Functional Interpretations of Intuitionistic Linear Logic

Abstract

Abstract. We present three different functional interpretations of intuitionistic linear logic and show how these correspond to well-known functional interpretations of intuitionistic logic via embeddings of IL ω into ILL ω . The main difference from previous work of the second author is that in intuitionistic linear logic (as opposed to classical linear logic) the interpretations of !A are simpler and simultaneous quantifiers are no longer needed for the characterisation of the interpretations. We then compare our approach in developing these three proof interpretations with the one of de Paiva around the Dialectica category model of linear logic.

Proof interpretations with truth

“…More precisely, the q-variant of modified realizability corresponds to Girard's (·) * embedding, whereas the truth-variant corresponds to the (·) • embedding. Similar work was done for interpretations without truth in [5]. www.mlq-journal.org P r o o f. The proof is by induction on the structure of formulas.…”

“…The second author [15] has shown that modified realizability of IL ω can be decomposed into a modified realizability of classical linear logic together with Girard's embedding of IL ω into linear logic. Recently [5], this has been turned into a finer decomposition in terms of intuitionistic linear logic ILL ω . In this section we present a truth variant for this functional interpretation of ILL ω .…”

“…(restricted to atomic formulas A at , but can be generalised to arbitrary formulas). We assume only equality between booleans, for which we require the following axioms [5]:…”

“…The following definition presents (in a uniform way) both the modified realizability interpretation of intuitionistic linear logic [5] and its truth variant.…”

“…, A n (z) B(z). Since the truth interpretation coincides with the standard interpretation for pure ILL ω , we only need to check the rules for the modality !A (see [5] for the soundness proof for the pure fragment of ILL ω ).…”