The Foundations of Computability Theory

2020 ◽  
Author(s):  
Borut Robič
Keyword(s):  
Computability Theory

scholarly journals Difference sets and computability theory

1998 ◽  
Vol 93 (1-3) ◽  
pp. 63-72
Author(s):  
Rod Downey ◽  
Zoltán Füredi ◽  
Carl G. Jockusch ◽  
Lee A. Rubel
Keyword(s):  
Difference Sets ◽  
Computability Theory

Classifying model-theoretic properties

2008 ◽  
Vol 73 (3) ◽  
pp. 885-905 ◽  
Author(s):  
Chris J. Conidis
Keyword(s):  
Model Theory ◽  
Computability Theory ◽  
Prime Model ◽  
Decidable Theory ◽  
Dense Sets ◽  
Equivalent Properties ◽  
The Relationship

AbstractIn 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

scholarly journals PERMUTATIONS OF THE INTEGERS INDUCE ONLY THE TRIVIAL AUTOMORPHISM OF THE TURING DEGREES

2018 ◽  
Vol 24 (2) ◽  
pp. 165-174
Author(s):  
BJØRN KJOS-HANSSEN
Keyword(s):  
Open Problem ◽  
Main Idea ◽  
Computability Theory ◽  
Turing Degrees ◽  
Nontrivial Automorphism

AbstractIs there a nontrivial automorphism of the Turing degrees? It is a major open problem of computability theory. Past results have limited how nontrivial automorphisms could possibly be. Here we consider instead how an automorphism might be induced by a function on reals, or even by a function on integers. We show that a permutation of ω cannot induce any nontrivial automorphism of the Turing degrees of members of 2ω, and in fact any permutation that induces the trivial automorphism must be computable.A main idea of the proof is to consider the members of 2ω to be probabilities, and use statistics: from random outcomes from a distribution we can compute that distribution, but not much more.

The Basic Concepts of Computability Theory

Computability ◽  
1987 ◽  
pp. 138-143
Author(s):  
Klaus Weihrauch
Keyword(s):  
Computability Theory ◽  
Basic Concepts

The First Riddle of Induction

2020 ◽  
pp. 237-253
Author(s):  
Justin Vlasits
Keyword(s):  
Inductive Reasoning ◽  
Computability Theory ◽  
Formal Learning ◽  
Problem Of Induction ◽  
Sextus Empiricus ◽  
And Topology ◽  
Famous Argument

What, exactly, is puzzling about induction? While the so-called problem of induction is normally introduced through David Hume’s famous argument, this essay shows how Sextus Empiricus gets to the heart of the matter. When properly understood, Sextus’ argument shows how the very power of inductive reasoning—its ability to move from particulars to universals—is at the same time what makes it “totter.” The argument has only been analyzed in any detail by the formal learning theorist Kevin Kelly, who uses the formal tools of computability theory and topology to mount a principled response. It is shown that this response depends on questionable assumptions and thus that they have not resolved Sextus’ riddle of induction.

scholarly journals Computational Complexity Theory and the Philosophy of Mathematics†

2019 ◽  
Vol 27 (3) ◽  
pp. 381-439
Author(s):  
Walter Dean
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Complexity Theory ◽  
Philosophy Of Mathematics ◽  
Computability Theory ◽  
Computational Complexity Theory ◽  
Mathematical Problems ◽  
Np Problem

Abstract Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof.

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