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Page 3186 of Mathematical Reviews Vol. , Issue 2003e
[page]

2003
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Mathematical Reviews
*

D. (1-WI; Madison, WI

*Finite**computable**dimension**does**not**relativize*. (English summary) Arch. Math. Logic 41 (2002), no. 4, 309-320. ... In the present paper, Mc- Coy proves that this notion is trivial in that the notion of*finite**computable**dimension*(4 1)*does**not**relativize*. ...##
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Page 5961 of Mathematical Reviews Vol. , Issue 2004h
[page]

2004
*
Mathematical Reviews
*

Nevertheless the particular model theory of several classes of structures

*does**not*seem to admit the kinds of re- ductions used in that paper. ... the algebraic closure of a*finite*field and consequently to function fields whose constant fields are*not*algebraically closed. ...##
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Physical Relativism as an Interpretation of Existence
[article]

2013
*
arXiv
*
pre-print

In this manuscript, we propose a new interpretation of existence that we call physical

arXiv:1306.5484v2
fatcat:nkratr2lorgevejq6irvqkuuqu
*relativism*. ... Under physical*relativism*, the difference between mathematical existence and physical existence is clarified, and Wheeler's 'it from bit' viewpoint can be objectively evaluated. ... the existence of an arbitrarily long*computer*program that*does*halt. ...##
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Computability and Randomness

2019
*
Notices of the American Mathematical Society
*

For certain well-behaved sets ,

doi:10.1090/noti1905
fatcat:pg66obggazgwpj6dwy2buq6oze
*relativization*is actually*not*needed, and the classical*dimension*of is the supremum of the effective*dimensions*of its points. ... It has been known for some time that entropies of subshifts of*finite*type for*dimensions*≥ 2 are in general*not**computable*, but the following result gives a precise characterization. ...##
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Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

2018
*
ACM Transactions on Computation Theory
*

Other effectivizations, e.g.,

doi:10.1145/3201783
fatcat:hkadgadcinbbxj5jlemteg7qiu
*computable**dimensions*, polynomial time*dimensions*, and*finite*-state*dimensions*, have been investigated, but only the constructive*dimensions*are discussed here. ... We prove a point-to-set principle that enables one to use the (*relativized*, constructive)*dimension*of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff ... First, while the left-hand side is the classical Hausdorff*dimension*, which is a global property of E that*does**not*involve the theory of*computing*, the right-hand side is a pointwise property of the set ...##
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Unified characterizations of lowness properties via Kolmogorov complexity

2014
*
Archive for Mathematical Logic
*

2) We also show that X ∈ Low ⋆ (SR, WR) if and only if X is

doi:10.1007/s00153-014-0413-8
fatcat:udown43nmnhl5dtc2tlpyd2tv4
*computably*i.o. tt-traceable if and only if X is*not*totally complex if and only if X is Schnorr Hausdorff measure zero with respect to all*computable*... that X ∈ Low ⋆ (MLR, SR) if and only if X is c.e. tt-traceable if and only if X is anticomplex if and only if X is Martin-Löf packing measure zero with respect to all*computable**dimension*functions. ( ... In contrast, van Lambalgen's theorem with the usual*relativization**does**not*hold for Schnorr randomness,*computable*randomness [37, 55] , Kurtz randomness [21] or weak 2-randomness [1] . ...##
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Uniform Kurtz randomness

2013
*
Journal of Logic and Computation
*

For instance, van Lambalgen's theorem holds for uniform Kurtz randomness while

doi:10.1093/logcom/ext054
fatcat:iw2nb2syw5aaxlbrsgykbmwvaq
*not*for (the usual*relativization*of) Kurtz randomness. ... We propose studying uniform Kurtz randomness, which is the uniform*relativization*of Kurtz randomness. This notion has more natural properties than the usual*relativization*. ... In fact, van Lambalgen's theorem holds for uniform Schnorr randomness (the uniform*relativization*of Schnorr randomness) [22, 23] while it*does**not*hold for the usual*relativization*of Schnorr randomness ...##
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Page 6352 of Mathematical Reviews Vol. , Issue 89K
[page]

1989
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Mathematical Reviews
*

The author defines a language L contained in {0,1}* to be un- provably intractable if L

*does**not*belong to P, but T*does**not*prove “L*does**not*belong to P”, in the sense of the theorem above. ... Theorem: If P 4 NP then for any formula A(n) representing NP —P over the theory 7, there is a language LeéNP-—P such that, for all Turing machines M, accepting L, T*does**not*prove A(e). ...##
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Page 8788 of Mathematical Reviews Vol. , Issue 2002M
[page]

2002
*
Mathematical Reviews
*

Let G,, be the class of those cylindric-

*relativized*set algebras of*dimension*n whose unit is a union of Cartesian spaces. I. ... (H-AOS; Budapest) A*finite*axiomatization of locally square cylindric-*relativized*set algebras. (English summary) Studia Sci. Math. Hungar. 38 (2001), 1-11. ...##
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Finite Self-Information

2012
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Computability - The Journal of the Assosiation
*

We answer a question of Levin by showing that the converse

doi:10.3233/com-2012-003
dblp:journals/computability/HirschfeldtW12
fatcat:jyocltznffefdlath6xc7micmm
*does**not*hold. ... In particular, we show that our proof can be adapted to produce a set that is low for both effective Hausdorff*dimension*and effective packing*dimension*, but*not*K-trivial. ... Since the*relativized*complexity K A (σ) may be thought of as measuring what A*does**not*know about σ, we subtract it from I(σ : τ ), and similarly we subtract K B (τ ). ...##
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Algorithmic Fractal Dimensions in Geometric Measure Theory
[article]

2020
*
arXiv
*
pre-print

We survey these developments, with emphasis on connections with

arXiv:2007.14346v1
fatcat:qllrmblw7fgvfmc347ngxfsvny
*computable*functions on the reals, recent uses of algorithmic*dimensions*in proving new theorems in classical (non-algorithmic) fractal geometry ... The development of algorithmic fractal*dimensions*in this century has had many fruitful interactions with geometric measure theory, especially fractal geometry in Euclidean spaces. ... National Science Foundation research grants 1247051 and 1545028 and is based in part on lectures that he gave at the New Zealand Mathematical Research Institute Summer School on Mathematical Logic and*Computability*...##
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Page 3794 of Mathematical Reviews Vol. , Issue 95g
[page]

1995
*
Mathematical Reviews
*

dominated by the arithmetical functions, and A

*does**not**compute*a generic set. ... class of constructivizations of which has only*finitely*many pairwise nonautoequivalent mem- bers. (2) A new class of constructive models with effectively infinite algorithmic*dimension*is introduced; ...##
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The Point-to-Set Principle, the Continuum Hypothesis, and the Dimensions of Hamel Bases
[article]

2021
*
arXiv
*
pre-print

The statement of our theorem is classical; it

arXiv:2109.10981v2
fatcat:4fr2h2zsy5akbeidnz6efjfrfq
*does**not*involve the theory of*computing*. ... However, our proof makes essential use of algorithmic fractal*dimension*--a*computability*-theoretic construct--and the point-to-set principle of J. Lutz and N. Lutz (2018). ... Introduction This brief paper is an intellectual export from the theory of*computing*to classical mathematics, i.e., mathematics*not*ostensibly involving the theory of*computing*. ...##
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Lowness for effective Hausdorff dimension

2014
*
Journal of Mathematical Logic
*

We examine the sequences A that are low for

doi:10.1142/s0219061314500111
fatcat:yqpp7nx6urhm3gj7xcxs542noa
*dimension*, i.e., those for which the effective (Hausdorff)*dimension*relative to A is the same as the unrelativized effective*dimension*. ... We show that there is a perfect Π 0 1 -class of low for*dimension*sequences. Since there are only countably many low for random sequences, many more sequences are low for*dimension*. ... Question 7. 3 . 3*Does*lowness for packing*dimension*imply lowness for*dimension*?*Finite*self-information. ...##
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The Typical Constructible Object
[chapter]

2016
*
Lecture Notes in Computer Science
*

Hence if A is 1-generic, A

doi:10.1007/978-3-319-40189-8_12
fatcat:fhm7jydns5ffnnehgidcv5yzn4
*does**not*belong to the boundary of U and as A 1 is easily co-infinite, M*does**not**compute*A 1 relative to A 0 . ...*finite*then it is*computable*so as before {A} is meager as its complement is a dense effective open set, (ii) if A is infinite then the class U = {B ⊆ N : A B} is a dense effective open set that*does**not*...
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