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Section: New Results

Computation in Focused Intuitionistic Logic

Participants : Taus Brock-Nannestad, Nicolas Guenot, Daniel Gustafsson.

Focusing is a proof-theoretical tecnique for eliminating unnecessary nondeterminism in proofs. Because it cuts down on nondeterminism, focusing is particularly useful for directing proof search. Focusing thus plays a key role in explaining the meaning and behaviour of logic programs.

Despite this success in clarifying the operational semantics of logic programming, focusing has not been as widely studied in the Curry-Howard style “proofs as programs” interpretation. Early results in this area established that $\lambda$-calculi associated with the focused calculi LJT and LJQ had evaluation strategies corresponding to call-by-name and call-by-value respectively. For the LJF calculus — which contains both LJT and LJQ as fragments — no such correspondence was known.

In [27] (PPDP'15) we show how a proof-term assignment to (a variant of) Liang and Miller's focused sequent calculus LJF permits a uniform treatment of the call-by-value and call-by-name reduction strategies of the $\lambda$-calculus, as well as combinations of these strategies. Additionally, we show how to extract an abstract machine from LJF by considering machine states as certain configurations of instances of the cut rule. The aforementioned correspondence extends to this setting, and we show that well-known abstract machines for call-by-value and call-by-name are in fact exactly the abstract machines that one gets when considering certain fragments of LJF.

In the seminal work of Paul Blain Levy, the call-by-push-value language was introduced as a way of subsuming the call-by-value and call-by-name strategies of the $\lambda$-calculus. It was later on conjectured that call-by-push-value was simply implementing a notion of focusing, and indeed this turns out to be the case, as we show in the aforementioned paper.