Soare proposes that the origination of "primitive recursion" began formally with the axioms of Peano, although "Well before the nineteenth century mathematicians used the principle of defining a function by induction. Based on this work of Dedekind, Peano and wrote the familiar five [sic] axioms for the positive integers. This leaves the five axioms that have become universally known as "the Peano axioms

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.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 computer, that is, a person who follows .

neither knew of the other’s work in regardbouddhiste.com published in the demonstrated equivalence of their formalisms strengthened both their claims to validity, expressed as the Church-Turing Thesis.

In computability theory the Church–Turing thesis (also known as Church's thesis, Church's conjecture and Turing's thesis) is a combined hypothesis about the nature of effectively calculable Church, A., , "An Unsolvable Problem of Elementary Number Theory", American Journal of .

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.

A Thesis and an Antithesis The origin of my article lies in the appearance of Copeland and Proudfoot's feature article in Scientific American, April This preposterous paper, as described on another page, suggested that Turing was the prophet of 'hypercomputation'.

In their references, the authors listed Copeland's entry on 'The Church-Turing thesis. 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.

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Church-Turing Thesis -- from Wolfram MathWorld