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

2003 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.  ... 

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.  ... 

Physical Relativism as an Interpretation of Existence [article]

Stuart Heinrich
2013 arXiv   pre-print
In this manuscript, we propose a new interpretation of existence that we call physical 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.  ... 
arXiv:1306.5484v2 fatcat:nkratr2lorgevejq6irvqkuuqu

Computability and Randomness

Rod Downey, Denis R. Hirschfeldt
2019 Notices of the American Mathematical Society  
For certain well-behaved sets , 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.  ... 
doi:10.1090/noti1905 fatcat:pg66obggazgwpj6dwy2buq6oze

Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

Jack H. Lutz, Neil Lutz
2018 ACM Transactions on Computation Theory  
Other effectivizations, e.g., 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  ... 
doi:10.1145/3201783 fatcat:hkadgadcinbbxj5jlemteg7qiu

Unified characterizations of lowness properties via Kolmogorov complexity

Takayuki Kihara, Kenshi Miyabe
2014 Archive for Mathematical Logic  
2) We also show that X ∈ Low ⋆ (SR, WR) if and only if X is 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] .  ... 
doi:10.1007/s00153-014-0413-8 fatcat:udown43nmnhl5dtc2tlpyd2tv4

Uniform Kurtz randomness

T. Kihara, K. Miyabe
2013 Journal of Logic and Computation  
For instance, van Lambalgen's theorem holds for uniform Kurtz randomness while 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  ... 
doi:10.1093/logcom/ext054 fatcat:iw2nb2syw5aaxlbrsgykbmwvaq

Page 6352 of Mathematical Reviews Vol. , Issue 89K [page]

1989 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).  ... 

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.  ... 

Finite Self-Information

Denis R. Hirschfeldt, Rebecca Weber
2012 Computability - The Journal of the Assosiation  
We answer a question of Levin by showing that the converse 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 ).  ... 
doi:10.3233/com-2012-003 dblp:journals/computability/HirschfeldtW12 fatcat:jyocltznffefdlath6xc7micmm

Algorithmic Fractal Dimensions in Geometric Measure Theory [article]

Jack H. Lutz, Elvira Mayordomo
2020 arXiv   pre-print
We survey these developments, with emphasis on connections with 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  ... 
arXiv:2007.14346v1 fatcat:qllrmblw7fgvfmc347ngxfsvny

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;  ... 

The Point-to-Set Principle, the Continuum Hypothesis, and the Dimensions of Hamel Bases [article]

Jack H. Lutz
2021 arXiv   pre-print
The statement of our theorem is classical; it 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.  ... 
arXiv:2109.10981v2 fatcat:4fr2h2zsy5akbeidnz6efjfrfq

Lowness for effective Hausdorff dimension

Steffen Lempp, Joseph S. Miller, Keng Meng Ng, Daniel D. Turetsky, Rebecca Weber
2014 Journal of Mathematical Logic  
We examine the sequences A that are low for 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.  ... 
doi:10.1142/s0219061314500111 fatcat:yqpp7nx6urhm3gj7xcxs542noa

The Typical Constructible Object [chapter]

Mathieu Hoyrup
2016 Lecture Notes in Computer Science  
Hence if A is 1-generic, A 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  ... 
doi:10.1007/978-3-319-40189-8_12 fatcat:fhm7jydns5ffnnehgidcv5yzn4
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