In computability theory, an undecidable problem is a type of computational problem that requires a yes/no answer, but where there cannot possibly be any computer program that always gives the correct answer; that is, any possible program would sometimes give the wrong answer or run forever without giving any answer. More formally, an undecidable problem is a problem whose language is not a recursive set; see the article Decidable language. There are uncountably many undecidable problems, so the list below is necessarily incomplete. Though undecidable languages are not recursive languages, they may be subsets of Turing recognizable languages: i.e., such undecidable languages may be recursively enumerable.
Many, if not most, undecidable problems in mathematics can be posed as word problems: determining when two distinct strings of symbols (encoding some mathematical concept or object) represent the same object or not.
For undecidability in axiomatic mathematics, see List of statements undecidable in ZFC.
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such undecidable languages may be recursively enumerable. Many, if not most, undecidableproblems in mathematics can be posed as word problems: determining...
computability theory and computational complexity theory, an undecidableproblem is a decision problem for which it is proved to be impossible to construct an...
The halting problem is an important undecidable decision problem; for more examples, see listofundecidableproblems. Decision problems can be ordered...
Product Puzzle", which is not impossible -gry, a word puzzle Listofundecidableproblems, no algorithm can exist to answer a yes–no question about the...
problem is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. A key part of the...
Recursive set, a "decidable set" in recursion theory Decision problemListofundecidableproblems Decision (disambiguation) Decide (disambiguation) This disambiguation...
Diophantine equations. In principle, many problems can be reduced to the halting problem. See the listofundecidableproblems. Gödel's incompleteness theorems...
correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the halting problem and the...
decidabilities. On the top are the undecidableproblems. Below it are the decidable problems. Furthermore, the decidable problems can be divided into a complexity...
Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") is a paper in mathematical...
answer the question of whether a given Oracle machine will ever halt. Automata theory Abstract machine Listofundecidableproblems Computational complexity...
History of the Church–Turing thesis Lambda calculus Listofundecidableproblems Post correspondence problem Post's theorem Primitive recursive function Recursion...
in the integers. The listofundecidableproblems gives additional examples ofproblems with no computable solution. The study of which mathematical constructions...
exponential run time. Even more difficult are the undecidableproblems, such as the halting problem. They cannot be completely solved by any algorithm...
ISBN 978-0-85664-464-1. Davis, Martin (1965). The Undecidable: Basic Papers On Undecidable Propositions, Unsolvable Problems and Computable Functions. New York: Raven...
the first purely algebraic problem to be proved undecidable. Shelah later showed that the Whitehead problem remains undecidable even if one assumes the continuum...
in a single operation. The problem can be of any complexity class. Even undecidableproblems, such as the halting problem, can be used. An oracle machine...
proved that the problem Given g and n, does the sequence of iterates gk(n) reach 1? is undecidable, by representing the halting problem in this way. Closer...
constructions of classical geometry, and solving the general quintic equation algebraically. Also provably unsolvable are so-called undecidableproblems, such...
Journal of Symbolic Logic, 1, 103–105, 1936. Reprinted in The Undecidable, pp. 289ff. Emil Post (1947), "Recursive Unsolvability of a Problemof Thue",...
those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm, with one of the more prominent...
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