Computability and Recursion

  title={Computability and Recursion},
  author={Robert Irving Soare},
  journal={Bulletin of Symbolic Logic},
  pages={284 - 321}
  • R. Soare
  • Published 1 September 1996
  • Computer Science
  • Bulletin of Symbolic Logic
Abstract We consider the informal concept of “computability” or “effective calculability” and two of the formalisms commonly used to define it, “(Turing) computability” and “(general) recursiveness”. We consider their origin, exact technical definition, concepts, history, general English meanings, how they became fixed in their present roles, how they were first and are now used, their impact on nonspecialists, how their use will affect the future content of the subject of computability theory… 
The History and Concept of Computability
  • R. Soare
  • Philosophy
    Handbook of Computability Theory
  • 1999
Implicit and Explicit Examples of the Phenomenon of Deviant Encodings
  • Paula Quinon
  • Philosophy, Computer Science
    Studies in Logic, Grammar and Rhetoric
  • 2020
The idea that Carnapian explications provide a much more adequate framework for understanding the concept of computation, than the classical philosophical analysis is developed.
Gödel on Turing on Computability 1. Introduction
In section 9 of his paper, “On Computable Numbers,” Alan Turing outlined an argument for his version of what is now known as the Church–Turing thesis: “the [Turing machine] ‘computable’ numbers
Contents Incomputability, Turing Functionals, and Open Computing
One of the most important concepts in all of computability, Turing’s notion 1939 §4 of an oracle machine (o-machine) defines the computability of one set B relative to another set A which during the computation can be interrogated as a oracle (database).
Computation, hypercomputation, and physical science
Mathematical and Technological Computability
This chapter begins with a brief history of computations and the technical means used to support them, and a discussion of the Turing machine, including a detailed explanation of why it can be said to cover all systems of rule-bound symbol manipulation.
The Representational Foundations of Computation
Computability theory studies notations for various non-linguistic domains and illuminates how different ways of representing a domain support different finite mechanical procedures over that domain.
Are Gandy Machines Really Local
A precise definition of the realization of a Turing-computable algorithm into a physical situation is given and Gandy machines, intended in a physical sense, are analysed as a case study and an inaccuracy in Gandy’s analysis with respect to the locality notion is shown, showing the epistemological relevance of this realization concept.
Turing and the discovery of computability
The sections from §2—§9 challenge the conventional wisdom and traditional ideas found in many books and papers on computability theory and are based on a half century of study of the subject beginning with Church at Princeton in the 1960s and on a careful rethinking of these traditional ideas.


Origins of Recursive Function Theory
  • S. Kleene
  • Mathematics
    Annals of the History of Computing
  • 1981
The notion of "?-definability" was the first of what are now accepted as equivalent exact mathematical descriptions of the class of the functions for which algorithms exist and traces the investigation in 1931-1933 by which the notion was quite unexpectedly so accepted.
Computability and λ-Definability
  • A. Turing
  • Computer Science, Mathematics
    J. Symb. Log.
  • 1937
The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene.
K-graph machines : generalizing Turing's machines and arguments
K-graph machines are introduced and used to give a detailed mathematical explication of the first two aspects of Turing's considerations for general configurations, i.e. boundedness and locality conditions and mechanical operations, which provide a significant strengthening of Turing’s argument for his central thesis.
Post Emil L.. Finite combinatory processes—formulation 1. The journal of symbolic logic, vol. 1 (1936), pp. 103–105.
structions, can be regarded as a kind of Turing machine. I t is thus immediately clear that computability, so defined, can be identified with (especially, is no less general than) the notion of
Recursively enumerable sets of positive integers and their decision problems
Introduction. Recent developments of symbolic logic have considerable importance for mathematics both with respect to its philosophy and practice. That mathematicians generally are oblivious to the
The Upper Semi-Lattice of Degrees of Recursive Unsolvability
The concept 'degree of recursive unsolvability' was introduced briefly in Post [16]. In his abstract [17] the concept was formulated precisely via an extension of [15], and a resulting partial scale
Computability and logic
This book discusses Computability Theory, Modal logic and provability, and its applications to first-order logic, which aims to clarify and clarify the role of language in the development of computability.
Although the English mathematician Alan Mathison Turing (1912-1954) is remembered today primarily for his work in mathematical logic (Turing machines and the " Entscheidungsproblem"), machine
Church's Thesis and Principles for Mechanisms
Arithmetical predicates and function quantifiers
The extended class of number-theoretic predicates obtained by applying the two-sorted predicate calculus to recursive predicates of number variables and variables for one-place number-thoretic functions is considered, and it is shown that similarly these fall into a hierarchy according to the sequences of alternating function quantifiers by which they can be defined from arithmetical predicates.