Church's thesis without tears

1983 ◽  
Vol 48 (3) ◽  
pp. 797-803 ◽  
Author(s):  
Fred Richman
Keyword(s):  
Function Theory ◽  
Computable Function ◽  
Modern Theory ◽  
Central Feature ◽  
Partial Functions ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Church's Thesis ◽  
The Church ◽  
Computable Functions

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

The Mathematical Work of S.C.Kleene

1995 ◽  
Vol 1 (1) ◽  
pp. 9-43 ◽  
Author(s):  
J.R. Shoenfield
Keyword(s):  
Recursion Theory ◽  
Mathematical Work ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Close Relationship ◽  
Church's Thesis ◽  
Mathematical Career ◽  
Definable Functions ◽  
Class Of Functions ◽  
Computable Functions

§1. The origins of recursion theory. In dedicating a book to Steve Kleene, I referred to him as the person who made recursion theory into a theory. Recursion theory was begun by Kleene's teacher at Princeton, Alonzo Church, who first defined the class of recursive functions; first maintained that this class was the class of computable functions (a claim which has come to be known as Church's Thesis); and first used this fact to solve negatively some classical problems on the existence of algorithms. However, it was Kleene who, in his thesis and in his subsequent attempts to convince himself of Church's Thesis, developed a general theory of the behavior of the recursive functions. He continued to develop this theory and extend it to new situations throughout his mathematical career. Indeed, all of the research which he did had a close relationship to recursive functions.Church's Thesis arose in an accidental way. In his investigations of a system of logic which he had invented, Church became interested in a class of functions which he called the λ-definable functions. Initially, Church knew that the successor function and the addition function were λ-definable, but not much else. During 1932, Kleene gradually showed1 that this class of functions was quite extensive; and these results became an important part of his thesis 1935a (completed in June of 1933).

What is Computable?

2019 ◽  
pp. 31-42
Author(s):  
Giuseppe Primiero
Keyword(s):  
Computable Function ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Recursive Definitions ◽  
Computable Functions

This chapter defines formally the notion of computable function through its inductive and recursive definitions. It covers the construction schemas to define total and partial computable functions. It concludes by explaining how recursive definitions are general and equivalent to other formulations, a result known as Church’s Thesis.

scholarly journals Elements of concrete algorithmics: computability and solvability

2020 ◽  
pp. 198-207
Author(s):  
O.I. Provotar ◽  
◽  
O.O. Provotar ◽  
Keyword(s):  
Church’S Thesis ◽  
Recursive Functions ◽  
Church's Thesis

An approach to proving the fundamental results of the theory of recursive functions using specific algorithms is consider. For this, the basic constructions of the algorithm are describing exactly and Church's thesis for more narrow classes of algorithmically computational functions is specified (concretized). Using this approach, the belonging of functions to classes of algorithmically computable is argued by the construction of the corresponding algorithms.

Gödel numberings of partial recursive functions

1958 ◽  
Vol 23 (3) ◽  
pp. 331-341 ◽  
Author(s):  
Hartley Rogers
Keyword(s):  
Mathematical Logic ◽  
Recursive Function ◽  
Function Theory ◽  
Recursive Structure ◽  
Partial Functions ◽  
Logical Calculus ◽  
Recursive Functions ◽  
Several Variables ◽  
Partial Recursive Functions ◽  
Made In

In § 1 we present conceptual material concerning the notion of a Gödel numbering of the partial recursive functions. § 2 presents a theorem about these concepts. § 3 gives several applications. The material in § 1 and § 2 grew out of attempts by the author to find routine solutions to some of the problems discussed in § 3. The author wishes to acknowledge his debt in § 2 to the fruitful methods of Myhill in [M] and to thank the referee for an abbreviated and improved version of the proof for Lemma 3 in § 2.In the literature of mathematical logic, “Gödel numbering” usually means an effective correspondence between integers and the well-formed formulas of some logical calculus. In recursive function theory, certain such associations between the non-negative integers and instructions for computing partial recursive functions have been fundamental. In the present paper we shall be concerned only with numberings of the latter, more special, sort. By numbers and integers we shall mean non-negative integers. Our notation is, in general, that of [K]. If ϕ and ψ are two partial functions, ϕ = ψ shall mean that (∀x)[ϕ(x)≃(ψx)], i.e., that ϕ and ψ are defined for the same arguments and are equal on those arguments. We consider partial recursive functions of one variable; applications of the paper to the case of several variables, or to the case of all partial recursive functions in any number of variables, can be made in the usual way using the coordinate functions (a)i of [K, p. 230]. It will furthermore be observed that we consider only concepts that are invariant with respect to general recursive functions; more limited notions of Gödel numbering, taking into account, say, primitive recursive structure, are beyond the scope of the present paper.

Can there be no nonrecursive functions?

1971 ◽  
Vol 36 (2) ◽  
pp. 309-315 ◽  
Author(s):  
Joan Rand Moschovakis
Keyword(s):  
Free Choice ◽  
Church’S Thesis ◽  
Recursive Functions ◽  
Choice Sequence ◽  
Church's Thesis ◽  
Theoretic Function ◽  
Ordinary Number ◽  
Definable Function

In 1936 Alonzo Church proposed the following thesis: Every effectively computable number-theoretic function is general recursive. The classical mathematician can easily give examples of nonrecursive functions, e.g. by diagonalizing a list of all general recursive functions. But since no such function has been found which is effectively computable, there is as yet no classical evidence against Church's Thesis.The intuitionistic mathematician, following Brouwer, recognizes at least two notions of function: the free-choice sequence (or ordinary number-theoretic function, thought of as the ever-finite but ever-extendable sequence of its values) and the sharp arrow (or effectively definable function, all of whose values can be specified in advance).

scholarly journals A Natural Axiomatization of Computability and Proof of Church's Thesis

2008 ◽  
Vol 14 (3) ◽  
pp. 299-350 ◽  
Author(s):  
Nachum Dershowitz ◽  
Yuri Gurevich
Keyword(s):  
Effective Means ◽  
State Machine ◽  
Church’S Thesis ◽  
Abstract State Machine ◽  
Additional Requirement ◽  
Classical Algorithm ◽  
Church's Thesis ◽  
Computable Functions

AbstractChurch's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and—in particular—the effectively-computable functions on string representations of numbers.

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