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Metric Structures and Probabilistic Computation
[article]

2008
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arXiv
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pre-print

Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) can play an analogous role for structures described in continuous first-order logic.

arXiv:0806.0398v1
fatcat:ymupc3jdorhp5h7xplxcdkuive
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... main result of this paper is an effective completeness theorem, showing that every decidable continuous first-order theory has a probabilistically decidable model. Later sections give examples of the application of this framework to various classes of structures, and to some problems of computational complexity theory.##
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Structural Highness Notions
[article]

2021
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arXiv
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pre-print

We introduce several highness notions on degrees related to the problem of computing isomorphisms between structures, provided that isomorphisms exist. We consider variants along axes of uniformity, inclusion of negative information, and several other problems related to computing isomorphisms. These other problems include Scott analysis (in the form of back-and-forth relations), jump hierarchies, and computing descending sequences in linear orders.

arXiv:2109.07619v1
fatcat:et3e3c7knbd45ce3tgfepeubvi
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Classification from a Computable Viewpoint
[article]

2008
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arXiv
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pre-print

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

arXiv:0803.3293v1
fatcat:v64gbjnw4jghreao7w4nszzlxu
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... aper, we describe some recent work on classification in computable structure theory.##
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PAC Learning, VC Dimension, and the Arithmetic Hierarchy
[article]

2014
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arXiv
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pre-print

We compute that the index set of PAC-learnable concept classes is m-complete Σ^0_3 within the set of indices for all concept classes of a reasonable form. All concept classes considered are computable enumerations of computable Π^0_1 classes, in a sense made precise here. This family of concept classes is sufficient to cover all standard examples, and also has the property that PAC learnability is equivalent to finite VC dimension.

arXiv:1406.1111v1
fatcat:6ate4oddpndfpgz6w25l6fj3p4
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Generically Computable Equivalence Structures and Isomorphisms
[article]

2018
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arXiv
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pre-print

We define notions of generically and coarsely computable relations and structures and functions between structures. We investigate the existence and uniqueness of equivalence structures in the context of these definitions

arXiv:1808.02782v1
fatcat:y5e6mkumcvehzc5xuffvukdwme
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Comparing Classes of Finite Structures
[article]

2008
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arXiv
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pre-print

We introduce a reducibility on classes of structures, essentially a uniform enumeration reducibility. This reducibility is inspired by the Friedman-Stanley paper on using Borel reductions to compare classes of countable structures. This reducibility is calibrated by comparing several classes of structures. The class of cyclic graphs and the class of finite prime fields are equivalent, and are properly below the class of arbitrary finite graphs. The class of finite graphs and the class of finite

arXiv:0803.3291v1
fatcat:xxrl4kc5hnd43hn66zgh5knneu
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... linear orders are maximal among all classes of finite structures. We also prove some general characterizations of reducibility to certain classes. Examples of large chains and antichains of classes are constructed.##
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Effective completeness for real computation
[article]

2009
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arXiv
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pre-print

The main result of this paper, as previously presented to arxiv, was incorrect. See the full text for details and for reference to the remaining results.

arXiv:0807.3558v2
fatcat:h7dezctafbebzjoayqxg2mqjzy
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Index Sets of Computable Structures
[article]

2008
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arXiv
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pre-print

The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Q-vector spaces, Archimedean real closed ordered fields, reduced Abelian p-groups of length less than ω^2, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be m-complete Π_n^0, d-Σ_n^0, or Σ_n^0, for various n. In each

arXiv:0803.3294v1
fatcat:aoinimw3yjdknbvaodjidhchfa
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... , the calculation involves finding an " optimal" sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Π_n, d-Σ_n, or Σ_n) yields a bound on the complexity of the index set. When we show m the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.##
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The Distance Function on a Computable Graph
[article]

2011
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arXiv
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pre-print

We apply the techniques of computable model theory to the distance function of a graph. This task leads us to adapt the definitions of several truth-table reducibilities so that they apply to functions as well as to sets, and we prove assorted theorems about the new reducibilities and about functions which have nonincreasing computable approximations. Finally, we show that the spectrum of the distance function can consist of an arbitrary single btt-degree which is approximable from above, or of

arXiv:1111.2480v1
fatcat:xydxjnck5jhbvks4k66fiodcki
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... all such btt-degrees at once, or of the bT-degrees of exactly those functions approximable from above in at most n steps.##
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Turing Degrees of Isomorphism Types of Algebraic Objects
[article]

2005
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arXiv
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pre-print

VALENTINA HARIZANOV,

arXiv:math/0507128v1
fatcat:qg5i3nezencrlmyoi5u4slk7wu
*WESLEY**CALVERT*, AND ALEXANDRA SHLAPENTOKH VALENTINA HARIZANOV,*WESLEY**CALVERT*, AND ALEXANDRA SHLAPENTOKH VALENTINA HARIZANOV,*WESLEY**CALVERT*, AND ALEXANDRA SHLAPENTOKH ... VALENTINA HARIZANOV,*WESLEY**CALVERT*, AND ALEXANDRA SHLAPENTOKH ...##
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Metric structures and probabilistic computation

2011
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Theoretical Computer Science
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Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of mathematical objects that can be described in classical first-order logic. The present paper shows that probabilistic computation (sometimes called randomized computation) and continuous logic stand in a similar close relationship. The main result of this

doi:10.1016/j.tcs.2011.02.005
fatcat:7bydv5rc6jaejeyqfx5ln4kpai
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... r is an effective completeness theorem, showing that every decidable continuous first-order theory has a probabilistically decidable model. We also show that probabilistically computable structures give rise to a model of ACA 0 in a natural way, and describe a connection with complexity theory.##
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The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length
[article]

2004
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arXiv
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pre-print

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider only countable members. This paper explores such a notion for classes of computable structures by working out a sequence of examples. We follow recent work by Goncharov and Knight in using the degree of the isomorphism problem for a class to distinguish

arXiv:math/0406505v1
fatcat:o6ghjx2wszgkndeonsy5pg4pgu
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... iable classes from non-classifiable. In this paper, we calculate the degree of the isomorphism problem for Abelian p-groups of bounded Ulm length. The result is a sequence of classes whose isomorphism problems are cofinal in the hyperarithmetical hierarchy. In the process, new back-and-forth relations on such groups are calculated.##
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Three notions of effective computation on R
[article]

2008
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arXiv
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pre-print

We compare three notions of effectiveness on uncountable structures. The first notion is that of a -computable structure, based on a model of computation proposed by Blum, Shub, and Smale, which uses full-precision real arithmetic. The second notion is that of an F-parameterizable structure, defined by Morozov and based on Mal'tsev's notion of a constructive structure. The third is Σ-definability over HF(), defined by Ershov as a generalization of the observation that the computably enumerable

arXiv:0803.3073v2
fatcat:okdo6b2gjbfbbfj5agguaym6yu
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... ets are exactly those Σ_1-definable in HF(N). We show that every -computable structure has an F-parameterization, but that the expansion of the real field by the exponential function is F-parameterizable but not -computable. We also show that the structures with -computable copies are exactly the structures with copies Σ-definable over HF(). One consequence of this equivalence is a method of approximating certain -computable structures by Turing computable structures.##
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Some results on R-computable structures
[article]

2009
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arXiv
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pre-print

Lemma 4.8 (

arXiv:0803.3404v2
fatcat:widpt6exaba3jbsi23vabhhpsq
*Calvert*-Miller [8]). Let M be a R-computable manifold. ... Nevertheless, the answer to the question of computing a fundamental group from a manifold is largely negative: Theorem 4.6 (*Calvert*-Miller [8] ). ...##
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Real Computable Manifolds and Homotopy Groups
[chapter]

2009
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Lecture Notes in Computer Science
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Using the model of real computability developed by Blum, Cucker, Shub, and Smale, we investigate the difficulty of determining the answers to several basic topological questions about manifolds. We state definitions of real-computable manifold and of real-computable paths in such manifolds, and show that, while BSS machines cannot in general decide such questions as nullhomotopy and simple connectedness for such structures, there are nevertheless real-computable presentations of paths and

doi:10.1007/978-3-642-03745-0_16
fatcat:ktkwh4gkxze6lcke6dolxv5jte
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... py equivalence classes under which such computations are possible.
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