Effectively calculable Rosser addresses the notion of "effective computability" as follows: Thus the adverb-adjective "effective" is used in a sense of "1a: In the following, the words "effectively calculable" will mean "produced by any intuitively 'effective' means whatsoever" and "effectively computable" will mean "produced by a Turing-machine or equivalent mechanical device". Turing's "definitions" are virtually the same:

In Church's original formulation Church, the thesis says that real-world calculation can be done using the lambda calculuswhich is equivalent to using general recursive functions. The Church-Turing thesis encompasses more kinds of computations than those originally envisioned, such as those involving cellular automatacombinatorsregister machinesand substitution systems.

It also applies to other kinds of computations found in theoretical computer science such as quantum computing and probabilistic computing.

There are conflicting points of view about the Church-Turing thesis. One says that it can be proven, and the other says that it serves as a definition for computation. There has never been a proof, but the evidence for its validity comes from the fact that every realistic model of computation, yet discovered, has been shown to be equivalent.

If there were a device which could answer questions beyond those that a Turing machine can answer, then it would be called an oracle.

Some computational models are more efficient, in terms of computation time and memory, for different tasks.

For example, it is suspected that quantum computers can perform many common tasks with lower time complexitycompared to modern computers, in the sense that for large enough versions of these problems, a quantum computer would solve the problem faster than an ordinary computer.

In contrast, there exist questions, such as the halting problemwhich an ordinary computer cannot answer, and according to the Church-Turing thesis, no other computational device can answer such a question. The Church-Turing thesis has been extended to a proposition about the processes in the natural world by Stephen Wolfram in his principle of computational equivalence Wolframwhich also claims that there are only a small number of intermediate levels of computing power before a system is universal and that most natural systems are universal.The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent computation involving a Turing machine.

In Church's original formulation (Church , ), the thesis says that . neither knew of the other’s work in tranceformingnlp.com published in the demonstrated equivalence of their formalisms strengthened both their claims to validity, expressed as the Church-Turing Thesis.

Turing’s Thesis Solomon Feferman 2NOTICES OF THE AMS VOLUME 53, NUMBER 10 I n the sole extended break from his life and var-ied career in England, Alan Turing spent the years – doing graduate work at. the Church-Turing thesis, as it emerged in when Church en-dorsed Turing’s characterization of the concept of eﬀective calcula-bility.

Jan 08, · When the Church-Turing thesis is expressed in terms of the replacement concept proposed by Turing, it is appropriate to refer to the thesis also as ‘Turing’s thesis’, and as ‘Church’s thesis’ when expressed in terms of one or another of the formal replacements proposed by Church. There are various equivalent formulations of the Turing-Church thesis (which is also known as Turing's thesis, Church's thesis, and the Church-Turing thesis). One formulation of the thesis is that every effective computation can be carried out by a Turing machine. What is the Church–Turing thesis?In , the English mathematician Alan Turing published a ground-breaking paper entitled “On computable numbers, with an application to the Entscheidungsproblem”.In this paper, Turing introduced the notion of an abstract model of computation as an idealisation of the practices and capabilities of a human .

(The article by Sieg in this volume details this history. It is valuable also to note from Krajewski, also in this volume, that the.

There are various equivalent formulations of the Turing-Church thesis (which is also known as Turing's thesis, Church's thesis, and the Church-Turing thesis).

One formulation of the thesis is that every effective computation can be carried out by a Turing machine. The history of the Church–Turing thesis ("thesis") involves the history of the development of the study of the nature of functions whose values are effectively calculable; or, in more modern terms, functions whose values are algorithmically computable.

It is an important topic in modern mathematical theory and computer science, particularly associated with the work of Alonzo Church .

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Church-Turing Thesis -- from Wolfram MathWorld