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In computability theory the Church-Turing thesis, Church's thesis, Church's conjecture or Turing's thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, like computers, and what kind of algorithms they can perform. It is generally assumed that an algorithm must satisfy the following requirements:
An example of such a method is the Euclidean algorithm for determining the greatest common divisor of two natural numbers. The notion of algorithm is intuitively clear but is not formally defined since it is not exactly clear what a "simple and precise instruction" is, and what exactly the "required intelligence to execute these instructions" is. (See for example effective results in number theory for cases well beyond the Euclidean algorithm.) Informally the thesis states that our notion of algorithm can be made precise (in the form of computable function) and computers can run those algorithms. Furthermore any computer can theoretically run any algorithm, that is the theoretic computational power of each computer is the same and it is not possible to build a calculation device which is more powerful than a computer. (Note that this formulation has implicit in it the idea that memory/storage is separate from device; any actual computer has finite memory, but the formulation always assumes that memory can be added at will.) The thesis may be regarded as a physical law as it cannot be mathematically proven.
Church-Turing thesisThe thesis, in Turing's own words, can be stated as:
Due to the vagueness of the concept of a "function which would naturally be regarded as computable", the thesis can neither be proven nor disproven formally. Any computer program can be translated into a Turing machine, and any Turing machine can be translated into any general-purpose programming language, so the thesis is equivalent to saying that any general-purpose programming language is sufficient to express any algorithm. Various variations of the thesis exist; for example, the Physical Church-Turing thesis (PCTT) states:
This stronger statement was proven false in 2002 when William Fouché discovered that a Turing machine cannot effectively approximate any of the values of one-dimensional Brownian motion at rational points in time almost surely (with respect to Wiener measure; see reference below) Another variation is the Strong Church-Turing thesis (SCTT), which states (cf. Bernstein, Vazirani 1997):
HistoryThe thesis is named after mathematicians Alonzo Church and Alan Turing. In his 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" Alan Turing tried to capture the notion of algorithm (then called "effective computability"), with the introduction of Turing machines. In that paper he showed that the 'Entscheidungsproblem' could not be solved. A few months earlier Alonzo Church had proven a similar result in "A Note on the Entscheidungsproblem" but he used the notions of recursive functions and Lambda-definable functions to formally describe effective computability. Lambda-definable functions were introduced by Alonzo Church and Stephen Kleene (Church 1932, 1936a, 1941, Kleene 1935) and recursive functions by Kurt Gödel and Jacques Herbrand (Gödel 1934, Herbrand 1932). These two formalisms describe the same set of functions, as was shown in the case of functions of positive integers by Church and Kleene (Church 1936a, Kleene 1936). When hearing of Church's proposal, Turing was quickly able to show that his Turing machines in fact describe the same set of functions (Turing 1936, 263ff). Success of the thesisSince that time many other formalisms for describing effective computability have been proposed, including recursive functions, the lambda calculus, register machines, Post systems, combinatory logic, and Markov algorithms. All these systems have been shown to compute essentially the same functions as Turing machines; systems like this are called Turing-complete. Because all these different attempts of formalizing the concept of algorithm have yielded equivalent results, it is now generally assumed that the Church-Turing thesis is correct. However, the thesis is a definition and not a theorem, and hence cannot be proven true. It could, however, be disproven if a method could be exhibited which is universally accepted as being an effective algorithm but which cannot be performed on a Turing machine. In the early twentieth century, mathematicians often used the informal phrase effectively computable, so it was important to find a good formalization of the concept. Modern mathematicians instead use the well-defined term Turing computable (or computable for short). Since the undefined terminology has faded from use, the question of how to define it is now less important. The success of the Church–Turing thesis prompted supertheses that extend the thesis, including the conjecture that there is a polynomial transformation from the representation of computable functions in one formalization to their representation in another, and the conjecture that every model of computation can be step-by-step simulated by a Turing machine. Philosophical implicationsThe Church-Turing thesis has some profound implications for the philosophy of mind. There are also some important open questions which cover the relationship between the Church-Turing thesis and physics, and the possibility of hypercomputation. When applied to physics, the thesis has several possible meanings:
There are actually many technical possibilities which fall outside or between these three categories, but these should serve to illustrate the concept. Additional readingReferences
External links
See also
de:Church-Turing-These es:Tesis de Church-Turing fr:Thèse de Church-Turing zh:邱奇-图灵论题 |
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