Examples of using Halting problem in English and their translations into Serbian
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This is the undecidability of the halting problem.
It is easy to prove that the halting problem is NP-hard but not NP-complete.
This means that this gives us an algorithm to decide the halting problem.
No oracle machine is capable of solving its own halting problem(a variation of Turing's proof applies).
But, many of these index sets are even more complicated than the halting problem.
In computability theory, the halting problem is a decision problem which can be stated as follows.
Much of computability theory builds on the halting problem result.
This is due to the fact that the halting problem is unsolvable, which has major implications for the theoretical limits of computing.
This fact is closely related to the algorithmic unsolvability of the Halting problem.
Alan turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist.
There are decision problems that are NP-hard but not NP-complete,for example the halting problem.
Thus this problem is strictly more difficult than the Halting problem, which asks whether the machine with index e halts on input 0.
In fact, a weaker form of the First Incompleteness Theorem is an easy consequence of the undecidability of the halting problem.
The halting problem, which is the set of(descriptions of) Turing machines that halt on input 0, is a well-known example of a noncomputable set.
In particular, we often show that a problem P is undecidable by showing that the halting problem reduces to P.
Although the halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. .
Alan Turing proved in 1936 that there isno general method or algorithm which can solve the halting problem for all possible inputs.
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, while the halting problem, in general, is undecidable.
Thus, if we had adecider 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.
For example, the Boolean satisfiability problem can be reduced to the halting problem by transforming it to the description of a Turing machine that tries all truth value assignments and when it finds one that satisfies the formula it halts and otherwise it goes into an infinite loop.
Harrington gave a further example of an automorphic property: that of the creative sets,the sets which are many-one equivalent to the halting problem.
Finding an upper bound on the busy beaver function is equivalent to solving the halting problem, a problem known to be unsolvable by Turing machines.
This fact creates a hierarchy of machines, called the arithmetical hierarchy, each with a morepowerful halting oracle and an even harder halting problem.
In particular, we often show that a problem P is undecidable by showing that the halting problem reduces to P. The complexity classes P, NP and PSPACE are closed under(many-one,"Karp") polynomial-time reductions.
The natural examples of sets that are not computable,including many different sets that encode variants of the halting problem, have two properties in common.
An example of a distNP-complete problem is the Bounded Halting Problem, BH, defined as follows: BH={(M, x, 1t): M is a non-deterministic Turing machine that accepts x in≤ t steps.} In his original paper, Levin showed an example of a distributional tiling problem that is average-case NP-complete.
Intuitively, this difference in unsolvability is because each instance of the"total machine" problem represents infinitely many instances of the Halting problem.
After ten years, Kleene andPost showed in 1954 that there are intermediate Turing degrees between those of the computable sets and the halting problem, but they failed to show that any of these degrees contains a recursively enumerable set.
Many degrees with special properties were constructed: hyperimmune-free degrees where every function computable relative to that degree is majorized by a(unrelativized) computable function; high degrees relative to which one can compute a function f which dominates every computable function g in the sense that there is a constant c depending on g such that g(x)< f(x) for all x> c; random degrees containing algorithmically random sets; 1-generic degrees of 1-generic sets;and the degrees below the halting problem of limit-recursive sets.
The natural examples of sets that are not computable,including many different sets that encode variants of the halting problem, have two properties in common: They are recursively enumerable, and Each can be translated into any other via a many-one reduction.