scholarly journals Effectivity and reducibility with ordinal Turing machines

Computability ◽  
2021 ◽  
pp. 1-16
Author(s):  
Merlin Carl
Keyword(s):  
Classical Theory ◽  
Turing Machines ◽  
Turing Computability ◽  
Sample Application

This article expands our work in (LNCS 9709 (2016), 225–233). By its reliance on Turing computability, the classical theory of effectivity, along with effective reducibility and Weihrauch reducibility, is only applicable to objects that are either countable or can be encoded by countable objects. We propose a notion of effectivity based on Koepke’s Ordinal Turing Machines (OTMs) that applies to arbitrary set-theoretical Π 2 -statements, along with according variants of effective reducibility and Weihrauch reducibility. As a sample application, we compare various choice principles with respect to effectivity.

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.

Let all social partners in the social support process count: New perspectives on classical theory

2013 ◽  
Author(s):  
Liu-Qin Yang ◽  
Robert R. Wright ◽  
Liu-Qin Yang ◽  
Lisa M. Kath ◽  
Michael T. Ford ◽  
...  
Keyword(s):  
Social Support ◽  
Classical Theory ◽  
The Social ◽  
Social Partners

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Methods of the Classical Theory of Elastodynamics

1993 ◽  
Author(s):  
Vladimir B. Poruchikov
Keyword(s):  
Classical Theory

Some Assumptions about Problem Solving Representation in Turing’s Model of Intelligence

1970 ◽  
Vol 4 (2) ◽  
pp. 136-142 ◽  
Author(s):  
Raymundo Morado ◽  
Francisco Hernández-Quiroz
Keyword(s):  
Problem Solving ◽  
Computational Models ◽  
Turing Machines

Turing machines as a model of intelligence can be motivated under some assumptions, both mathematical and philosophical. Some of these are about the possibility, the necessity, and the limits of representing problem solving by mechanical means. The assumptions about representation that we consider in this paper are related to information representability and availability, processing as solving, nonessentiality of complexity issues, and finiteness, discreteness and sequentiality of the representation. We discuss these assumptions and particularly something that might happen if they were to be rejected or weakened. Tinkering with these assumptions sheds light on the import of alternative computational models.

The Essence of Quantum Computing

2018 ◽  
Author(s):  
Rajendra K. Bera
Keyword(s):  
Hilbert Space ◽  
Quantum Computing ◽  
Quantum Physics ◽  
Quantum Algorithms ◽  
Quantum Computers ◽  
Turing Machines ◽  
Classical Physics ◽  
Core Set ◽  
The World ◽  
Subordinate Role

It now appears that quantum computers are poised to enter the world of computing and establish its dominance, especially, in the cloud. Turing machines (classical computers) tied to the laws of classical physics will not vanish from our lives but begin to play a subordinate role to quantum computers tied to the enigmatic laws of quantum physics that deal with such non-intuitive phenomena as superposition, entanglement, collapse of the wave function, and teleportation, all occurring in Hilbert space. The aim of this 3-part paper is to introduce the readers to a core set of quantum algorithms based on the postulates of quantum mechanics, and reveal the amazing power of quantum computing.

Turing Machines Based on Quantum Logic and Their Universality

2012 ◽  
Vol 35 (7) ◽  
pp. 1407 ◽  
Author(s):  
Yong-Ming LI ◽  
Ping LI
Keyword(s):  
Quantum Logic ◽  
Turing Machines

A Neo-Classical Theory of Distribution and Wealth

1986 ◽  
Author(s):  
Hans Ulrich Buhl
Keyword(s):  
Classical Theory

Modeling Multitape Minsky and Turing Machines by Three-Tape Minsky Machines

2020 ◽  
Vol 46 (6) ◽  
pp. 428-432
Author(s):  
S. S. Marchenkov ◽  
S. D. Makeev
Keyword(s):  
Turing Machines
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