Degrees of presentability of structures. I

Stukachev, A. I.

doi:10.1007/s10469-007-0041-z

Cited by 38 publications

(21 citation statements)

References 13 publications

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“…P The next proposition holds that a class of locally constructivizable (of level 1) countable structures is closed downward w.r.t. w , which is weakest among the reducibilities under consideration; this result follows immediately from Proposition 1 and from [1].…”

mentioning

confidence: 65%

“…Thus an s-expansion of the structure M defined by the presentation C 0 , and hence the structure (M,m), will be ∆-definable without parameters in HF(M). P An immediate consequence of [1,Thm. 7] is the following:…”

Section: * -Homogeneous Structuresmentioning

confidence: 89%

“…As noted in [1], card(S Σ ) = 2 α for every infinite cardinal α. Now we consider the question whether maximal antichains exist in S Σ (2 ω ).…”

Section: Propositionmentioning

confidence: 89%

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Degrees of presentability of structures. II

Stukachev

2008

Algebra Logic

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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 *

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confidence: 65%

Section: * -Homogeneous Structuresmentioning

confidence: 89%

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Degrees of presentability of structures. II

Stukachev

2008

Algebra Logic

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“…Considering the problems of enumerability, in [15] we obtain, by applying results and techniques due to J.F. Knight [5], the following result, which is in some way analogous to Selman-Rozinas Theorem.…”

Section: Medvedev and Muchnik Reducibilities In The Case Of Problems mentioning

confidence: 94%

“…In [15] we show that the requirement that a structure N have a degree in the Theorem 5 is essential and can not be dropped. For this we use the fact ( obtained independently by S. Wehner [17] and T. Slaman [12]) that there exist structures which mass problems of presentability belongs to the least non-zero degree of difficulty in the Medvedev lattice.…”

Section: ⇒ 2 Suppose M Is ∆-Definable In Hf(m)mentioning

confidence: 99%

On Mass Problems of Presentability

Stukachev

2006

Lecture Notes in Computer Science

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Abstract. We consider the notion of mass problem of presentability for countable structures, and study the relationship between Medvedev and Muchnik reducibilities on such problems and possible ways of syntactically characterizing these reducibilities. Also, we consider the notions of strong and weak presentability dimension and characterize classes of structures with presentability dimensions 1. Basic notions and factsThe main problem we consider in this paper is the relationship between presentations of countable structures on natural numbers and on admissible sets. Most of notations and terminology we use here are standard and corresponds to [4,1,13]. We denote the domains of a structures M, N, . . . by M, N. . . .. For any arbitrary structure M the hereditary finite superstructure HF(M), which is the least admissible set containing the domain of M as a subset, enables us to study effective (computable) properties of M by means of computability theory for admissible sets. The exact definition is as follows: the hereditary finite superstructure HF(M) over a structure M of signature σ is a structure of signature σ = σ ∪ {U 1 , ∈ 2 }, whose universe is HF (M ) = n∈ω H n (M ), where, card(a) < ω}, the predicate U distinguish the set of the elements of the structure M (regarded as urelements), while the relation ∈ has the usual set theoretic meaning.In the class of all formulas of signature σ we define the subclass of ∆ 0 -formulas as the closure of the class of atomic formulas under ∧, ∨, ¬, →, ∃x ∈ y, ∀x ∈ y; the class of Σ-formulas is the closure of the class of ∆ 0 -formulas under ∧, ∨, ¬, →, ∃x ∈ y, ∀x ∈ y, and the quantifier ∃x; the class of Π-formulas is defined in the same way, allowing the quantifier ∀x instead of ∃x. A relation on HF(M) is called Σ-definable (Π-definable) if it is defined by a corresponding formula, possibly with parameters; it is called ∆-definable if it is Σ-and Π-definable at the same time.

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On Processes and Structures

Stukachev

2013

Lecture Notes in Computer Science

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Degrees of presentability of structures. I

Cited by 38 publications

References 13 publications

Degrees of presentability of structures. II

Degrees of presentability of structures. II

On Mass Problems of Presentability

On Processes and Structures

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