used in proving negative results on Σ ), we consider some relevant notions. Definition 1. Structure M is locally constructivizable of level n (1 < n ω) if for every tuplem ∈ M <ω there exist a constructivizable structure N and a tuplen ∈ N <ω such that (M,m) ≡ HF n (N,n). A countable *

Rice sequences of relations

“…For him, the domain of A is HF A , and the added relation is the satisfaction relation for S 1 -formulae, namely hK (A). The importance of the role of S-reducibility in this context was previously shown by Stukachev [22,23]. Another independent definition of the jump is due to Puzarenko [24].…”

“…On the other hand, it is not hard to prove that A ≤ I E, because the only rice subsets of B n in E are either finite or co-finite. There are various proofs that A ≤ w B does not imply A ≤ S B; see, for instance, Stukachev [22] and Kalimullin [26,27]. For one such example: let A be the structure coding the family of the graphs of all computable functions, i.e.…”

“…The notion of S-reducibility between abstract structures was introduced independently by Khisamiev [28] and Stukachev [22], and it is based on the notion of S-definability of a structure inside an admissible set by Ershov. Stukachev [22,23] and Puzarenko and co-workers [24,27] showed the importance of this reducibility and compared it with various other reducibilities between structures (including ≤ w ). The definitions of earlier studies [22,28] used, of course, the notions of S-definability rather than rice.…”

“…Stukachev [22,23] and Puzarenko and co-workers [24,27] showed the importance of this reducibility and compared it with various other reducibilities between structures (including ≤ w ). The definitions of earlier studies [22,28] used, of course, the notions of S-definability rather than rice. The equivalence with the definition given here follows from theorems 4.2 and 4.4.…”

A fixed point for the jump operator on structures

“…In [Stu], Question 1 was asked for Σ-equivalence, a notion of equivalence between structures that is much stronger than Muchnik equivalence. The notion of Σ-equivalence between abstract structures was introduced and studied in detail in [Khi04,Stu07,Stu08]. Our proof of Theorem 1.3 does answer this question, as it actually shows, in ZFC+ "0 # exists," that there is a structure A such that A ≡ Σ A ′ .…”