Examples of using Halting problem in English and their translations into Romanian
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Much of computability theory builds on the halting problem result.
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. .
There are decision problems that are NP-hard but not NP-complete,for example the halting problem.
It is easy to prove that the halting problem is NP-hard but not NP-complete.
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.
Post showed that these sets are strictly between the computable sets and the halting problem with respect to many-one reducibility.
Post's original motivation in the study of this lattice was to find a structural notion such that every set which satisfies this property is neither in the Turing degree of the recursive sets norin the Turing degree of the halting problem.
Its main purpose was to prove that there were problems(namely the halting problem) that could not be solved by any sequential process.
Turing reduced the question of the existence of an'algorithm' or'general method' able to solve the Entscheidungsproblem to the question of the existence of a'general method' which decides whether any given Turing Machine halts or not(the halting problem).
Given a set A,the Turing jump of A is a set of natural numbers encoding a solution to the halting problem for oracle Turing machines running with oracle A.
The statement that the halting problem cannot be solved by a Turing machine[7] is one of the most important results in computability theory, as it is an example of a concrete problem that is both easy to formulate and impossible to solve using a Turing machine.
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.
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.
The Turing jump of any set is always of higher Turing degree than the original set, anda theorem of Friedburg shows that any set that computes the Halting problem can be obtained as the Turing jump of another set.
This result preceded Alan Turing's work on the halting problem, which also demonstrated the existence of a problem unsolvable by mechanical means.
Fred Cohen, a computer scientist who formulated the definition of a computer virus, went one step further anddemonstrated that this so-called“halting problem” applies to cybersecurity as well.
Church and Turing then showed that the lambda calculus andthe Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative"mechanical processes for computation.".
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.
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.
Given a set A,the Turing jump of A is a set of natural numbers encoding a solution to the halting problem for oracle Turing machines running with oracle A. The Turing jump of any set is always of higher Turing degree than the original set, and a theorem of Friedburg shows that any set that computes the Halting problem can be obtained as the Turing jump of another set.
Although the natural examples of noncomputable sets are all many-one equivalent, it is possible to construct recursively enumerable sets A and B such that A is Turing reducible to B butnot many-one reducible to B. It can be shown that every recursively enumerable set is many-one reducible to the halting problem, and thus the halting problem is the most complicated recursively enumerable set with respect to many-one reducibility and with respect to Turing reducibility.
Church and Turing then showed that the lambda calculus and the Turing machine used in Turing's halting problem were equivalent in capabilities, and subsequently demonstrated a variety of alternative"mechanical processes for computation." This resulted in the Church- Turing thesis.
Rice showed that for every nontrivial class C( which contains some but not all r. e. sets) the index set E={ e: the eth r. e. set We is in C}has the property that either the halting problem or its complement is many-one reducible to E, that is, can be mapped using a many-one reduction to E( see Rice 's theorem for more detail). But, many of these index sets are even more complicated than the halting problem.