Efficient Self-Interpretations in lambda Calculus

We start by giving a compact representation schema for λ-terms and show how this leads to an exceedingly small and elegant self-interpreter. We then define the notion of a self-reducer, and show how this too can be written as a small λ-term. Both the self-interpreter and the self-reducer are proved correct. We finally give a constructive proof for the second fixed point theorem for the representation schema. All the constructions have been implemented on a computer, and experiments verify their correctness. Timings show that the self-interpreter and self-reducer are quite efficient, being about 35 and 50 times slower than direct execution using a call-byneed reduction strategy.