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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.

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