An introduction to computable model theory on groups and fields

2011 ◽  
Vol 3 (1) ◽  
Author(s):  
Russell Miller
Keyword(s):  
Model Theory ◽  
Computable Model ◽  
Computable Model Theory

scholarly journals Infinite Time Computable Model Theory

2008 ◽  
pp. 521-557 ◽  
Author(s):  
Joel David Hamkins ◽  
Russell Miller ◽  
Daniel Seabold ◽  
Steve Warner
Keyword(s):  
Model Theory ◽  
Computable Model ◽  
Infinite Time ◽  
Computable Model Theory

scholarly journals A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

2018 ◽  
Vol 83 (1) ◽  
pp. 326-348 ◽  
Author(s):  
RUSSELL MILLER ◽  
BJORN POONEN ◽  
HANS SCHOUTENS ◽  
ALEXANDRA SHLAPENTOKH
Keyword(s):  
Model Theory ◽  
Open Problem ◽  
Category Theory ◽  
Computable Model ◽  
Computable Model Theory ◽  
Arbitrary Characteristic ◽  
Faithful Functor ◽  
New Construction ◽  
Image Position ◽  
Countable Structure

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

Bounding prime models

2004 ◽  
Vol 69 (4) ◽  
pp. 1117-1142 ◽  
Author(s):  
Barbara F. Csima ◽  
Denis R. Hirschfeldt ◽  
Julia F. Knight ◽  
Robert I. Soare
Keyword(s):  
Model Theory ◽  
Computable Function ◽  
Prime Model ◽  
Computable Model ◽  
Principal Type ◽  
Combinatorial Properties ◽  
Computable Model Theory ◽  
Monotonic Functions ◽  
Limitwise Monotonic Functions

Abstract.A set X is prime bounding if for every complete atomic decidable (CAD) theory T there is a prime model of T decidable in X. It is easy to see that X = 0′ is prime bounding. Denisov claimed that every X <T 0′ is not prime bounding, but we discovered this to be incorrect. Here we give the correct characterization that the prime bounding sets X ≤τ 0′ are exactly the sets which are not low2. Recall that X is low2 if X″ ≤τ 0″. To prove that a low2 set X is not prime bounding we use a 0′ -computable listing of the array of sets {Y : Y ≤τX } to build a CAD theory T which diagonalizes against all potential X-decidable prime models of T, To prove that any non-low2X is indeed prime bounding. we fix a function f ≤TX that is not dominated by a certain 0′-computable function that picks out generators of principal types. Given a CAD theory T. we use f to eventually find, for every formula φ(x̄) con sistent with T. a principal type which contains it. and hence to build an X-decidable prime model of T. We prove the prime bounding property equivalent to several other combinatorial properties, including some related to the limitwise monotonic functions which have been introduced elsewhere in computable model theory.

scholarly journals Applications of Kolmogorov complexity to computable model theory

2007 ◽  
Vol 72 (3) ◽  
pp. 1041-1054 ◽  
Author(s):  
Bakhadyr Khoussainov ◽  
Pavel Semukhin ◽  
Frank Stephan
Keyword(s):  
Model Theory ◽  
Kolmogorov Complexity ◽  
Isomorphism Type ◽  
Computable Model ◽  
Categorical Theory ◽  
Computable Model Theory ◽  
Open Question ◽  
Saturated Structure ◽  
Computable Isomorphism

AbstractIn this paper we answer the following well-known open question in computable model theory. Does there exist a computable not ℵ0-categorical saturated structure with a unique computable isomor-phism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an ℵ1-categorical but not ℵ0-categorical saturated -structure with a unique computable isomorphism type. In addition, using the construction we give an example of an ℵ1-categorical but not ℵ0-categorical theory whose only non-computable model is the prime one.

Degree Spectra of Relations on Computable Structures

2000 ◽  
Vol 6 (2) ◽  
pp. 197-212 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Model Theory ◽  
Large Body ◽  
Point Of View ◽  
Linear Ordering ◽  
Computable Model ◽  
Computable Model Theory ◽  
Degree Spectra ◽  
Successor Relation ◽  
The Subject ◽  
Cover Material

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. (A valuable reference is the handbook [7]. In particular, the introduction and the articles by Ershov and Goncharov and by Harizanov give useful overviews, while the articles by Ash and by Goncharov cover material related to the topic of this communication.)Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable (c. e.) degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough.

scholarly journals Computability-Theoretic Complexity of Countable Structures

2002 ◽  
Vol 8 (4) ◽  
pp. 457-477 ◽  
Author(s):  
Valentina S. Harizanov
Keyword(s):  
Model Theory ◽  
Classical Result ◽  
Classical Model ◽  
Algebraic Closure ◽  
Computable Function ◽  
Vector Spaces ◽  
Computable Model ◽  
Mathematical Structures ◽  
Computable Model Theory ◽  
Countable Models

Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures.

Computable model theory

2014 ◽  
pp. 124-194 ◽  
Author(s):  
Ekaterina B. Fokina ◽  
Valentina Harizanov ◽  
Alexander Melnikov
Keyword(s):  
Model Theory ◽  
Computable Model ◽  
Computable Model Theory
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