Step by Recursive Step: Church's Analysis of Effective Calculability

1997 ◽  
Vol 3 (2) ◽  
pp. 154-180 ◽  
Author(s):  
Wilfried Sieg
Keyword(s):  
Mathematical Work ◽  
Church’S Thesis ◽  
Alan Turing ◽  
Church's Thesis

AbstractAlonzo Church's mathematical work on computability and undecidability is well-known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was “Church's Thesis” put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's recursiveness for his Thesis. In Section 5, I contrast Church's analysis with that of Alan Turing and explore, in the very last section, an analogy with Dedekind's investigation of continuity.

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).

Church’s thesis

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Turing Computability ◽  
Church’S Thesis ◽  
The Third ◽  
Alan Turing ◽  
Rigorous Formulation ◽  
Person Following ◽  
Mechanical Procedure ◽  
Church's Thesis

An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A eventually produces fn as output. A function is ‘computable’ if there is an algorithm that computes it. A set S is ‘decidable’ if there is an algorithm that decides membership in S: if, given any appropriate n as input, the algorithm would output ‘yes’ if n∈S, and ‘no’ if n∉S. The notions of ‘algorithm’, ‘computable’ and ‘decidable’ are informal (or pre-formal) in that they have meaning independently of, and prior to, attempts at rigorous formulation. Church’s thesis, first proposed by Alonzo Church in a paper published in 1936, is the assertion that a function is computable if and only if it is recursive: ‘We now define the notion…of an effectively calculable function…by identifying it with the notion of a recursive function….’ Independently, Alan Turing argued that a function is computable if and only if there is a Turing machine that computes it; and he showed that a function is Turing-computable if and only if it is recursive. Church’s thesis is widely accepted today. Since an algorithm can be ‘read off’ a recursive derivation, every recursive function is computable. Three types of ‘evidence’ have been cited for the converse. First, every algorithm that has been examined has been shown to compute a recursive function. Second, Turing, Church and others provided analyses of the moves available to a person following a mechanical procedure, arguing that everything can be simulated by a Turing machine, a recursive derivation, and so on. The third consideration is ‘confluence’. Several different characterizations, developed more or less independently, have been shown to be coextensive, suggesting that all of them are on target. The list includes recursiveness, Turing computability, Herbrand–Gödel derivability, λ-definability and Markov algorithm computability.

Proving Church's Thesis

2000 ◽  
Vol 8 (3) ◽  
pp. 244-258 ◽  
Author(s):  
ROBERT BLACK
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

scholarly journals Reflections on Church's thesis.

1987 ◽  
Vol 28 (4) ◽  
pp. 490-498 ◽  
Author(s):  
Stephen C. Kleene
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis

Trial and error predicates and the solution to a problem of Mostowski

1965 ◽  
Vol 30 (1) ◽  
pp. 49-57 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Finite Number ◽  
Correct Answer ◽  
Decision Procedure ◽  
Finite Sequence ◽  
Decision Procedures ◽  
Turing Machines ◽  
Trial And Error ◽  
Church’S Thesis ◽  
Church's Thesis ◽  
Empirical Means

The purpose of this paper is to present two groups of results which have turned out to have a surprisingly close interconnection. The first two results (Theorems 1 and 2) were inspired by the following question: we know what sets are “decidable” — namely, the recursive sets (according to Church's Thesis). But what happens if we modify the notion of a decision procedure by (1) allowing the procedure to “change its mind” any finite number of times (in terms of Turing Machines: we visualize the machine as being given an integer (or an n-tuple of integers) as input. The machine then “prints out” a finite sequence of “yesses” and “nos”. The last “yes” or “no” is always to be the correct answer.); and (2) we give up the requirement that it be possible to tell (effectively) if the computation has terminated? I.e., if the machine has most recently printed “yes”, then we know that the integer put in as input must be in the set unless the machine is going to change its mind; but we have no procedure for telling whether the machine will change its mind or not.The sets for which there exist decision procedures in this widened sense are decidable by “empirical” means — for, if we always “posit” that the most recently generated answer is correct, we will make a finite number of mistakes, but we will eventually get the correct answer. (Note, however, that even if we have gotten to the correct answer (the end of the finite sequence) we are never sure that we have the correct answer.)

Epistemic Church’s Thesis and Absolute Undecidability

2016 ◽  
pp. 254-272
Author(s):  
Marianna Antonutti Marfori ◽  
Leon Horsten
Keyword(s):  
Church’S Thesis ◽  
Church's Thesis
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