# Self reduction for fully introspective lambda calculus representations

For some representation scheme $$\ulcorner \cdot \urcorner$$, a self interpreter $$R$$ is a lambda expression where $$R \ulcorner A \urcorner \underset{\beta}{=} A$$, while a self reducer $$E$$ is a lambda expression where $$E \ulcorner A \urcorner \underset{\beta}{=} \ulcorner {NF}_A \urcorner$$ ($${NF}_A$$ being the normal form of $$A$$).

For compact representation schemes like Mogensen-Scott encoding, I can find research on both self interpreters and self reducers (like "Efficient Self-Interpretation in Lambda Calculus" by Mogensen, 1994). However, I'm looking for research on self reducers for fully introspective representation schemes like that from Barendregt ("Self-interpretation in lambda calculus" by Barendregt, 1991).

So far, I can only find research on self interpreters for this representation. So, I would really be glad if someone knew some research papers on lambda calculus self reducers for the less compact and fast strain of lambda expression representations.