Computation in focused intuitionistic logic

Abstract

We investigate the control of evaluation strategies in a variant of the λ-calculus derived through the Curry-Howard correspondence from LJF, a sequent calculus for intuitionistic logic implementing the focusing technique. The proof theory of focused intuitionistic logic yields a single calculus in which a number of known λ-calculi appear as subsystems obtained by restricting types to a certain fragment of LJF. In particular, standard λ-calculi as well as the call-by-pushvalue calculus are analysed using this framework, and we relate cut elimination for LJF to a new abstract machine subsuming wellknown machines for these different strategies.

Multi-focused cut elimination

“…For standard linear logic, there is a compact argument (Chaudhuri 2008), and the intuitionistic and classical cases have also been studied (Liang and Miller 2009). Note that a more detailed analysis of cut elimination in the intuitionistic system LJF leads to a highly principled proof, which is not only useful to prove the focalisation result in a structural fashion (Simmons 2014) but also yields a natural computational interpretation which involves reduction strategies in the λ-calculus (Brock-Nannestad et al 2015). Similarly, the proof can be made principled enough in a focused sequent calculus for standard linear logic to enjoy a simple computational interpretation (Brock-Nannestad and Guenot 2015b).…”

Sequent Calculus and Equational Programming

“…This is not absolutely necessary, but it clarifies the definition of a focused system by linking the focus and blur rules to actual connectives. Note that this was also used in the presentation of a computational interpretation of the full LJF system [7]. The rules we use in this system are shown in Figure 1, where the term assignment is indicated in red and several turnstiles are used to distinguish an inversion phase from a focused phase .…”

“…and it is reminiscent of the pattern using the same syntax in Haskell -which is meant to exist in Agda as well, but this not yet implemented. However, in Haskell, this is restricted to the form x @ p so that it can only serve to name an assumption before decomposing it, and we could allow for such a use by avoiding maximal inversion, which is not strictly necessary in a focused system [7]. This rule is not necessary for the completeness of the calculus, and there are other ways to obtain the same result.…”

Proofs as Terms, Terms as Graphs

“…Various focused proof systems have been proposed for LJ [21,18,23] and LK [15,23]. Several authors also designed term calculi for LJT [21], LJQ [18], and LJF [10]. In [27], Miller and Wu use synthetic inference rules built using the focused proof system LJF to study term structures.…”