Infinitary self-reference in learning theory

Abstract Kleene's second recursion theorem provides a means for transforming any program p into a program e(p) which first creates a quiescent self-copy and then runs p on that self-copy together with any externally given input. e(p), in effect, has complete (low level), self-knowledge, and p represents how e(p) uses its self-knowledge (and its knowledge of the external world). Infinite regress is not required since e(p) creates its self-copy outside of itself. One mechanism to achieve this creation is a self-replication trick isomorphic to that employed by single-celled organisms. Another is for e(p) to look in a mirror to see which program it is. In 1974 the author published an infinitary generalization of Kleene's theorem which he called the operator recursion theorem. It provides a means for obtaining an (algorithmically) growing collection of programs which, in effect, share a common (also growing) mirror from which they can obtain complete low-level models of themselves and the other programs in the...

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