Problem of determining whether a given program will finish running or continue forever
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In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is undecidable, meaning that no general algorithm exists that solves the halting problem for all possible program–input pairs. The problem comes up often in discussions of computability since it demonstrates that some functions are mathematically definable but not computable.
A key part of the formal statement of the problem is a mathematical definition of a computer and program, usually via a Turing machine. The proof then shows, for any program f that might determine whether programs halt, that a "pathological" program g exists for which f makes an incorrect determination. Specifically, g is the program that, when called with some input, passes its own source and its input to f and does the opposite of what f predicts g will do. The behavior of f on g shows undecidability as it means no program f will solve the halting problem in every possible case.
In computability theory, the haltingproblem is the problem of determining, from a description of an arbitrary computer program and an input, whether the...
an algorithm that always leads to a correct yes-or-no answer. The haltingproblem is an example: it can be proven that there is no algorithm that correctly...
descriptions, then the problem of verifying every potential entry is undecidable, because it is equivalent to the well-known haltingproblem — there would be...
epistemology of that which we cannot know. Some related concepts include the haltingproblem, the limits of knowledge, the unknown unknowns, and chaos theory. Nicholas...
in a single operation. The problem can be of any complexity class. Even undecidable problems, such as the haltingproblem, can be used. An oracle machine...
Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's haltingproblem. In particular, no program P computing a lower bound for each text's...
correspondence problem is an undecidable decision problem that was introduced by Emil Post in 1946. Because it is simpler than the haltingproblem and the...
the terminology. Not every set of natural numbers is computable. The haltingproblem, which is the set of (descriptions of) Turing machines that halt on...
Hence, the problem is known to need more than exponential run time. Even more difficult are the undecidable problems, such as the haltingproblem. They cannot...
are so-called undecidable problems, such as the haltingproblem for Turing machines. Some well-known difficult abstract problems that have been solved relatively...
them. The haltingproblem is an important undecidable decision problem; for more examples, see list of undecidable problems. Decision problems can be ordered...
method' which decides whether any given Turing machine halts or not (the haltingproblem). If 'algorithm' is understood as meaning a method that can be represented...
^{2}t)} time. This version of the haltingproblem is among the simplest, most-easily described undecidable decision problems: Given an arbitrary positive integer...
proved that the problem Given g and n, does the sequence of iterates gk(n) reach 1? is undecidable, by representing the haltingproblem in this way. Closer...
order Horn clauses. The haltingproblem (determining whether a Turing machine halts on a given input) and the mortality problem (determining whether it...
undecidability of the haltingproblem. It is, however, decidable for context-free grammars. Intersection non-emptiness problem Sipser, Michael (2012)...
not Turing-computable. For example, a machine that could solve the haltingproblem would be a hypercomputer; so too would one that could correctly evaluate...
the condition is not acceptable, as it would allow us to solve the Haltingproblem. To see how, consider the statement H(M) stating "Turing machine M...
whether M will eventually produce s. This is due to the fact that the haltingproblem is unsolvable, which has major implications for the theoretical limits...
truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum...
computably enumerable (cf. picture for a fixed x). This set encodes the haltingproblem as it describes the input parameters for which each Turing machine...
well-defined characterisation under this definition. This includes the haltingproblem and the busy beaver game. It remains an open question as to whether...
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