Gilda Ferreira

On Various Negative Translations

odel, Gentzen, Kuroda and Krivine (in chronological order). In this paper we propose a framework for explaining how these different translations are related to each other. More precisely, we define a notion of a (modular) simplification starting from Kolmogorov translation, which leads to a partial order between different negative translations. In this derived ordering, Kuroda and Krivine are minimal elements. Two new minimal translations are introduced, with G¨ odel and Gentzen translations sitting in between Kolmogorov and one of these new translations.

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.

Commuting Conversions vs. the Standard Conversions of the “Good” Connectives

Commuting conversions were introduced in the natural deduction calculus as ad hoc devices for the purpose of guaranteeing the subformula property in normal proofs. In a well known book, Jean-Yves Girard commented harshly on these conversions, saying that ‘one tends to think that natural deduction should be modified to correct such atrocities.’ We present an embedding of the intuitionistic predicate calculus into a second-order predicative system for which there is no need for commuting conversions. Furthermore, we show that the redex and the conversum of a commuting conversion of the original calculus translate into equivalent derivations by means of a series of bidirectional applications of standard conversions.

Interpretability in Robinson's Q

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson’s theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson’s position. §

Functional interpretations of intuitionistic linear logic

We present three functional interpretations of intuitionistic linear logic and show how these correspond to well-known functional interpretations of intuitionistic logic via embeddings of ILω into ILLω. The main difference from previous work of the second author is that in intuitionistic linear logic the interpretations of !A are simpler (at the cost of an asymmetric interpretation of pure ILLω) and simultaneous quantifiers are no longer needed for the characterisation of the interpretations.

The Faithfulness of Fat: A Proof-Theoretic Proof

It is known that there is a sound and faithful translation of the full intuitionistic propositional calculus into the atomic polymorphic system Fat, a predicative calculus with only two connectives: the conditional and the second-order universal quantifier. The faithfulness of the embedding was established quite recently via a model-theoretic argument based in Kripke structures. In this paper we present a purely proof-theoretic proof of faithfulness. As an application, we give a purely proof-theoretic proof of the disjunction property of the intuitionistic propositional logic in which commuting conversions are not needed.

On bounded functional interpretations

Abstract Bounded functional interpretations are variants of functional interpretations where bounds (rather than precise witnesses) are extracted from proofs. These have been particularly useful in computationally interpreting non-computational principles such as weak Konig’s lemma. This paper presents a family of bounded functional interpretations — in the form of a parametrized interpretation — of both intuitionistic logic and (a fragment of) intuitionistic linear logic. We show how three different instantiations of the parameters give rise to three recently developed bounded interpretations: the bounded functional interpretation, bounded modified realizability and confined modified realizability.

Confined Modified Realizability

We present a refinement of the bounded modified realizability which provides both upper and lower bounds for witnesses. Our interpretation is based on a generalisation of Howard/Bezem’s notion of strong majorizability. We show how the bounded modified realizability coincides with (a weak version of) our interpretation in the case when least elements exist (e.g. natural numbers). The new interpretation, however, permits the extraction of more accurate bounds, and provides an ideal setting for dealing directly with data types whose natural ordering is not well-founded. Copyright line will be provided by the publisher

Harrington’s conservation theorem redone

Leo Harrington showed that the second-order theory of arithmetic WKL0 is