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Church turing thesis quantum computing

The 10th was to "devise a process according to which it can be determined by a finite number of operations whether a polynomial has an integral root".

As you might have noticed everything can be implemented by logic functions. This is important: To be able to do any logic function means to be Church-Turing mighty.

The Church–Turing thesis is a statement that characterizes the nature of computation and cannot be formally proven. Even though the three processes mentioned above proved to be equivalent, the fundamental premise behind the thesis—the notion of what it means for a function to be "effectively calculable" (computable)—is "a somewhat vague intuitive one". [ 3 ] Thus, the "thesis" remains a hypothesis. [ 3 ]

A well-known example of an effective method is the truth table test for tautologousness. In practice, this test is unworkable for formulae containing a large number of propositional variables, but in principle one could apply it successfully to any formula of the propositional calculus, given sufficient time, tenacity, paper, and pencils.

· The Church-Turing thesis (formerly commonly known simply as Church's thesis) says that any real-world computation can be translated into an equivalent ...

A well-known example of an effective method is the truth table test for tautologousness. In principle, a human being who works by rote could apply this test successfully to any formula of the propositional calculus—given sufficient time, tenacity, paper, and pencils (although the test is unworkable in practice for any formula containing more than a few propositional variables).

The Church-Turing thesis (also known as Church's thesis , Church's conjecture and Turing's thesis ) is a statement about computers . It says that a very simple kind of computer now named a “ Turing machine ” is able to compute all computable functions. The Church-Turing thesis is linked to Gödel's incompleteness theorems . When a programming language is able to do what a Turing machine can do, that language is called Turing complete . If a problem is solvable in one such language then it is solvable in all of those.

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 .

Observing the Church-Turing thesis one may find that the concept of algorithms has no mathematical definition. It implies that the statement has no mathematical proof, although it has a truth value. It is rather unusual, but only disproof may exist - if the thesis is wrong. To do this, it is enough to find a task, which can be solved by some method - regarded as an algorithm - and can be proven to be unsolvable by a Turing machine.

Copeland's argument comes to a crunch when he writes, near the end of the article, ... when Church writes (in a review of Post (1936)): To define effectiveness as computability by an arbitrary machine, subject to restrictions of finiteness, would seem to be an adequate representation of the ordinary notion (Church 1937b: 43), he is to be understood not as entertaining some form of thesis M but as endorsing the identification of the effectively calculable functions with those functions that can be calculated by an arbitrary machine whose principles of operation are such as to mimic the actions of a human computer.

Since its inception, variations on the original thesis have arisen, including statements about what can physically be realized by a computer in our universe ( physical Church-Turing thesis ) and what can be efficiently computed ( Church–Turing thesis (complexity theory) ). These variations are not due to Church or Turing, but arise from later work in complexity theory and digital physics . The thesis also has implications for the philosophy of mind (see below ).

In the following, the words " effectively [16] calculable " will mean "produced by any intuitively 'effective' means whatsoever" and " computable " will mean "produced by a Turing-machine or equivalent mechanical device".

In 1889, Giuseppe Peano presented his The principles of arithmetic, presented by a new method , based on the work of Dedekind . Soare proposes that the origination of "primitive recursion" began formally with the axioms of Peano, although

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The Church-Turing Thesis (Stanford Encyclopedia of Philosophy)

Church-Turing Thesis -- from Wolfram MathWorld

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