Kleene's recursion theorem

Kleene's recursion theorem isresultcomputability theory first proved by Stephen Kleene;allowsconstruct programs, Turing machinesrecursive functions that refer backtheir own description.

The theorem can be formulatedany ofequivalent modelscomputation listed incomputability theory article. Here we will use Pascal programs.

If p isstring that describesPascal program which readssingle input stringproducessingle output string, then we will denotecorresponding partial function from input stringsoutput strings by f_p. (This ispartial function becausesome input stringsprogram described by p may run into an infinite loopnever produce an output.)

The statement oftheoremas follows: If Q isPascal program which takes two input strings xyproducessingle output string Q(x,y) (which again may be undefined if Q runs into an infinite loop), then there always existsstring p describing some Pascal program such thatall input strings y:

Q(p,y) = f_p(y).

This latter equality isbe read as an equalitypartial functions: Q(p,y)defined ifonly if f_p(y)defined,in this casetwo values aresame.

As an application, we can show that there must existPascal program which takessingle input, ignores that inputprints itself as output: Define Q(x,y) = x, thentheorem yieldsPascal program p such thatall y: f_p(y) = Q(p,y) = p. Itan amusing exerciseactually write suchprogram p.