Stephen C. Kleene. Origins of recursive function theory. Annals of the history of computing, vol. 3 (1981), pp. 52– 67. - Martin Davis. Why Gödel didn't have Church's thesis. Information and control, vol. 54 (1982), pp. 3– 24. - Stephen C. Kleene. Reflections on Church's thesis. Notre Dame journal of formal logic, vol. 28 (1987), pp. 490– 498.

1990 ◽  
Vol 55 (1) ◽  
pp. 348-350 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Recursive Function ◽  
Function Theory ◽  
Formal Logic ◽  
Church’S Thesis ◽  
History Of Computing ◽  
History Of ◽  
Church's Thesis ◽  
And Control ◽  
Martin Davis

scholarly journals Computability and Recursion

1996 ◽  
Vol 2 (3) ◽  
pp. 284-321 ◽  
Author(s):  
Robert I. Soare
Keyword(s):  
Recursive Function ◽  
Conceptual Analysis ◽  
Function Theory ◽  
Computability Theory ◽  
Turing Machines ◽  
Church’S Thesis ◽  
The Future ◽  
Church's Thesis ◽  
The Subject

AbstractWe 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, and its connection to other related areas.After a careful historical and conceptual analysis of computability and recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of “computability” and “recursion.” Specifically we recommend that: the term “recursive” should no longer carry the additional meaning of “computable” or “decidable;” functions defined using Turing machines, register machines, or their variants should be called “computable” rather than “recursive;” we should distinguish the intensional difference between Church's Thesis and Turing's Thesis, and use the latter particularly in dealing with mechanistic questions; the name of the subject should be “Computability Theory” or simply Computability rather than “Recursive Function Theory.”

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.

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 RISK FACTORS OF MOLAR PREGNANCY

2017 ◽  
pp. 53-58
Author(s):  
Lam Huong Le
Keyword(s):  
High Risk ◽  
Gestational Trophoblastic Disease ◽  
Living Standard ◽  
Control Group ◽  
Uterine Perforation ◽  
Molar Pregnancy ◽  
Health It ◽  
Pregnancy Risk ◽  
History Of ◽  
And Control

Objectives: Molar pregnancy is the gestational trophoblastic disease and impact on the women’s health. It has several complications such as toxicity, infection, bleeding. Molar pregnancy also has high risk of choriocarcinoma which can be dead. Aim: To assess the risks of molar pregnancy. Materials and Methods: The case control study included 76 molar pregnancies and 228 pregnancies in control group at Hue Central Hospital. Results: The average age was 32.7 ± 6.7, the miximum age was 17 years old and the maximum was 46 years old. The history of abortion, miscarriage in molar group and control group acounted for 10.5% and 3.9% respectively, with the risk was higher 2.8 times; 95% CI = 1.1-7.7 (p<0.05). The history of molar pregnancy in molar pregnancy group was 9.2% and the molar pregnancy risk was 11.4 times higher than control group (95% CI = 2.3-56.4). The women having ≥ 4 times births accounted for 7.9% in molar group and 2.2% in control group, with the risk was higher 3.8 times, 95% CI= 1.1-12.9 (p<0.05). The molar risk of women < 20 and >40 years old in molar groups had 2.4 times higher than (95% CI = 1.1 to 5.2)h than control group. Low living standard was 7.9% in molar group and 1.3% in the control group with OR= 6.2; 95% CI= 1.5-25.6. Curettage twice accounted for 87.5%, there were 16 case need to curettage three times. There was no case of uterine perforation and infection after curettage. Conclusion: The high risk molar pregnancy women need a better management. Pregnant women should be antenatal cared regularly to dectect early molar pregnancy. It is nessecery to monitor and avoid the dangerous complications occuring during the pregnancy. Key words: Molar pregnancy, pregnancy women

Work and Technological Change

2020 ◽  
Author(s):  
Stephen R. Barley
Keyword(s):  
Artificial Intelligence ◽  
Historical Context ◽  
Energy Transformation ◽  
History Of ◽  
Work And Organizations ◽  
Role Based ◽  
Control Technologies ◽  
And Control ◽  
Work And Employment

The four chapters of this book summarize the results of thirty-five years dedicated to studying how technologies change work and organizations. The first chapter places current developments in artificial intelligence into the historical context of previous technological revolutions by drawing on William Faunce’s argument that the history of technology is one of progressive automation of the four components of any production system: energy, transformation, and transfer and control technologies. The second chapter lays out a role-based theory of how technologies occasion changes in organizations. The third chapter tackles the issue of how to conceptualize a more thorough approach to assessing how intelligent technologies, such as artificial intelligence, can shape work and employment. The fourth chapter discusses what has been learned over the years about the fears that arise when one sets out to study technical work and technical workers and methods for controlling those fears.

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