In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, the Entscheidungsproblem (pronounced [ɛntˈʃaɪ̯dʊŋspʀoˌbleːm], German German (Deutsch, [ˈdɔʏtʃ] ) is a West Germanic language, thus related to and classified alongside English and Dutch. It is one of the world's major languages and the most widely spoken first language in the European Union. Globally, German is spoken by approximately 120 million native speakers and also by about 80 million non-native speakers for 'decision problem In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem. The answer can be either 'yes' or 'no', and') is a challenge posed by David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of in 1928. The Entscheidungsproblem asks for an algorithm In mathematics, computer science, and related subjects, an 'algorithm' is an effective method for solving a problem expressed as a finite sequence of instructions. Algorithms are used for calculation, data processing, and many other fields that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false. The algorithm need not justify its answer, nor provide a proof, so long as it is always correct. Such an algorithm would be able to decide, for example, whether statements such as Goldbach's conjecture Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. It states: or the Riemann hypothesis In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann , is a conjecture about the distribution of the zeros of the Riemann zeta-function which states that all non-trivial zeros of the Riemann zeta function have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over are true, even though no proof or disproof of these statements is known. The Entscheidungsproblem has often been identified in particular with the decision problem for first-order logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each (that is, the problem of algorithmically determining whether a first-order statement is universally valid).
In 1936 and 1937 Alonzo Church Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem and Alan Turing Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst and computer scientist. He was influential in the development of computer science and providing a formalisation of the concept of the algorithm and computation with the Turing machine, playing a significant role in the creation of the modern computer respectively[1], published independent papers showing that it is impossible to decide algorithmically whether statements in arithmetic Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers. In common usage, it refers to the simpler properties when are true or false, and thus a general solution to the Entscheidungsproblem is impossible. This result is now known as Church's Theorem or the Church–Turing Theorem (not to be confused with the Church–Turing thesis).
Contents |
History of the problem
The origin of the Entscheidungsproblem goes back to Gottfried Leibniz Gottfried Wilhelm Leibniz (German pronunciation: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪpnɪts]; born 1 July 1646 in Leipzig [OS: 21 June] – died in Hannover 14 November 1716) was a German mathematician and philosopher. Leibniz wrote primarily in Latin and French, who in the seventeenth century, after having constructed a successful mechanical calculating machine, dreamt of building a machine that could manipulate symbols in order to determine the truth values In logic and mathematics, a logical value, also called a truth value, is a value indicating the relation of a proposition to truth of mathematical statements (Davis 2000: pp. 3–20). He realized that the first step would have to be a clean formal language A formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar ; accordingly, words that belong to a formal language are sometimes called well-formed words (, and much of his subsequent work was directed towards that goal. In 1928, David Hilbert David Hilbert /ˈdaːfɪt ˈhɪlbʌt/ was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. He discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of and Wilhelm Ackermann Wilhelm Friedrich Ackermann was a German mathematician best known for the Ackermann function, an important example in the theory of computation posed the question in the form outlined above.
In continuation of his "program" with which he challenged the mathematics community in 1900, at a 1928 international conference David Hilbert asked three questions, the third of which became known as "Hilbert's Entscheidungsproblem" (Hodges p. 91). As late as 1930 he believed that there would be no such thing as an unsolvable problem (Hodges p. 92, quoting from Hilbert).
Negative answer
Before the question could be answered, the notion of "algorithm" had to be formally defined. This was done by Alonzo Church Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem in 1936 with the concept of "effective calculability" based on his λ calculus In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. It was introduced by Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics. After the original system was shown to be logically inconsistent , and by Alan Turing in the same year with his concept of Turing machines A Turing machine is a theoretical device that manipulates symbols contained on a strip of tape. Despite its simplicity, a Turing machine can be adapted to simulate the logic of any computer algorithm, and is particularly useful in explaining the functions of a CPU inside a computer. The "Turing" machine was described by Alan Turing in 193. It was later recognized that these are equivalent models of computation.
The negative answer to the Entscheidungsproblem was then given by Alonzo Church in 1935–36 and independently shortly thereafter by Alan Turing in 1936–37. Church proved that there is no computable function Computable functions are the basic objects of study in computability theory. The set of computable functions is equivalent to the set of Turing-computable functions and partial recursive functions. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any which decides for two given λ calculus expressions whether they are equivalent or not. He relied heavily on earlier work by Stephen Kleene Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science. One of many distinguished students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory. Kleene's work grounds the study of. Turing reduced In computability theory and computational complexity theory, a reduction is a transformation of one problem into another problem. Depending on the transformation used this can be used to define complexity classes on a set of problems the halting problem In computability theory, the halting problem is a decision problem which can be stated as follows: given a description of a program, decide whether the program finishes running or will run forever. This is equivalent to the problem of deciding, given a program and an input, whether the program will eventually halt when run with that input, or will for Turing machines to the Entscheidungsproblem. The work of both authors was heavily influenced by Kurt Gödel Kurt Gödel (German pronunciation: [kʊʁt ˈɡøːdl̩] ; April 28, 1906, Brno, Moravia – January 14, 1978, Princeton, New Jersey, USA) was an Austrian-American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century,'s earlier work on his incompleteness theorem Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that, especially by the method of assigning numbers (a Gödel numbering) to logical formulas in order to reduce logic to arithmetic.
Turing's argument is as follows. Suppose we had a general decision algorithm for statements in a first-order First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic is distinguished from propositional logic by its use of quantifiers; each language. The question whether a given Turing machine halts or not can be formulated as a first-order statement, which would then be susceptible to the decision algorithm. But Turing had proven earlier that no general algorithm can decide whether a given Turing machine halts.
The Entscheidungsproblem is related to Hilbert's tenth problem, which asks for an algorithm In mathematics, computer science, and related subjects, an 'algorithm' is an effective method for solving a problem expressed as a finite sequence of instructions. Algorithms are used for calculation, data processing, and many other fields to decide whether Diophantine equations In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations. In more technical language, they define an algebraic curve, algebraic surface, or more have a solution. The non-existence of such an algorithm, established by Yuri Matiyasevich in 1970, also implies a negative answer to the Entscheidungsproblem.
Some first-order theories are algorithmically decidable; examples of this include Presburger arithmetic, real closed fields In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers and static type systems In computer science, a type system may be defined as a tractable syntactic framework for classifying phrases according to the kinds of values they compute. A type system associates types with each computed value. By examining the flow of these values, a type system attempts to prove that no type errors can occur. The type system in question of (most) programming languages A programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human communication. The general first-order theory of the natural numbers In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1, 2, 3, ...} according to the traditional definition; or the set of non-negative integers {0, 1, 2, ...} according to a definition first appearing in the nineteenth century expressed in Peano's axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental cannot be decided with such an algorithm, however.
See also
- Hilbert's second problem
- Oracle machine In complexity theory and computability theory, an oracle machine is an abstract machine used to study decision problems. It can be visualized as a Turing machine with a black box, called an oracle, which is able to decide certain decision problems in a single operation. The problem can be of any complexity class. Even undecidable problems, like
Notes
- ^ Church's paper was presented to the American Mathematical Society on 19 April 1935 and published on 15 April 1936. Turing, who had made substantial progress in writing up his own results, was disappointed to learn of Church's proof upon its publication (see correspondence between Max Newman and Church in Alonzo Church papers). Turing quickly completed his paper and rushed it to publication; it was received by the Proceedings of the London Mathematical Society on 28 May 1936, read on 12 November 1936, and published in January 1937 series 2, volume 42 (1936-1937); Turing added corrections in volume 43(1937) pp. 544-546. See Davis 1965:116.
References
- Alonzo Church, "An unsolvable problem of elementary number theory", American Journal of Mathematics, 58 (1936), pp 345–363
- Alonzo Church, "A note on the Entscheidungsproblem", Journal of Symbolic Logic, 1 (1936), pp 40–41.
- Martin Davis, 2000, Engines of Logic, W.W. Norton & Company, London, ISBN 0-393-32229-7 pbk.
- Alan Turing, "On computable numbers In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. Equivalent definitions can be given using μ-recursive functions, Turing, with an application to the Entscheidungsproblem", Proceedings of the London Mathematical Society The London Mathematical Society is one of the UK's learned societies for mathematics (the others being the Royal Statistical Society (RSS) and the Institute of Mathematics and its Applications (IMA)), Series 2, 42 (1937), pp 230–265. Online versions: from journal website, from Turing Digital Archive, from abelard.org. Errata appeared in Series 2, 43 (1937), pp 544–546.
- Martin Davis Martin David Davis, is an American mathematician, known for his work on Hilbert's tenth problem (Jackson 2008, p. 560). He received his Ph.D. from Princeton University in 1950, where his adviser was Alonzo Church (Jackson 2008, p. 560). He is Professor Emeritus at New York University. He is the co-inventor of the Davis-Putnam and the DPLL, "The Undecidable, Basic Papers on Undecidable Propositions, Unsolvable Problems And Computable Functions", Raven Press, New York, 1965. Turing's paper is #3 in this volume. Papers include those by Godel, Church, Rosser, Kleene, and Post.
- Andrew Hodges, Alan Turing: The Enigma, Simon and Schuster, New York, 1983. Allen M. Turing's biography. Cf Chapter "The Spirit of Truth" for a history leading to, and a discussion of, his proof.
- Toulmin, Stephen Stephen Edelston Toulmin was a British philosopher, author, and educator. Influenced by the Austrian born British philosopher Ludwig Wittgenstein, Toulmin devoted his works to the analysis of moral reasoning. Throughout his writings, he sought to develop practical arguments which can be used effectively in evaluating the ethics behind moral issues, "Fall of a Genius", a book review of "Alan Turing: The Enigma by Andrew Hodges", in The New York Review of Books, January 19, 1984, p. 3ff.
- Alfred North Whitehead Alfred North Whitehead, OM was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education. He co-authored the epochal Principia Mathematica with Bertrand Russell and Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, agnostic, socialist, pacifist, and social critic. Although he spent most of his life in England, he was born in Wales where he also died, aged 97, Principia Mathematica to *56, Cambridge at the University Press, 1962. Re: the problem of paradoxes, the authors discuss the problem of a set not be an object in any of its "determining functions", in particular "Introduction, Chap. 1 p. 24 "...difficulties which arise in formal logic", and Chap. 2.I. "The Vicious-Circle Principle" p. 37ff, and Chap. 2.VIII. "The Contradictions" p. 60 ff.
Categories: Theory of computation | Recursion theory | Mathematical logic Mathematical logic has several meanings in common usage. It originally referred to symbolic or formal logic, and then came to be associated with the study of the logical foundations of mathematics. In contemporary use by mathematical logicians, the term refers to several branches of pure mathematics whose study involves careful attention to formal | Metatheorems | German words and phrases Categories: Words and phrases by language | German language
