Simple set

Simple and immune sets are classes of subsets of the natural numbers and provide important counterexamples in computability theory. They are closely related to the concept of recursive enumerable (RE) connected: While immune quantities exactly are the infinite sets that do not have infinite enumerable subset, the simple sets are recursively enumerable complements immune quantities.

Emil Post suspected already in the 1940s, the existence of simple quantities, but not until 1956 or 1957, this will be ( independently) also proved by Richard Friedberg and Albert Muchnik.

Definition

It was an effective enumeration of all recursively enumerable sets.

A set of natural numbers hot immune, if the following holds:

If so is infinite but has no infinite recursively enumerable subset. Here denotes the cardinality of a set.

A lot of hot now just in case it is itself recursively enumerable and its complement immune.

History

As a post 1944 began to compare problems after their decidability, quickly raised the question whether every recursively enumerable set which is not decidable automatically is already full for the enumerable sets. Well complements have RE- complete sets (more precisely all productive quantities ) an infinite enumerable subset. That is, the existence of simple quantities is already sufficient for the negative answer to the above question. Post even suspected this existence, they could not prove, however. Only when Friedberg and Muchnik 1956/57, the ( general ) problem solved Postsche, they constructed en passant the first simple quantity. This was also the first demonstration in the history of computability theory, which was performed using the priority method. Today, however, there is significantly simpler constructions simpler quantities.

Example

By assumption there is a Turing machine ( or algorithm in an alternative accountability model), which lists the amount when entering. Now let that partial payment function at the always is the first of found item in which it applies, if such exists. Then obviously ( partially ) predictable and their range of values ​​is easy.

Analogously, so also construct another simple quantities.

Properties

• Immune quantities are not recursively enumerable, otherwise they would indeed themselves contain an infinite enumerable subset. Simple volumes are therefore not decidable.

• After Myhill simple sets are therefore not RE -complete.

Hyper Simple and hyperimmune quantities

They say that an ordered set of natural numbers will outvoted by a number function if applies. Thus, the function value is always at least as large as the smallest - value.

A lot of hot hyperimmune if it is majorized by any total computable function.

A lot of hot hyper just then, if it is recursively enumerable hyperimmune and its complement.

• Hyper Simple volumes are necessarily infinite.
• Hyperimmune quantities involved in immune and hyper- simple sets therefore always easy.
• There are simple sets that are not hyper simple.