The halting problem is an early example of a decision problem , and also a good example of the limits of determinism in computer science.
Is the halting problem undecidable?
Alan Turing proved in 1936 that a general algorithm running on a Turing machine that solves the halting problem for all possible program-input pairs necessarily cannot exist. Hence, the halting problem is undecidable for Turing machines .
What type of problem is the halting problem?
unsolvable algorithmic problem is the halting problem, which states that no program can be written that can predict whether or not any other program halts after a finite number of steps. The unsolvability of the halting problem has immediate practical bearing on software development.
Is the halting problem solved?
Halting problem is perhaps the most well-known problem that has been proven to be undecidable; that is, there is no program that can solve the halting problem for general enough computer programs.
Who discovered the halting problem?
A decision problem that was discovered and investigated by Alan Turing in 1936. Suppose M is a Turing machine and let x be an input to M. If we start the machine running two things might happen: after a finite number of steps the machine might stop, or it might run on forever.
Is halting problem computable?
Example: The halting problem is partially computable . To determine HALTS(P,D), simply call P(D). Then, HALTS(P,D) halts and outputs Yes if P(D) halts, and loops otherwise. ... If a problem is not even partially computable, there is no way of checking even a YES answer.
Why is the halting problem so important?
The Halting problem lets us reason about the relative difficulty of algorithms . It lets us know that, there are some algorithms that don’t exist, that sometimes, all we can do is guess at a problem, and never know if we’ve solved it.
Is the halting problem in P?
It is also easy to see that the halting problem is not in NP since all problems in NP are decidable in a finite number of operations, but the halting problem, in general, is undecidable . There are also NP-hard problems that are neither NP-complete nor Undecidable.
What makes a problem decidable?
A problem is said to be Decidable if we can always construct a corresponding algorithm that can answer the problem correctly . ... It is also important to know that these problems are termed as Turing Decidable since a Turing machine always halts on every input, accepting or rejecting it.
How do you prove a problem is undecidable?
- For the halting problem instance (N, y), create a new machine M for the L problem.
- On input x, M simulates (N, y) for length(x) steps.
- If the simulation halted within that number of steps, then M halts. Otherwise, M deliberately goes into an infinite loop.
Is halting problem NP hard?
– If we had a polynomial time algorithm for the halting problem, then we could solve the satisfiability problem in polynomial time using A and X as input to the algorithm for the halting problem . – Hence the halting problem is an NP-hard problem which is not in NP . – So it is not NP-complete.
Why is halting problem unsolvable?
It asks the following question: Given an arbitrary Turing machine M over alphabet = { a , b } , and an arbitrary string w over , does M halt when it is given w as an input ? It can be shown that the halting problem is not decidable , hence unsolvable. Theorem 1 : The halting problem is undecidable.
Is halting problem recursively enumerable?
The language HALT corresponding to the Halting problem is recursively enumerable , but not recursive. In particular, the universal TM accepts HALT, but no TM can decide HALT. There are languages which are not recursively enumerable, in particular the language NOTRE in the proof.
Is first order logic decidable?
First-order logic is not decidable in general; in particular, the set of logical validities in any signature that includes equality and at least one other predicate with two or more arguments is not decidable. Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable.
What is halting problem explain?
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 .
What is meant by halting problem?
The halting problem, commonly applied to Turing-complete programs and models, is the problem of finding out whether, with the given input, a program will halt at some time or continue to run indefinitely .
