Searching for Applicable Versions of Computable Structures

2021 ◽  
pp. 1-11
Author(s):  
P. E. Alaev ◽  
V. L. Selivanov
Keyword(s):  
Computable Structures

scholarly journals Π10 classes and strong degree spectra of relations

2007 ◽  
Vol 72 (3) ◽  
pp. 1003-1018 ◽  
Author(s):  
John Chisholm ◽  
Jennifer Chubb ◽  
Valentina S. Harizanov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch ◽  
...  
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Truth Table ◽  
Linear Ordering ◽  
Computable Structures ◽  
Degree Spectra

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

scholarly journals Some new computable structures of high rank

2018 ◽  
Vol 146 (7) ◽  
pp. 3097-3109 ◽  
Author(s):  
Matthew Harrison-Trainor ◽  
Gregory Igusa ◽  
Julia F. Knight
Keyword(s):  
High Rank ◽  
Computable Structures

Computable structures of Scott rank 𝜔₁^{𝐶𝐾} in familiar classes

2007 ◽  
pp. 49-66 ◽  
Author(s):  
W. Calvert ◽  
S. S. Goncharov ◽  
J. F. Knight
Keyword(s):  
Computable Structures ◽  
Scott Rank

Recursive elements and constructive extensions of computable local integral domains

1973 ◽  
Vol 38 (2) ◽  
pp. 272-290 ◽  
Author(s):  
Glen H. Suter
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Integral Domain ◽  
Rational Numbers ◽  
Algebraic Structures ◽  
Computable Structures ◽  
Integral Domains ◽  
Local Integral ◽  
Algebraic Operations ◽  
Axiom Systems

With reservations, one can think of abstract algebra as the study of what consequences can be drawn from the axioms associated with certain concrete algebraic structures. Two important examples of such concrete algebraic structures are the integers and the rational numbers. The integers and the rational numbers have two properties which are not in general mirrored in the abstract axiom systems associated with them. That is, the integers and the rational numbers both have effectively computable metrics and their algebraic operations are effectively computable. The study of abstract algebraic systems which possess effectively computable algebraic operations has produced many interesting results. One can think of a computable algebraic structure as one whose elements have been labeled by the set of positive integers and whose operations are effectively computable. The formal definition of computable local integral domain will be given in §3. Some specific computable structures which have been studied are the integers, the rational numbers, and the rational numbers with p-adic valuation. Computable structures were studied in general by Rabin [12]. This paper concerns computable local integral domains as exemplified by the local integral domain Zp. Zp is the localization of the integers with respect to the maximal prime ideal generated by the positive prime p. We should note that the concept of local integral domain is not first order.Let the ordered pair (Q, M) stand for a local ring, where Q is the local ring and M is the unique maximal prime ideal of Q. Since most of my results are proving the existence of certain effective procedures, the assumption that Q has a principal maximal ideal M (rather than M has n generators) greatly simplifies many of the proofs.

scholarly journals Hanf number for Scott sentences of computable structures

2018 ◽  
Vol 57 (7-8) ◽  
pp. 889-907
Author(s):  
S. S. Goncharov ◽  
J. F. Knight ◽  
I. Souldatos
Keyword(s):  
Computable Structures ◽  
Hanf Number ◽  
Scott Sentences

scholarly journals Classification from a Computable Viewpoint

2006 ◽  
Vol 12 (2) ◽  
pp. 191-218 ◽  
Author(s):  
Wesley Calvert ◽  
Julia F. Knight
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Large Body ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Computable Structures ◽  
Structure Theorems ◽  
Sample Structure ◽  
Descriptive Set ◽  
Scott Rank

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work.

Degree spectra of relations on computable structures in the presence of Δ20 isomorphisms

2002 ◽  
Vol 67 (2) ◽  
pp. 697-720 ◽  
Author(s):  
Denis R. Hirschfeldt
Keyword(s):  
Point Of View ◽  
The Other ◽  
Computable Structure ◽  
Sufficient Condition ◽  
Computable Structures ◽  
Linear Orderings ◽  
Degree Spectra ◽  
Other Hand ◽  
Infinite Degree ◽  
Degree Spectrum

AbstractWe give some new examples of possible degree spectra of invariant relations on Δ20-categorical computable structures, which demonstrate that such spectra can be fairly complicated. On the other hand, we show that there are nontrivial restrictions on the sets of degrees that can be realized as degree spectra of such relations. In particular, we give a sufficient condition for a relation to have infinite degree spectrum that implies that every invariant computable relation on a Δ20-categorical computable structure is either intrinsically computable or has infinite degree spectrum. This condition also allows us to use the proof of a result of Moses [23] to establish the same result for computable relations on computable linear orderings.We also place our results in the context of the study of what types of degree-theoretic constructions can be carried out within the degree spectrum of a relation on a computable structure, given some restrictions on the relation or the structure. From this point of view we consider the cases of Δ20-categorical structures, linear orderings, and 1-decidable structures, in the last case using the proof of a result of Ash and Nerode [3] to extend results of Harizanov [14].

Orbit complexity by computable structures

Nonlinearity ◽  
2000 ◽  
Vol 13 (5) ◽  
pp. 1531-1546 ◽  
Author(s):  
Stefano Galatolo
Keyword(s):  
Computable Structures

COMPUTABLE STRUCTURES OF RANK $\omega_{1}^{{\rm CK}}$

2010 ◽  
Vol 10 (01n02) ◽  
pp. 31-43 ◽  
Author(s):  
J. F. KNIGHT ◽  
J. MILLAR
Keyword(s):  
Internal Model ◽  
Computable Structure ◽  
Computable Structures ◽  
Scott Rank ◽  
Countable Structure

For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks.

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