Information About ™Church-turing Thesis |
| CATEGORIES ABOUT CHURCH–TURING THESIS | |
| recursion theory | |
| alan turing | |
| theory of computation | |
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The Thesis claims that any calculation that is possible can be performed by an Algorithm running on a computer, provided that sufficient time and storage space are available. It is generally assumed that an algorithm must satisfy the following requirements: #The algorithm consists of a finite set of simple and precise instructions that are described with a finite number of symbols. #The algorithm will always produce the result in a finite number of steps. #The algorithm can in principle be carried out by a human being with only paper and pencil. #The execution of the algorithm requires no intelligence of the human being except that which is needed to understand and execute the instructions. The Euclidean Algorithm for determining the Greatest Common Divisor of two Natural Number s is an example of such an algorithm. This description of algorithm is intuitively clear but lacks formal rigor, 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 s) and computers can run those algorithms. Furthermore, a computer can theoretically run any algorithm; that is, all ordinary computers (that is, all Turing Machines ) are equivalent to each other in terms of theoretical computational power, and it is not possible to build a calculation device that is more powerful than a computer. (Note that this formulation of power disregards practical factors such as speed or memory capacity; it considers all that is theoretically possible, given unlimited time and memory.) The thesis may be regarded as a Physical Law or as a definition, as it has not been mathematically proven. Stephen Kleene considered it a definition (Kleene in Undecidable, p. 274). See History Section below. CHURCH–TURING THESIS The thesis can be stated as: "Every ' Function which would naturally be regarded as Computable ' can be computed by a Turing Machine ." Due to the vagueness of the concept of a "function which would naturally be regarded as computable", the thesis cannot formally be proven. Disproof would be possible only if humanity found ways of building Hypercomputer s whose results should "naturally be regarded as computable". 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: "Every function that can be physically computed can be computed by a Turing machine." This stronger statement may have been proven false in 2002 when Willem Fouché discovered that a Turing machine probably cannot effectively approximate any of the values of one-dimensional Brownian Motion at rational points in time (with respect to Wiener Measure ; see reference below). Another variation is the Strong Church–Turing thesis (SCTT), which states (cf. Bernstein, Vazirani 1997): "Any 'reasonable' model of computation can be efficiently simulated on a probabilistic Turing machine." HISTORY In his 1943 paper ''Recursive Predicates and Quantifiers'' (reprinted in ''The Undecidable'', p. 255) Stephen Kleene first proposed his "THESIS I": "This heuristic fact recursive functions are effectively calculable ...led Church to state the following thesis (Kleene's footnote 22). The same thesis is implicit in Turing's description of computing machines (Kleene's footnote 23). ::"THESIS I. ''Every effectively calculable function (effectively decidable predicate) is general recursive'' italics "Since a precise mathematical definition of the term effectively calculable (effectively decidable) has been wanting, we can take this thesis ... as a definition of it..." (Kleene in Undecidable, p. 274) Kleene's footnote 22 references the paper by Alonzo Church and his footnote 23 references the paper by Alan Turing . In his 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem " 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 Church had proven a similar result in "A Note on the Entscheidungsproblem" but he used the notions of Recursive Function s and Lambda-definable Function s 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 were introduced 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). After 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 THESIS Since that time, many other formalisms for describing effective computability have been proposed, including Recursive Function s, the Lambda Calculus , Register Machine s, Post System s, Combinatory Logic , and Markov Algorithm s. All these systems have been shown to compute 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 proved true. The physical version could, however, be disproved 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 IMPLICATIONS The Church–Turing thesis has been alleged to have 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: #The universe is equivalent to a Turing machine (and thus, computing non-recursive functions is physically impossible). This has been termed the ''strong Church–Turing thesis'' (not to be confused with the previously mentioned SCTT) and is a foundation of Digital Physics . #The universe is not a Turing machine (i.e., the laws of physics are not Turing-computable), but incomputable physical events are not "harnessable" for the construction of a Hypercomputer . For example, a universe in which physics involves Real Numbers , as opposed to Computable Real s, might fall into this category. #The universe is a Hypercomputer , and it is possible to build physical devices to harness this property and calculate non-recursive functions. For example, it is an open question whether all Quantum Mechanical events are Turing-computable, although it has been proved that any system built out of Qubit s is (at best) Turing-complete. John Lucas (and more famously, Roger Penrose ) have suggested that the human mind might be the result of quantum hypercomputation, although there is little scientific evidence for this theory. There are many other technical possibilities which fall outside or between these three categories, but these serve to illustrate the range of the concept. REFERENCES
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