Thursday, January 12, 2012

Structural focalization updated

I've uploaded to both ArXiV and my webpage a significantly revised draft of the paper Structural focalization, which I've spoken about here before. Feedback is welcome!

One of the points I make about the structural focalization technique is that, because it is all so nicely structurally inductive, it can be formalized in Twelf. As part of a separate project, I've now also repeated the whole structural focalization development in Agda! The code is available from GitHub. While a structural focalization proof has some more moving parts than a simple cut-and-identity proof, it also has one significant advantage over every Agda proof of cut admissibility that I'm aware of: it requires no extra structural metrics beyond normal structural induction! (My favorite structural metric is the totally nameless representation, but there are other ways of threading that needle, including, presumably, these "sized types" that everyone seems to talk about.)

In regular, natural-deduction substitution, you can get away without structural metrics by proving the statement that if \(\Gamma \vdash A\) and \(\Gamma, A, \Gamma' \vdash C\) then \(\Gamma, \Gamma' \vdash C\); the extra "slack" given by \(\Gamma'\) means that you operate by structural induction on the second given derivation without ever needing to apply weakening or exchange. Most cut-elimination proofs are structured in such a way that you have to apply left commutative and right commutative cuts on both of the given derivations, making this process tricky at the best; I've never gotten it to work at all, but you might be able to do something like "if \(\Gamma, \Gamma' \longrightarrow A\) and \(\Gamma, A, \Gamma'' \longrightarrow C\) then \(\Gamma, \Gamma', \Gamma'' \longrightarrow C\)." If someone can make this work let me know!

A focused sequent calculus, on the other hand, has three separate phases of substitution. The first phase is principal substitution, where the type gets smaller and you can do whatever you want to the derivations, including weakening them. The second phase is rightist substitution, which acts much like natural-deduction substitution, and where you can similarly get away with adding "slack" to the second derivation. The third phase is leftist substitution, and you can get by in this phase by adding "slack" to the first derivation: the leftist cases read something like "if \(\Gamma, \Gamma' \longrightarrow A\) and \(\Gamma, A \longrightarrow C\) then \(\Gamma, \Gamma' \longrightarrow C\)."

In Structural focalization, I note that the structural focalization technique could be seen as a really freaking complicated way of proving the cut and identity for an unfocused sequent calculus. But in Agda, there's a reason you might actually want to do things the "long way" - not only do you have something better when you finish (a focalization result), but you get cut and identity without needing an annoying structural metric.

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