scholarly journals An incomplete set of shortest descriptions

2012 ◽  
Vol 77 (1) ◽  
pp. 291-307 ◽  
Author(s):  
Frank Stephan ◽  
Jason Teutsch
Keyword(s):  
Recursion Theory ◽  
Truth Table ◽  
Outstanding Problem ◽  
Free Constructions

AbstractThe truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree by identifying an acceptable set of domain-random strings within each degree.

Degree theoretic definitions of the low2 recursively enumerable sets

1995 ◽  
Vol 60 (3) ◽  
pp. 727-756 ◽  
Author(s):  
Rod Downey ◽  
Richard A. Shore
Keyword(s):  
Recursion Theory ◽  
Truth Table ◽  
Major Theme ◽  
Syntactic Complexity ◽  
Access To Information ◽  
Turing Machines ◽  
Recursively Enumerable ◽  
Rates Of Growth ◽  
The Given ◽  
Effective Function

The primary relation studied in recursion theory is that of relative complexity: A set or function A (of natural numbers) is reducible to one B if, given access to information about B, we can compute A. The primary reducibility is that of Turing, A ≤TB, where arbitrary (Turing) machines, φe, can be used; access to information about (the oracle) B is unlimited and the lengths of computations are potentially unbounded. Many other interesting reducibilities result from restricitng one or more of these facets of the procedure. Thus, for example, the strongest notion considered is one-one reducibility on sets: A ≤1B iff there is a one-one recursive (= effective) function f such that x Є A ⇔ f(x) Є B. Many-one (≤m) reducibility simply allows f to be many-one. Other intermediate reducibilities include truth-table (≤tt) and weak truth-table (≤wtt). The latter imposes a recursive bound f(x) on the information about B that can be used to compute A(x). The former also bounds the length of computations by requiring that the computation of A(x) from B halt in at most f(x) many steps.Each such reducibility r defines a notion of degree, degr(A) = {B : A ≤rB ∧ B ≤rA}, and a corresponding structure of the r-degrees ordered by r-reducibility. (We typically denote the degree of A by a.) A major theme in recursion theory has been the investigation of the relation between a set's place in these orderings (the algebraic properties of its degree) and other algorithmic, set-theoretic or definability type notions of complexity. Important examples of such other notions include rates of growth of functions, the types of approximation procedures which converge to the given function or set and the (syntactic) complexity of defining the set (or function) in arithmetic or analysis.

Structural analysis of a Σ5 grain boundary in YBa2Cu3O7δ

1993 ◽  
Vol 51 ◽  
pp. 942-943
Author(s):  
J.-Y. Wang ◽  
Y. Zhu ◽  
A.H. King ◽  
M. Suenaga
Keyword(s):  
Grain Boundary ◽  
Lattice Mismatch ◽  
Weak Link ◽  
Coincidence Site Lattice ◽  
Large Angle ◽  
Dislocation Core ◽  
Superconducting Order Parameter ◽  
Outstanding Problem ◽  
Transmission Electron

One outstanding problem in YBa2Cu3O7−δ superconductors is the weak link behavior of grain boundaries, especially boundaries with a large-angle misorientation. Increasing evidence shows that lattice mismatch at the boundaries contributes to variations in oxygen and cation concentrations at the boundaries, while the strain field surrounding a dislocation core at the boundary suppresses the superconducting order parameter. Thus, understanding the structure of the grain boundary and the grain boundary dislocations (which describe the topology of the boundary) is essential in elucidating the superconducting characteristics of boundaries. Here, we discuss our study of the structure of a Σ5 grain boundary by transmission electron microscopy. The characterization of the structure of the boundary was based on the coincidence site lattice (CSL) model.Fig.l shows two-beam images of the grain boundary near the projection. An array of grain boundary dislocations, with spacings of about 30nm, is clearly visible in Fig. 1(a), but invisible in Fig. 1(b).

Combinational Logic Analysis Using Laser Voltage Probing

2015 ◽  
Author(s):  
Venkat Krishnan Ravikumar ◽  
Winson Lua ◽  
Seah Yi Xuan ◽  
Gopinath Ranganathan ◽  
Angeline Phoa
Keyword(s):  
Failure Analysis ◽  
Continuous Wave ◽  
Truth Table ◽  
Combinational Logic ◽  
Infra Red ◽  
Analysis Design ◽  
Logic State

Abstract Laser Voltage Probing (LVP) using continuous-wave near infra-red lasers are popular for failure analysis, design and test debug. LVP waveforms provide information on the logic state of the circuitry. This paper aims to explain the waveforms observed from combinational circuitries and use it to rebuild the truth table.

scholarly journals RSPCN: Super-Resolution of Digital Elevation Model Based on Recursive Sub-Pixel Convolutional Neural Networks

2021 ◽  
Vol 10 (8) ◽  
pp. 501
Author(s):  
Ruichen Zhang ◽  
Shaofeng Bian ◽  
Houpu Li
Keyword(s):  
Neural Networks ◽  
Digital Elevation Model ◽  
Convolutional Neural Networks ◽  
Adaptive Learning ◽  
Nearest Neighbor ◽  
Super Resolution ◽  
Recursion Theory ◽  
Research Issues ◽  
Digital Elevation ◽  
Elevation Model

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.

scholarly journals FINDING THE LIMIT OF INCOMPLETENESS I

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.

scholarly journals Π10 classes and strong degree spectra of relations

2007 ◽  
Vol 72 (3) ◽  
pp. 1003-1018 ◽  
Author(s):  
John Chisholm ◽  
Jennifer Chubb ◽  
Valentina S. Harizanov ◽  
Denis R. Hirschfeldt ◽  
Carl G. Jockusch ◽  
...  
Keyword(s):  
Kolmogorov Complexity ◽  
Initial Segment ◽  
Truth Table ◽  
Linear Ordering ◽  
Computable Structures ◽  
Degree Spectra

AbstractWe study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable subsets of 2ω and Kolmogorov complexity play a major role in the proof.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
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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