Computable model theory

Fokina, Ekaterina B.; Harizanov, Valentina S.; Melnikov, Alexander G.

doi:10.1017/cbo9781107338579.006

Cited by 47 publications

(9 citation statements)

References 268 publications

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“…As usual, η denotes the (standard ordering of) rationals. Zubkov and Frolov [20] proved that for any countable graph G, there exists a binary relation R on (η, <) such that the spectrum DgSp (η,<) (R) is equal to the degree spectrum of the structure G. For further results on degree spectra of relations, we refer an interested reader to, e.g., Section 7 of [8].…”

Section: Related Workmentioning

confidence: 99%

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Degrees of relations on canonically ordered natural numbers and integers

Wrocławski

Bazhenov

Kalociński

2023

Preprint

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We investigate the degree spectra of computable relations on canonically ordered natural numbers (ω,<) and integers (ζ,<). As for (ω,<), we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all Δ2 degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to [1], we obtain a more general solution to the problem regarding possible degree spectra on (ω,<), answering the question whether there are infinitely many such spectra. As for (ζ,<), we prove the following dichotomy result: given an arbitrary computable relation R on (ζ,<), its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for (ω,<) obtained in [2], and provide initial insight to Wright's question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of [1]. MSC Classification: 03C57

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Section: Related Workmentioning

confidence: 99%

“…One of the research programs in computable structure theory consists in the study of how the complexity of a relation on a given structure behaves under isomorphisms (see, e.g., [5][6][7][8]). Recall that a structure is computable if its domain and basic relations, functions and constants are uniformly computable.…”

Section: Introductionmentioning

confidence: 99%

Degrees of relations on canonically ordered natural numbers and integers

Wrocławski

Bazhenov

Kalociński

2023

Preprint

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“…The systematic investigations of degree spectra were initiated by Richter [37]. The reader is referred to [12] for a detailed survey of the results on degree spectra.…”

Section: Introductionmentioning

confidence: 99%

On the effective universality of mereological theories

Bazhenov

Tsai

2021

Mathematical Logic Qtrly

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Mereological theories are based on the binary relation “being a part of”. The systematic investigations of mereology were initiated by Leśniewski. More recent authors (including Simons, Casati and Varzi, Hovda) formulated a series of first‐order mereological axioms. These axioms give rise to a plenitude of theories, which are of great philosophical interest. The paper considers first‐order mereological theories from the point of view of computable (or effective) algebra. Following the approach of Hirschfeldt, Khoussainov, Shore, and Slinko, we isolate two important computability‐theoretic properties P (namely, degree spectra of structures, and effective dimensions), and consider the following problem: for a given mereological theory T, is it true that its models can realize every possible variant of the property P? If the answer is positive, then we say that the theory T is DSED$\mathit {DSED}$‐universal. We obtain the following results about known mereological theories. Any theory T which is weaker than Extensional Closure Mereology (CEM) is DSED$\mathit {DSED}$‐universal. A similar fact is true for the theory GM2. On the other hand, any theory stronger that CEM + (C) + (G) is not DSED$\mathit {DSED}$‐universal. In particular, General Extensional Mereology is not DSED$\mathit {DSED}$‐universal.

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“…The computable structure theory of pcas as partial c.e. structures was recently studied by Fokina and Terwijn [12]. Since pcas can be seen as abstract models of computation, it is only natural to consider their complexity as algebraic structures from the viewpoint of computability theory.…”

Section: Introductionmentioning

confidence: 99%