Equivalence Relations on Classes of Computable Structures

Fokina, Ekaterina B.; Friedman, Sy-David

doi:10.1007/978-3-642-03073-4_21

Cited by 19 publications

(23 citation statements)

References 9 publications

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“…The notion of F F -reducibility was first used in [9] and was called "tcreducibility". In the next section we will explain the relationship between F Freducibility and the notion of tc-reducibility introduced in [4] to compare the classes of countable structures.…”

Section: σ 1 1 Sets and Relationsmentioning

confidence: 99%

“…Thus, it will make sense to compare relations on classes of computable structures with relations on subsets of ω. The most studied cases are that of isomorphism and bi-embeddability relations, e.g., [2,6,9,16]. We are interested in studying the relations on classes that are nicely defined.…”

Section: Computable Characterization and Classificationmentioning

confidence: 99%

“…In [9] equivalence relations were compared not only via F F -reducibility but also via hyperarithmetical reducibility (h-reducibility):…”

Section: Open Problemsmentioning

confidence: 99%

See 2 more Smart Citations

Isomorphism relations on computable structures

Fokina¹,

Friedman²,

Harizanov³

et al. 2012

J. symb. log.

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Section: σ 1 1 Sets and Relationsmentioning

confidence: 99%

Section: Computable Characterization and Classificationmentioning

confidence: 99%

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Isomorphism relations on computable structures

Fokina¹,

Friedman²,

Harizanov³

et al. 2012

J. symb. log.

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“…Indeed, Coskey, Hamkins and Miller [4] and later Miller and Ng [11] studied the countable analogues of several standard equivalence relations arising in Borel theory. This reducibility has gone by many different names in the literature, having been called m-reducibility in [1,8,2] and F F -reducibility in [6], in addition to a version on first-order theories which was called Turing-computable reducibility [13,3]. Computable reducibility was also used in [9,11]; we prefer this name since we wish to avoid confusion with the usual m-reducibility and Turing-reducibility between sets of numbers.…”

Section: Introductionmentioning

confidence: 99%

On the Degree Structure of Equivalence Relations Under Computable Reducibility

Ng¹,

Yu²

2019

Notre Dame J. Formal Logic

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We study the degree structure of the ω-c.e., n-c.e. and Π 0 1 equivalence relations under the computable many-one reducibility. In particular we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-c.e. equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e. and Π 0 1 equivalence relations. We prove that for all the degree classes considered, upward density holds and downward density fails.

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“…Computable reducibility is readily defined. It has gone by many different names in the literature, having been called m-reducibility in [1,2,11] and FFreducibility in [7,8,9], in addition to a version on first-order theories which was called Turing-computable reducibility (see [3,4]). Definition 1.1 Let E and F be equivalence relations on ω.…”

Section: Introductionmentioning

confidence: 99%

Finitary Reducibility on Equivalence Relations

Miller

Ng²

2016

J. symb. log.

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Abstract. We introduce the notion of finitary computable reducibility on equivalence relations on the domain . This is a weakening of the usual notion of computable reducibility, and we show it to be distinct in several ways. In particular, whereas no equivalence relation can be Π 0 n+2 -complete under computable reducibility, we show that, for every n, there does exist a natural equivalence relation which is Π 0 n+2 -complete under finitary reducibility. We also show that our hierarchy of finitary reducibilities does not collapse, and illustrate how it sharpens certain known results. Along the way, we present several new results which use computable reducibility to establish the complexity of various naturally defined equivalence relations in the arithmetical hierarchy. §1. Introduction. Computable reducibility provides a natural way of measuring and comparing the complexity of equivalence relations on the natural numbers. Like most notions of reducibility on sets of natural numbers, it relies on the concept of Turing computability to rank objects according to their complexity, even when those objects themselves may be far from computable. It has found particular usefulness in computable model theory, as a measurement of the classical property of being isomorphic: if one can computably reduce the isomorphism problem for computable models of a theory T 0 to the isomorphism problem for computable models of another theory T 1 , then it is reasonable to say that isomorphism on models of T 0 is no more difficult than on models of T 1 . The related notion of Borel reducibility was famously applied this way by Friedman and Stanley in [10], to study the isomorphism problem on all countable models of a theory. Yet computable reducibility has also become the subject of study in pure computability theory, as a way of ranking various well-known equivalence relations arising there.Recently, as part of our study of this topic, we came to consider certain reducibilities weaker than computable reducibility. This article introduces these new, finitary notions of reducibility on equivalence relations and explains some of their uses. We believe that researchers familiar with computable reducibility will find finitary reducibility to be a natural and appropriate measure of complexity, not to supplant computable reducibility but to enhance it and provide a finer analysis of situations in which computable reducibility fails to hold.Computable reducibility is readily defined. It has gone by many different names in the literature, having been called m-reducibility in [1, 2, 11] and FF-reducibility

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Equivalence Relations on Classes of Computable Structures

Cited by 19 publications

References 9 publications

Isomorphism relations on computable structures

Isomorphism relations on computable structures

On the Degree Structure of Equivalence Relations Under Computable Reducibility

Finitary Reducibility on Equivalence Relations

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