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In A Self-Interpreter for F-omega, Brown and Palsberg construct for each term of each type $x:T$ a representation $\bar x : \Box T$ (a metatheoretical function which can't be represented in the language), and an interpretation function $u_T : \Box T \to T$. This is reminiscent of GL provability logic, which in particular abstracts how PA can represent its own proof theory. But while the representation looks like the axiom $\vdash T \implies \vdash \Box T$, the second $\vdash \Box T \to T$ is inconsistent with GL.

So what sort of gadget is this, in logical terms?

Can it be used at all to talk about the metatheory of F-omega?

What would a representation operation that satisfies the axioms of GL look like?

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    $\begingroup$ This might be closer to Artemov's "logic of proofs" than to GL. $\endgroup$
    – Emil Jeřábek
    Nov 21, 2016 at 11:25

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There is no proof significance, since already Gödel's $T$ has a Brown-Palsberg self-interpreter for free. We need a better definition of a typed self-interpreter.

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    $\begingroup$ But what about their "deep" interpreter? They seem to be able to do non-trivial things with it. For example, they have a normalization checker, which usually implies a proof of consistency of a type theory. $\endgroup$
    – fhyve
    Nov 21, 2016 at 20:39
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    $\begingroup$ Their interpreter seems to have some structurally desirable properties, but their definition of interpreter allows for trivial solutions, such as the one linked to in my answer. As far as I can tell nothing much is going on here. For instance, the normalization checker is just a simple recursion on the structure of the term. $\endgroup$
    – Andrej Bauer
    Nov 21, 2016 at 21:06

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