Oracle Machines and Relativized Recursion Theory

The digital elevation model (DEM) is known as one kind of the most significant fundamental geographical data models. The theory, method and application of DEM are hot research issues in geography, especially in geomorphology, hydrology, soil and other related fields. In this paper, we improve the efficient sub-pixel convolutional neural networks (ESPCN) and propose recursive sub-pixel convolutional neural networks (RSPCN) to generate higher-resolution DEMs (HRDEMs) from low-resolution DEMs (LRDEMs). Firstly, the structure of RSPCN is described in detail based on recursion theory. This paper explores the effects of different training datasets, with the self-adaptive learning rate Adam algorithm optimizing the model. Furthermore, the adding-“zero” boundary method is introduced into the RSPCN algorithm as a data preprocessing method, which improves the RSPCN method’s accuracy and convergence. Extensive experiments are conducted to train the method till optimality. Finally, comparisons are made with other traditional interpolation methods, such as bicubic, nearest-neighbor and bilinear methods. The results show that our method has obvious improvements in both accuracy and robustness and further illustrate the feasibility of deep learning methods in the DEM data processing area.

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FINDING THE LIMIT OF INCOMPLETENESS I

Bulletin of Symbolic Logic ◽

10.1017/bsl.2020.9 ◽

2020 ◽

Vol 26 (3-4) ◽

pp. 268-286

Author(s):

YONG CHENG

Keyword(s):

Model Theory ◽

Recursion Theory ◽

Minimal Degree ◽

Incompleteness Theorem ◽

Turing Degree ◽

Turing Reducibility ◽

Inseparable Pair

AbstractIn this paper, we examine the limit of applicability of Gödel’s first incompleteness theorem ($\textsf {G1}$ for short). We first define the notion “$\textsf {G1}$ holds for the theory $T$”. This paper is motivated by the following question: can we find a theory with a minimal degree of interpretation for which $\textsf {G1}$ holds. To approach this question, we first examine the following question: is there a theory T such that Robinson’s $\mathbf {R}$ interprets T but T does not interpret $\mathbf {R}$ (i.e., T is weaker than $\mathbf {R}$ w.r.t. interpretation) and $\textsf {G1}$ holds for T? In this paper, we show that there are many such theories based on Jeřábek’s work using some model theory. We prove that for each recursively inseparable pair $\langle A,B\rangle $, we can construct a r.e. theory $U_{\langle A,B\rangle }$ such that $U_{\langle A,B\rangle }$ is weaker than $\mathbf {R}$ w.r.t. interpretation and $\textsf {G1}$ holds for $U_{\langle A,B\rangle }$. As a corollary, we answer a question from Albert Visser. Moreover, we prove that for any Turing degree $\mathbf {0}< \mathbf {d}<\mathbf {0}^{\prime }$, there is a theory T with Turing degree $\mathbf {d}$ such that $\textsf {G1}$ holds for T and T is weaker than $\mathbf {R}$ w.r.t. Turing reducibility. As a corollary, based on Shoenfield’s work using some recursion theory, we show that there is no theory with a minimal degree of Turing reducibility for which $\textsf {G1}$ holds.

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α-degrees of α-theories

Journal of Symbolic Logic ◽

10.2307/2272413 ◽

1972 ◽

Vol 37 (4) ◽

pp. 677-682 ◽

Cited By ~ 1

Author(s):

George Metakides

Keyword(s):

Initial Segment ◽

Recursion Theory ◽

The Other ◽

Infinite Length ◽

Infinitary Logic ◽

Admissible Set ◽

Recursively Enumerable ◽

Limit Ordinal ◽

The Subject ◽

Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

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Friedberg Numbering in Fragments of Peano Arithmetic and α-Recursion Theory

Journal of Symbolic Logic ◽

10.2178/jsl.7804060 ◽

2013 ◽

Vol 78 (4) ◽

pp. 1135-1163 ◽

Cited By ~ 1

Author(s):

Wei Li

Keyword(s):

Recursion Theory ◽

Peano Arithmetic ◽

Fragments Of Peano Arithmetic

AbstractIn this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchy Lα, where α is Σ1 admissible. We prove that(1) Over P− + BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and(2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.

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