High and low Kleene degrees of coanalytic sets

1983 ◽  
Vol 48 (2) ◽  
pp. 356-368 ◽  
Author(s):  
Stephen G. Simpson ◽  
Galen Weitkamp
Keyword(s):  
Characteristic Function ◽  
Cross Section ◽  
Finite Time ◽  
Infinite Sequence ◽  
Turing Machines ◽  
Jump Operator ◽  
Halting Problem ◽  
Computing Machine ◽  
Definition Of ◽  
Countably Infinite

We say that a set A of reals is recursive in a real y together with a set B of reals if one can imagine a computing machine with an ability to perform a countably infinite sequence of program steps in finite time and with oracles for B and y so that decides membership in A for any real x input to by way of an oracle for x. We write A ≤ yB. A precise definition of this notion of recursion was first considered in Kleene [9]. In the notation of that paper, A ≤yB if there is an integer e so that χA(x) = {e}(x y, χB, 2E). Here χA is the characteristic function of A. Thus Kleene would say that A is recursive in (y, B, 2E), where 2E is the existential integer quantifier.Gandy [5] observes that the halting problem for infinitary machines such as , as in the case of Turing machines, gives rise to a jump operator for higher type recursion. Thus given a set B of reals, the superjump B′ of B is defined to be the set of all triples 〈e, x, y〉 such that the eth machine with oracles for y and B eventually halts when given input x. A set A is said to be semirecursive in y together with B if for some integer e, A is the cross section {x: 〈e, x, y 〉 ∈ B′}. In Kleene [9] it is demonstrated that a set A is semirecursive in y alone if and only if it is

Generalized Computational Systems

1997 ◽  
Author(s):  
Marco Giunti
Keyword(s):  
Turing Machine ◽  
Turing Machines ◽  
Computational System ◽  
Turing Computability ◽  
System A ◽  
Infinite Array ◽  
Definition Of ◽  
Countably Infinite ◽  
Computational Systems

The definition of a computational system that I proposed in chapter 1 (definition 3) employs the concept of Turing computability. In this chapter, however, I will show that this concept is not absolute, but instead depends on the relational structure of the support on which Turing machines operate. Ordinary Turing machines operate on a linear tape divided into a countably infinite number of adjacent squares. But one can also think of Turing machines that operate on different supports. For example, we can let a Turing machine work on an infinite checkerboard or, more generally, on some n-dimensional infinite array. I call an arbitrary support on which a Turing machine can operate a pattern field. Depending on the pattern field F we choose, we in fact obtain different concepts of computability. At the end of this chapter (section 6), I will thus propose a new definition of a computational system (a computational system on pattern field F) that takes into account the relativity of the concept of Turing computability. If F is a doubly infinite tape, however, computational systems on F reduce to computational systems. Turing (1965) presented his machines as an idealization of a human being that transforms symbols by means of a specified set of rules. Turing based his analysis on four hypotheses: 1. The capacity to recognize, transform, and memorize symbols and rules is finite. It thus follows that any transformation of a complex symbol must always be reduced to a series of simpler transformations. These operations on elementary symbols are of three types: recognizing a symbol, replacing a symbol, and shifting the attention to a symbol that is contiguous to the symbol which has been considered earlier. 2. The series of elementary operations that are in fact executed is determined by three factors: first, the subject’s mental state at a given time; second, the symbol which the subject considers at that time; third, a rule chosen from a finite number of alternatives.

scholarly journals A Mathematical Model of Quantum Computer by Both Arithmetic and Set Theory

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Mathematical Model ◽  
Set Theory ◽  
Quantum Computer ◽  
Turing Machines ◽  
Halting Problem ◽  
Actual Infinity ◽  
Formal Relation ◽  
Definition Of ◽  
The Mathematical Model ◽  
Gödel Incompleteness Theorems

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I.Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them.Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines.The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV.Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations.

scholarly journals Structure of the Balmer jump

2018 ◽  
Vol 613 ◽  
pp. A55
Author(s):  
F. Calvo ◽  
L. Belluzzi ◽  
O. Steiner
Keyword(s):  
Hydrogen Atom ◽  
Absorption Coefficient ◽  
Total Cross Section ◽  
Analytical Model ◽  
Cross Section ◽  
Quantum Structure ◽  
Balmer Series ◽  
Polarization Phenomena ◽  
Definition Of ◽  
Rapid Drop

Context.The spectrum of the hydrogen atom was explained by Bohr more than one century ago. We revisit here some of the aspects of the underlying quantum structure, with a modern formalism, focusing on the limit of the Balmer series.Aims.We investigate the behaviour of the absorption coefficient of the isolated hydrogen atom in the neighbourhood of the Balmer limit.Methods.We analytically computed the total cross-section arising from bound-bound and bound-free transitions in the isolated hydrogen atom at the Balmer limit, and established a simplified semi-analytical model for the surroundings of that limit. We worked within the framework of the formalism of Landi Degl’Innocenti & Landolfi (2004, Astrophys. Space Sci. Lib., 307), which permits an almost straight-forward generalization of our results to other atoms and molecules, and which is perfectly suitable for including polarization phenomena in the problem.Results.We analytically show that there is no discontinuity at the Balmer limit, even though the concept of a “Balmer jump” is still meaningful. Furthermore, we give a possible definition of the location of the Balmer jump, and we check that this location is dependent on the broadening mechanisms. At the Balmer limit, we compute the cross-section in a fully analytical way.Conclusions.The Balmer jump is produced by a rapid drop of the total Balmer cross-section, yet this variation is smooth and continuous when both bound-bound and bound-free processes are taken into account, and its shape and location is dependent on the broadening mechanisms.

scholarly journals A SOUNDNESS ANALYSIS OF ZENO’S OF ELEA DICHOTOMY

2021 ◽  
pp. 27-42
Author(s):  
Igor Berestov
Keyword(s):  
Finite Time ◽  
Infinite Sequence ◽  
Sequence Of Movements

We are studying three basic interpretations of the Dichotomy aporia, in which Zeno tries to prove the impossibility of movement. In all these interpretations, the key assumption is the dubious statement about the impossibility of performing an infinite sequence of actions in a finite time. However, we show that in the two interpretations of the Dichotomy it is possible to get rid of the dubious key assumption, replacing it with the seemingly much more reliable assumption that covering the distance is representable as a sequence of displacements. Our approach is based on the thesis proved by P. Benacerraf that completing an infinite sequence of movements in an interpretation of the Dichotomy is not sufficient to arrive to the end of the distance.

scholarly journals Computational Completeness of Interaction Machines and Turing Machines

10.29007/39jj ◽  
2018 ◽  
Author(s):  
Peter Wegner ◽  
Eugene Eberbach ◽  
Mark Burgin
Keyword(s):  
Computer Science ◽  
Complete Theory ◽  
Simple Computation ◽  
Turing Machines ◽  
Complete Model ◽  
Models Of Computation ◽  
Computational Completeness ◽  
Definition Of

In the paper we prove in a new and simple way that Interactionmachines are more powerful than Turing machines. To do thatwe extend the definition of Interaction machines to multiple interactivecomponents, where each component may perform simple computation.The emerging expressiveness is due to the power of interaction and allowsto accept languages not accepted by Turing machines. The mainresult that Interaction machines can accept arbitrary languages over agiven alphabet sheds a new light to the power of interaction. Despite ofthat we do not claim that Interaction machines are complete. We claimthat a complete theory of computer science cannot exist and especially,Turing machines or Interaction machines cannot be a complete model ofcomputation. However complete models of computation may and shouldbe approximated indefinitely and our contribution presents one of suchattempts.

scholarly journals Recommendations for Reporting Ion Mobility Mass Spectrometry Measurements

2018 ◽  
Author(s):  
Valerie Gabelica ◽  
Alexandre A. Shvartsburg ◽  
Carlos Afonso ◽  
Perdita E. Barran ◽  
Justin L. P. Benesch ◽  
...  
Keyword(s):  
Mass Spectrometry ◽  
Cross Section ◽  
Ion Mobility ◽  
Collision Cross Section ◽  
Ion Mobility Mass Spectrometry ◽  
Gas Temperature ◽  
Mobility Data ◽  
Community Effort ◽  
Primary Standards ◽  
Definition Of

Here we present a guide on ion mobility mass spectrometry experiments, which covers both linear and nonlinear methods: what is measured, how the measurements are done, and how to report the results, including the uncertainties on mobility and collision cross section values. The guide aims to clarify some possibly confusing concepts, and the reporting recommendations should help researchers, authors and reviewers to contribute comprehensive reports, so that the ion mobility data can be reused more confidently. Starting from the concept of the definition of the measurand, we emphasize that (i) mobility values (K0) depend intrinsically on ion structure, the nature of the bath gas, temperature, and E/N, (ii) ion mobility does not measure surfaces directly, but collision cross section (CCS) values are derived from mobility values using a physical model, (iii) methods relying on calibration are empirical (and thus may provide method-dependent results) only if the gas nature, temperature or E/N cannot match those of the primary method. Our analysis highlights the urgency of a community effort towards establishing primary standards and reference materials for ion mobility, and provides recommendations to do so. <br><br><br>

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

scholarly journals What is the dynamical hypothesis?

1998 ◽  
Vol 21 (5) ◽  
pp. 633-634 ◽  
Author(s):  
Nick Chater ◽  
Ulrike Hahn
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Turing Machines ◽  
Definition Of ◽  
Dynamical Hypothesis

Van Gelder's specification of the dynamical hypothesis does not improve on previous notions. All three key attributes of dynamical systems apply to Turing machines and are hence too general. However, when a more restricted definition of a dynamical system is adopted, it becomes clear that the dynamical hypothesis is too underspecified to constitute an interesting cognitive claim.

Geometric Indicators of Flexible Slice Lip

2015 ◽  
Vol 816 ◽  
pp. 188-194
Author(s):  
Krzysztof Talaśka ◽  
Ireneusz Malujda ◽  
Dominik Wilczyński ◽  
Jan Górecki ◽  
Paweł Tarkowski ◽  
...  
Keyword(s):  
Stress State ◽  
Cross Section ◽  
External Factors ◽  
Geometric Parameters ◽  
Geometric Features ◽  
Maximum Deflection ◽  
Paper Making ◽  
Definition Of ◽  
High Flexibility

The paper is thematically connected with paper making machines. This type of the machine is equipped with a head which has a flexible slice lip for skimming the pulp applied to produce the paper. The definition of geometric features of the slice lip is a key issue. The slice lip should be made of material which is resistant to external factors, and designers want to obtain a high flexibility of this element. It is possible to get a high flexibility by forming a proper shape of the slice lip. The paper presents the proposal of dimensionless geometric indicators which combine basic geometric parameters of the slice lip. These indicators are the following: indicator of the cross-section, indicator of the length and indicator of the length of the transverse notch. FEM strength analyses were done – the aim of these analyses was the determination of the stress state and maximum deflection of the slice lip. The authors have analysed the slice lip with holes for mounting the actuators and the slice lip and with transverse notches for increasing its flexibility. The results were presented in the form of graphs.

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