Unifying Functional Interpretations: Past and Future

This article surveys work done in the last six years on the unification of various functional interpretations including G\"odel's dialectica interpretation, its Diller-Nahm variant, Kreisel modified realizability, Stein's family of functional interpretations, functional interpretations "with truth", and bounded functional interpretations. Our goal in the present paper is twofold: (1) to look back and single out the main lessons learnt so far, and (2) to look forward and list several open questions and possible directions for further research.Comment: 18 page

Almost disjoint families and “never” cardinal invariants

summary:We define two cardinal invariants of the continuum which arise naturally from combinatorially and topologically appealing properties of almost disjoint families of sets of the natural numbers. These are the never soft and never countably paracompact numbers. We show that these cardinals must both be equal to $\omega_1$ under the effective weak diamond principle $\diamondsuit (\omega,\omega,<)$, answering questions of da Silva S.G., On the presence of countable paracompactness, normality and property $(a)$ in spaces from almost disjoint families, Questions Answers Gen. Topology 25(2007), no. 1, 1--18, and give some information about the strength of this principle

A Dialectica-Like Interpretation of a Linear MSO on Infinite Words

We devise a variant of Dialectica interpretation of intuitionistic linear logic for Open image in new window, a linear logic-based version MSO over infinite words. Open image in new window was known to be correct and complete w.r.t. Church’s synthesis, thanks to an automata-based realizability model. Invoking Büchi-Landweber Theorem and building on a complete axiomatization of MSO on infinite words, our interpretation provides us with a syntactic approach, without any further construction of automata on infinite words. Via Dialectica, as linear negation directly corresponds to switching players in games, we furthermore obtain a complete logic: either a closed formula or its linear negation is provable. This completely axiomatizes the theory of the realizability model of Open image in new window. Besides, this shows that in principle, one can solve Church’s synthesis for a given ∀∃ -formula by only looking for proofs of either that formula or its linear negation

Functional Interpretations of Intuitionistic Linear Logic

We present three different functional interpretations of intuitionistic linear logic ILL and show how these correspond to well-known functional interpretations of intuitionistic logic IL via embeddings of IL into ILL. 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

Linear logic for constructive mathematics

We show that numerous distinctive concepts of constructive mathematics arise automatically from an interpretation of "linear higher-order logic" into intuitionistic higher-order logic via a Chu construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of classical concepts using the choice between multiplicative and additive linear connectives. Linear logic thus systematically "constructivizes" classical definitions and deals automatically with the resulting bookkeeping, and could potentially be used directly as a basis for constructive mathematics in place of intuitionistic logic.Comment: 39 page