Proof interpretations with truth

Abstract

Key words Modified realizability with truth and q-variant, intuitionistic linear logic, Aczel and Kleene slashes. MSC (2010) 03F07, 03F10, 03F25This article systematically investigates so-called "truth variants" of several functional interpretations. We start by showing a close relation between two variants of modified realizability, namely modified realizability with truth and q-modified realizability. Both variants are shown to be derived from a single "functional interpretation with truth" of intuitionistic linear logic. This analysis suggests that several functional interpretations have truth and q-variants. These variants, however, require a more involved modification than the ones previously considered. Following this lead we present truth and q-variants of the Diller-Nahm interpretation, the bounded modified realizability and the bounded functional interpretation.

Functional Interpretations of Intuitionistic Linear Logic

“…In particular, Lemma 4.3 was first observed by Gaspar, and also appears in [5]. Many thanks also to the anonymous referees for the detailed revision and comments that so much improved the final version of this paper.…”

Functional Interpretations of Intuitionistic Linear Logic

“…Proof. Assertion (i) can be derived from axioms (4) and (6); (ii) follows easily from axiom 1.; (iii) can be deduced from axiom (5) and the forward implications in (iv) follow immediately from item (ii) and the inverse implications can easily be deduced using (iii). The forward implication in assertion (v) can be derived using assertions (i) and (iv), the other implication being trivial.…”

“…The second author [10,11,12,13] has recently shown how different functional interpretations of intuitionistic logic can be factored into a uniform family of interpretations of classical linear logic combined with Girard's standard embedding (·) * of intuitionistic logic into linear logic (see also [5]). In the symmetric context of classical linear logic each formula A is associated with a simultaneous one-move two-player game |A| x y .…”

A parametrised functional interpretation of Heyting arithmetic

“…These questions were addressed by the second author and various co-authors in a series of papers on unifying functional interpretations. Starting with a unification of interpretations of intuitionistic logic [21], which was followed by various analysis of functional interpretations within the finer setting of linear logic [13,22,23,24], a proposal on how functional interpretations could actually be combined in the so-called hybrid functional interpretations [17,25], and the inclusion of truth variants in the unification [15].…”

“…With this, one is able to define a "unifying functional interpretation" which when instantiated gave rise to several of the existing functional interpretations, including the Dialectica interpretation, modified realizability (its q-and truth variants), Stein's family of interpretations, and the Diller-Nahm interpretation [21,24]. This process also led to the design of a "Diller-Nahm with truth" interpretation [15], which at the time was thought not to be possible.…”