Breaking through the normalization barrier: a self-interpreter for f-omega.

According to conventional wisdom, a self-interpreter for a strongly normalizing lambda-calculus is impossible. We call this the normalization barrier. The normalization barrier stems from a theorem in computability theory that says that a total universal function for the total computable functions is impossible. In this paper we break through the normalization barrier and define a self-interpreter for System F_omega, a strongly normalizing lambda-calculus. After a careful analysis of the classical theorem, we show that static type checking in F_omega can exclude the proof's diagonalization gadget, leaving open the possibility for a self-interpreter. Along with the self-interpreter, we program four other operations in F_omega, including a continuation-passing style transformation. Our operations rely on a new approach to program representation that may be useful in theorem provers and compilers.