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The universal halting problem, also known (in recursion theory) as totality, is the problem of determining whether a given computer program will halt for every input (the name totality comes from the equivalent question of whether the computed function is total). This problem is not only undecidable, as the halting problem is, but highly ...
In computability theory, the halting problem is a decision problem which can be stated as follows: . Given the description of an arbitrary program and a finite input, decide whether the program finishes running or will run forever.
The halting problem (determining whether a Turing machine halts on a given input) and the mortality problem (determining whether it halts for every starting configuration). Determining whether a Turing machine is a busy beaver champion (i.e., is the longest-running among halting Turing machines with the same number of states and symbols).
Since the halting problem is known to be undecidable, this is a contradiction and the assumption that there is an algorithm P(a) that decides a non-trivial property for the function represented by a must be false.
For example, the conventional proof of the unsolvability of the halting problem is essentially a diagonal argument. Also, diagonalization was originally used to show the existence of arbitrarily hard complexity classes and played a key role in early attempts to prove P does not equal NP .
It is Turing equivalent to the halting problem and thus at level Δ 0 2 of the arithmetical hierarchy. Not every set that is Turing equivalent to the halting problem is a halting probability. A finer equivalence relation, Solovay equivalence, can be used to characterize the halting probabilities among the left-c.e. reals. [4]
In a sense, these are the "hardest" recursively enumerable problems. Generally, no constraint is placed on the reductions used except that they must be many-one reductions. Examples of RE-complete problems: Halting problem: Whether a program given a finite input finishes running or will run forever.
Thus the halting problem is an example of a computably enumerable (c.e.) set, which is a set that can be enumerated by a Turing machine (other terms for computably enumerable include recursively enumerable and semidecidable). Equivalently, a set is c.e. if and only if it is the range of some computable function.