scholarly journals From Bi-Immunity to Absolute Undecidability

2009 ◽  
Vol 78 (4) ◽  
pp. 1218-1228 ◽  
Author(s):  
Laurent Bienvenu ◽  
Adam R. Day ◽  
Rupert Hölzl
Keyword(s):  
Coding Theory ◽  
Binary Sequence ◽  
Positive Answer ◽  
Truth Table ◽  
Turing Degree ◽  
Upper Density ◽  
Hadamard Codes

AbstractAn infinite binary sequence A is absolutely undecidable if it is impossible to compute A on a set of positions of positive upper density. Absolute undecidability is a weakening of bi-immunity. Downey, Jockusch and Schupp [2] asked whether, unlike the case for bi-immunity, there is an absolutely undecidable set in every non-zero Turing degree. We provide a positive answer to this question by applying techniques from coding theory. We show how to use Walsh–Hadamard codes to build a truth-table functional which maps any sequence A to a sequence B, such that given any restriction of B to a set of positive upper density, one can recover A. This implies that if A is non-computable, then B is absolutely undecidable. Using a forcing construction, we show that this result cannot be strengthened in any significant fashion.

scholarly journals Amenable equivalence relations and Turing degrees

1991 ◽  
Vol 56 (1) ◽  
pp. 182-194 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Linear Order ◽  
Order Type ◽  
Positive Answer ◽  
Turing Degree ◽  
Partial Answer ◽  
Equivalence Relations ◽  
Turing Degrees

In [12] Slaman and Steel posed the following problem:Assume ZF + DC + AD. Suppose we have a function assigning to each Turing degree d a linear order <d of d. Then must the rationals embed order preservingly in <d for a cone of d's?They had already obtained a partial answer to this question by showing that there is no such d ↦ <d with <d of order type ζ = ω* + ω on a cone. Already the possibility that <d has order type ζ · ζ was left open.We use here, ideas and methods associated with the concept of amenability (of groups, actions, equivalence relations, etc.) to prove some general results from which one can obtain a positive answer to the above problem.

Contiguity and distributivity in the enumerable Turing degrees — Corrigendum

2002 ◽  
Vol 67 (4) ◽  
pp. 1579-1580
Author(s):  
Rodney G. Downey ◽  
Steffen Lempp
Keyword(s):  
Truth Table ◽  
Turing Degree ◽  
Turing Degrees ◽  
Computably Enumerable Set ◽  
Enumerable Degree ◽  
Computably Enumerable Sets ◽  
Image Position ◽  
Computably Enumerable Degree ◽  
Computably Enumerable

A computably enumerable Turing degree a is called contiguous iff it contains only a single computably enumerable weak truth table degree (Ladner and Sasso [2]). In [1], the authors proved that a nonzero computably enumerable degree a is contiguous iff it is locally distributive, that is, for all a1, a2, c with a1 ∪a2 = a and c ≤ a, there exist ci, ≤ ai with c1 ∪ c2 = c.To do this we supposed that W was a computably enumerable set and ∪ a computably set with a Turing functional Φ such that ΦW = U. Then we constructed computably enumerable sets A0, A1 and B together with functionals Γ0, Γ1, Γ, and Δ so thatand so as to satisfy all the requirements below.That is, we built a degree-theoretical splitting A0, A1 of W and a set B ≤TW such that if we cannot beat all possible degree-theoretical splittings V0, V1 of B then we were able to witness the fact that U ≤WW (via Λ).After the proof it was observed that the set U of the proof (page 1222, paragraph 4) needed only to be Δ20. It was then claimed that a consequence to the proof was that every contiguous computably enumerable degree was, in fact, strongly contiguous, in the sense that all (not necessarily computably enumerable) sets of the degree had the same weak truth table degree.

Undecidability and initial segments of the (r.e.) tt-degrees

1990 ◽  
Vol 55 (3) ◽  
pp. 987-1006 ◽  
Author(s):  
Christine Ann Haught ◽  
Richard A. Shore
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Truth Table ◽  
Turing Degree ◽  
General Notion ◽  
Basic Information ◽  
Recursive Procedure ◽  
Tabular Type

A notion of reducibility ≤r between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is r-reducible to a set B, A ≤rB, if one of the procedures applied to B gives A. Associated with any such reducibility notion is the structure of r-degrees, the equivalence classes of sets with respect to this reducibility, with the induced ordering. The most general notion of a computable reducibility is that of Turing, ≤T. Here we say that A ≤TB if there is a Turing machine φe which, when equipped with an oracle for B, computes A: φeB = A. Such Turing degree computations are characterized by the phenomenon that only during the computation itself do we discover which questions about B need to be answered to compute A(x). In contrast, for nearly all other computable reducibilities the set of questions needed is given in advance by a recursive procedure. Perhaps the most common example of such a procedure is many-one reducibility, ≤m: A ≤mB if there is a recursive function f such that x ∈ A ⇔ f(x) ∈ B.Reducibilities with the property that the output, A(x), is determined by the answers that B gives to a set of questions calculated recursively from x are said to be of tabular type. The most general tabular reducibility is called truth-table reducibility, ≤tt. The procedures [e] associated with this reducibility are specified by a recursive function f (= {e}) which, for each x, gives a set of n questions about the oracle and, for each of the possible 2n sets of answers, gives the corresponding output. As usual this defines A ≤ttB as “there is an e such that [e]B = A”. It is with this notion of reducibility and the associated tt-degrees that we shall be concerned in this paper. Basic information on several such strong reducibilities can be found in Rogers [28]. For more information we recommend the survey articles by Odifreddi [25] and Degtev [2] as well as Odifreddi's book [26].

Mars: was There an Ancient Eden?

1997 ◽  
Vol 161 ◽  
pp. 203-218 ◽  
Author(s):  
Tobias C. Owen
Keyword(s):  
Positive Answer ◽  
Biogenic Elements ◽  
Habitable Zones ◽  
The Face ◽  
The Origin Of Life ◽  
Early Mars ◽  
Favorable Conditions ◽  
Open Question ◽  
Rocky Planets ◽  
Impact Erosion

AbstractThe clear evidence of water erosion on the surface of Mars suggests an early climate much more clement than the present one. Using a model for the origin of inner planet atmospheres by icy planetesimal impact, it is possible to reconstruct the original volatile inventory on Mars, starting from the thin atmosphere we observe today. Evidence for cometary impact can be found in the present abundances and isotope ratios of gases in the atmosphere and in SNC meteorites. If we invoke impact erosion to account for the present excess of129Xe, we predict an early inventory equivalent to at least 7.5 bars of CO2. This reservoir of volatiles is adequate to produce a substantial greenhouse effect, provided there is some small addition of SO2(volcanoes) or reduced gases (cometary impact). Thus it seems likely that conditions on early Mars were suitable for the origin of life – biogenic elements and liquid water were present at favorable conditions of pressure and temperature. Whether life began on Mars remains an open question, receiving hints of a positive answer from recent work on one of the Martian meteorites. The implications for habitable zones around other stars include the need to have rocky planets with sufficient mass to preserve atmospheres in the face of intensive early bombardment.

Coding Theory

2004 ◽  
Author(s):  
San Ling ◽  
Chaoping Xing
Keyword(s):  
Coding Theory

The sense of presence in virtual environments:

2003 ◽  
Vol 15 (2) ◽  
pp. 69-71 ◽  
Author(s):  
Thomas W. Schubert
Keyword(s):  
Virtual Environments ◽  
Coding Theory ◽  
Model Building ◽  
Self Report ◽  
Survey Study ◽  
Spatial Presence ◽  
Component Structure ◽  
Sense Of Presence ◽  
Being There ◽  
Measure Sense

Abstract. The sense of presence is the feeling of being there in a virtual environment. A three-component self report scale to measure sense of presence is described, the components being sense of spatial presence, involvement, and realness. This three-component structure was developed in a survey study with players of 3D games (N = 246) and replicated in a second survey study (N = 296); studies using the scale for measuring the effects of interaction on presence provide evidence for validity. The findings are explained by the Potential Action Coding Theory of presence, which assumes that presence develops from mental model building and suppression of the real environment.

Efficient Steganography Algorithm Based On DCT And Entropy Thresholding Technique

2018 ◽  
Vol 8 (1) ◽  
pp. 45
Author(s):  
Satvir Singh
Keyword(s):  
Relative Entropy ◽  
Experimental Work ◽  
Random Function ◽  
Signal To Noise Ratio ◽  
Binary Sequence ◽  
Threshold Value ◽  
Absolute Difference ◽  
Secret Message ◽  
Signal To Noise ◽  
Confidential Information

Steganography is the special art of hidding important and confidential information in appropriate multimedia carrier. It also restrict the detection of  hidden messages. In this paper we proposes steganographic method based on dct and entropy thresholding technique. The steganographic algorithm uses random function in order to select block of the image where the elements of the binary sequence of a secret message will be inserted. Insertion takes place at the lower frequency  AC coefficients of the  block. Before we insert the secret  message. Image under goes dc transformations after insertion of the secret message we apply inverse dc transformations. Secret message will only be inserted into a particular block if  entropy value of that particular block is greater then threshold value of the entropy and if block is selected by the random function. In  Experimental work we calculated the peak signal to noise ratio(PSNR), Absolute difference , Relative entropy. Proposed algorithm give high value of PSNR  and low value of Absolute difference which clearly indicate level of distortion in image due to insertion of secret message is reduced. Also value of  relative entropy is close to zero which clearly indicate proposed algorithm is sufficiently secure. 

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