Markov algorithms

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, such as electronic computers. 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: ...

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• Church–Turing thesis - Church–Turing thesis
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In the following: H refers to the source "Hodges" U refers to the source "Undecidable" W refers to definitions from Websters Ninth New Collegiate Dictionary Marriam-Webster Inc., Springfield Mass, 1990 PM refers to the source Principia Mathematica circa B.C.-- Pythagoras shows the existence of numbers that are not rational, i.e. numbers exist that are not the natural numbers or ratios of the counting numbers. Numbers that are either natural numbers or ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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The thesis, in Turing's own words, 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 hypercomp ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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Take as axioms the axioms of the primitive recursive functions, but extend the definitions so as to allow for partial functions. Add one further operator, the unbounded search operator, defined as follows: If f(x,z1,z2,...,zn) is a partial function on the natural numbers with n+1 arguments x, z1,...,zn, then the function μx f is the partial function with arguments z ...

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Recursive function, Recursive function - Definition, Recursive function - Examples, Recursive function - External link

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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 hypercomputers whose results should "naturally be regarded as computable". Any computer program can be translated into a Turing machine, and any Turing machine can be ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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A partial function is called computable if the graph of f is a recursively enumerable set. The set of partial computable functions with one parameter is usually denoted or if the number of parameters is clear from the context. A total function is called computable if the graph of f is a recursive set. The set of total computable fun ...

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Computable function, Computable function - Definition, Computable function - Remarks, Computable function - Examples, Computable function - Properties

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Sometimes, for reasons of clarity, we write a computable function as We can easily encode g into a new function using a pairing function. ...

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Computable function, Computable function - Definition, Computable function - Remarks, Computable function - Examples, Computable function - Properties

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In his 1943 paper Recursive Predicates and Quantifiers (reprinted in The Undecidable, p. 255) Stephen Kleene first proposed his "THESIS I": "This heuristic fact [general 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 r ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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No program can solve the halting problem. There are programs that give correct answers for some instances of it, and run forever on all other instances. A program that returns answers for some instances of the halting problem might be called a partial halting solver (PHS). Can we recognize a correct PHS when we see it? Let the PHS recognition problem be this: given a PHS, determine whether it returns only correct answers. This problem sounds like it might be easier than the halting problem itself. It is not. It is ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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The concepts raised by Gödel's incompleteness theorems are very similar to those raised by the halting problem, and the proofs are quite similar. In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem. This weaker form differs from the standard statement of the incompleteness theorem by asserting that a complete, consistent and sound axiomatization of all statements about natural numbers is unachievable. The "sound" part is the weakening: it means that we require ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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One possible way of formally stating the halting problem is as follows: Given a Gödel numbering of the computable functions, with the Cantor pairing function, the set is called the halting set. The problem of deciding whether the halting set is recursive or not is called the halting problem. As the set is recursively enumerable the halting problem is not solvable by a computable function. Alternative equivalent formulations, for inst ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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The 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: 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 previous ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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Since 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 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, t ...

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Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference

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The importance of the halting problem lies in the fact that it is the first problem to be proved undecidable. Subsequently, many other such problems have been described; the typical method of proving a problem to be undecidable is with the technique of reduction. To do this, the computer scientist shows that if a solution to the new problem was found, it could be used to decide an undecidable problem (by transforming instances of the undecidable problem into instances of the new problem). Since we already know that no method can decid ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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The proof proceeds by reductio ad absurdum. We start with assuming that there is a function halt(a, i) that returns true if the algorithm represented by the string a halts when given as input the string i, and returns false otherwise. (The existence of the universal Turing machine proves that every possible algorithm corresponds to at least one such string.) Given this algorithm we can construct another algorithm trouble(s) as follows: function trouble(string s) if halt(s, s) = false r ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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In his original proof Turing formalized the concept of algorithm by introducing Turing machines. However, the result is in no way specific to them; it applies equally to any other model of computation that is equivalent in its computational power to Turing machines, such as Markov algorithms, Lambda calculus, Post systems or register machines. What is important is that the formalization allows a straightforward mapping of algorithms to some data type that the algorithm can operate upon. For example, if the formalism lets algori ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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Many students, upon analyzing the above proof, ask whether there might be an algorithm that can return a third option, such as "undecidable." This reflects a misunderstanding of decidability. It is easy to construct one algorithm that always answers "halts" and another that always answers "doesn't halt." For any specific program and input, one of these two algorithms answers correctly, even though nobody may know which one. The difficulty of the halting problem lies not in particular programs, but in the requ ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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It might seem like humans could solve the halting problem. After all, a programmer can often look at a program and tell whether it will halt. It is useful to understand why this cannot be true. For simplicity, we will consider the halting problem for programs with no input, which is also undecidable. To "solve" the halting problem means to be able to look at any program and tell whether it halts. It is not enough to be able to look at some programs and decide. Humans may also not be able to solve the halting problem, due ...

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Halting problem, Halting problem - Formal statement, Halting problem - Importance and consequences, Halting problem - Sketch of proof, Halting problem - Common pitfalls, Halting problem - Formalization of the halting problem, Halting problem - Relationship with Gödel's incompleteness theorem, Halting problem - Can humans solve the halting problem?, Halting problem - Recognizing partial solutions, Halting problem - History of the Halting Problem, Halting problem - Footnotes

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