The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

The domain is always Turing equivalent to the halting problem.

Knowing the first bits of, one could calculate the halting problem for all programs of a size up to.

Let the program for which the halting problem is to be solved be N bits long.

Thus, the halting problem would be solved for.

Because many outstanding problems in number theory, such as Goldbach's conjecture are equivalent to solving the halting problem for special programs ( which would basically search for counter-examples and halt if one is found ), knowing enough bits of Chaitin's constant would also imply knowing the answer to these problems.

* It is Turing equivalent to the halting problem and thus at level of the arithmetical hierarchy.

Not every set that is Turing equivalent to the halting problem is a halting probability.

The proof of this fact relies on an algorithm which, given the first n digits of Ω, solves Turing's halting problem for programs of length up to n. Since the halting problem is undecidable, Ω can not be computed.

Turing reduced the halting problem for Turing machines to the.

Most general purpose functional programming languages allow unrestricted recursion and are Turing complete, which makes the halting problem undecidable, can cause unsoundness of equational reasoning, and generally requires the introduction of inconsistency into the logic expressed by the language's type system.

Its main purpose was to prove that there were problems ( namely the halting problem ) that could not be solved by any sequential process.

Even undecidable problems, like the halting problem, can be used.

It is possible to posit the existence of an oracle which computes a non-computable function, such as the answer to the halting problem or some equivalent.

This fact creates a hierarchy of machines, called the arithmetical hierarchy, each with a more powerful halting oracle and an even harder halting problem.

As an example, consider the following variant of the halting problem.

The Councilman, who is the Administration floor leader, also criticized Bernard L. Werner, public works director, for `` halting snow operations '' on Tuesday night after the Sunday storm.

In the computer science subfield of algorithmic information theory, a Chaitin constant ( Chaitin omega number ) or halting probability is a real number that informally represents the probability that a randomly constructed program will halt.

Although there are infinitely many halting probabilities, it is common to use the letter Ω to refer to them as if there were only one.

Each halting probability is a normal and transcendental real number which is not computable, which means that there is no algorithm that enumerates its digits.

It is from this interpretation that halting probabilities take their name.

In this way, Ω < sub > F </ sub > represents the probability that a randomly selected infinite sequence of 0s and 1s begins with a bit string ( of some finite length ) that is in the domain of F. It is for this reason that Ω < sub > F </ sub > is called a halting probability.

No halting probability is computable.

By a straightforward reduction to the halting problem it is possible to prove that ( for any Turing complete language ) finding all possible run-time errors in an arbitrary program ( or more generally any kind of violation of a specification on the final result of a program ) is undecidable: there is no mechanical method that can always answer truthfully whether a given program may or may not exhibit runtime errors.

Also, the halting problem for Zeno machines is not solvable by a Zeno machine ( Potgieter, 2006 ).

The halting problem is therefore called non-computable or undecidable.

The halting language is therefore recursively enumerable.

The complement of the halting language is therefore not recursively enumerable.

The problem of finding WCET by analysis is equivalent to the halting problem and is therefore insoluble in the general case.

Ventura placed an order with Designline for more units, this time based on Scania K230UB chassis, but this order was cancelled when an earthquake in 2010 caused severe damage to the Designline factory, therefore halting production, this order was transferred to Custom Coaches and these units were delivered in the first half of 2011.

After finite time the first n bits of the output will never change any more ( it does not matter that this time itself is not computable by a halting program ).

In a synthesis environment, the synthesis tool usually operates with the policy of halting synthesis upon any violation.

This is not surprising, as the ability to make this determination would also imply the ability to solve the halting problem and perform any other non-algorithmic calculation that can be described in English.

Chico spoke only limited and halting English, so the joke centered on him responding to almost any question with his catch phrase: " Baseball ... been berra berra good ... to me.

The final solidification of Riverton ’ s boundaries came when the City of Herriman incorporated in 2000, halting any possible further expansion westward by Riverton.

* Penrose uses Gödel's incompleteness theorem ( which states that there are mathematical truths which can never be proven in a sufficiently strong mathematical system ; any sufficiently strong system of axioms will also be incomplete ) and Turing's halting problem ( which states that there are some things which are inherently non-computable ) as evidence for his position.

The couple agreed to spend 13 weeks in a drug-treatment program in exchange for halting any further prosecution of their drug-possession case.

An extension of the halting problem is called Rice's Theorem, which states that it is undecidable ( in general ) whether a given language possesses any specific nontrivial property.

Clarke then states that such a policy would not endanger privacy rights through the institution of a privacy advocate who could stop abuses or any activity that went beyond halting the theft of important files.

As it turned out, Foster's halting efforts to establish a separate power center within the UMWA had this effect in any case, as Howat, Brophy and his allies dropped out of the " Save the Union " movement as the CP's leadership in it became apparent.

Thus, if we had a decider R for E, we would be able to produce a decider S for the halting problem H ( M, w ) for any machine M and input w. Since we know that such an S cannot exist, it follows that the language E is also undecidable.

* January 15, 1971 – The District Court granted a preliminary injunction halting any further construction activities pending the outcome of the litigation.

To prove that term-rewriting terminates is not possible in any instance, as we know from the halting problem.

This practical problem is that autoconnection may automatically place the SSID in the wireless adapter operating profile, halting your ability to log any additional stations encountered later on.

In a November 2006 referendum, citizens of Jacksonville voted to leave the Cecil Field Airport and Commerce Center in civilian hands under the Jacksonville Aviation Authority, effectively halting any future plans of relocation.