In computability theory, a subset of the natural numbers is called simple if it is computably enumerable (c.e.) and co-infinite (i.e. its complement is infinite), but every infinite subset of its complement is not c.e.. Simple sets are examples of c.e. sets that are not computable.

Relation to Post's problem

Simple sets were devised by Emil Leon Post in the search for a non-Turing-complete c.e. set. Whether such sets exist is known as Post's problem. Post had to prove two things in order to obtain his result: that the simple set A is not computable, and that the K, the halting problem, does not Turing-reduce to A. He succeeded in the first part (which is obvious by definition), but for the other part, he managed only to prove nonexistence of a many-one reduction.

Post's idea was validated by Friedberg and Muchnik in the 1950s using a novel technique called the priority method. They give a construction for a set that is simple (and thus non-computable), but fails to compute the halting problem.

Formal definitions and some properties

denotes a standard uniformly c.e. listing of all the c.e. sets.

• is called immune that is c.e..

• is called simple if it is c.e. and its complement is immune.

• is called effectively immune.

• is called effectively simple if it is c.e. and its complement is effectively immune. Every effectively simple set is simple and Turing-complete.

• is called hyperimmune in order.

• is called hypersimple if it is simple and its complement is hyperimmune.

Notes

References