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- Analysis in weak systemsPublication . Fernandes, António M.; Ferreira, Fernando; Ferreira, GildaThe authors survey and comment their work on weak analysis. They describe the basic set-up of analysis in a feasible second-order theory and consider the impact of adding to it various forms of weak Konig's lemma. A brief discussion of the Baire categoricity theorem follows. It is then considered a strengthening of feasibility obtained (fundamentally) by the addition of a counting axiom and showed how it is possible to develop Riemann integration in the stronger system. The paper finishes with three questions in weak analysis.
- The computational content of atomic polymorphismPublication . Ferreira, Gilda; Vasconcelos, Vasco TWe show that the number-theoretic functions de nable in the atomic polymorphic system (Fat) are exactly the extended polynomials. Two proofs of the above result are presented: one reducing the functions' de n- ability problem in Fat to de nability in the simply typed lambda-calculus and other directly adapting Helmut Schwichtenberg's strategy for de nability in the simply typed lambda-calculus to the atomic polymorphic setting. The uniformity granted in the polymorphic system, when compared with the simply typed lambda-calculus, is emphasized.
- Techniques in weak analysis for conservation resultsPublication . Fernandes, António; Ferreira, Fernando; Ferreira, GildaWe review and describe the main techniques for setting up systems of weak analysis, i.e. formal systems of second-order arithmetic related to subexponential classes of computational complexity. These involve techniques of proof theory (e.g., Herbrand’s theorem and the cut-elimination theorem) and model theoretic techniques like forcing. The techniques are illustrated for the particular case of polytime computability. We also include a brief section where we list the known results in weak analysis.
- The Russell-Prawitz embedding and the atomization of universal instantiationPublication . Espírito Santo, José; Ferreira, GildaGiven the recent interest in the fragment of system F where universal instantiation is restricted to atomic formulas, a fragment nowadays named system Fat, we study directly in system F new conversions whose purpose is to enforce that restriction. We show some benefits of these new atomization conversions: (1) They help achieving strict simulation of proof reduction by means of the Russell-Prawitz embedding of IPC into system F; (2) They are not stronger than a certain “dinaturality” conversion known to generate a consistent equality of proofs; (3) They provide the bridge between the Russell-Prawitz embedding and another translation, due to the authors, of IPC directly into system Fat; (4) They give means for explaining why the Russell-Prawitz translation achieves strict simulation whereas the translation into Fat does not.
- Instantiation overflowPublication . Dinis, Bruno; Ferreira, GildaThe well-known embedding of full intuitionistic propositional calculus into the atomic polymorphic system Fat is possible due to the intriguing phenomenon of instantiation overflow. Instantiation overflow ensures that (in Fat) we can instantiate certain universal formulas by any formula of the system, not necessarily atomic. Until now only three types in Fat were identi ed with such property: the types that result from the Prawitz translation of the propositional connectives (\bot,\wedge, \vee) into Fat (or Girard's system F). Are there other types in Fat with instantiation overflow? In this paper we show that the answer is yes and we isolate a class of formulas with such property.
- Bounded theories for polyspace computabilityPublication . Bianconi, Ricardo; Ferreira, Gilda; Silva, EmmanuelWe present theories of bounded arithmetic and weak analysis whose provably total functions (with appropriate graphs) are the polyspace computable functions. More precisely, inspired in Ferreira’s systems PTCA, Sigma^b_1-NIA and BTFA in the polytime framework, we propose analogue theories concerning polyspace computability. Since the techniques we employ in the characterization of PSPACE via formal systems (e.g. Herbrand’s theorem, cut-elimination theorem and the expansion of models) are similar to the ones involved in the polytime setting, we focus on what is specific of polyspace and explains the lift from PTIME to PSPACE.
- Rasiowa–Harrop disjunction propertyPublication . Ferreira, GildaWe show that there is a purely proof-theoretic proof of the Rasiowa–Harrop disjunction property for the full intuitionistic propositional calculus (IPC), via natural deduction, in which commuting conversions are not needed. Such proof is based on a sound and faithful embedding of IPC into an atomic polymorphic system. This result strengthens a homologous result for the disjunction property of IPC (presented in a recent paper co-authored with Fernando Ferreira) and answers a question then posed by Pierluigi Minari.
- η-conversions of IPC implemented in atomic FPublication . Ferreira, GildaIt is known that the β-conversions of the full intuitionistic propositional calculus (IPC) translate into βη-conversions of the atomic polymorphic calculus Fat. Since Fat enjoys the property of strong normalization for βη-conversions, an alternative proof of strong normalization for IPC considering β-conversions can be derived. In the present article, we improve the previous result by analysing the translation of the η-conversions of the latter calculus into a technical variant of the former system (the atomic polymorphic calculus Fat^∧_at). In fact, from the strong normalization of Fat^∧_at we can derive the strong normalization of the full intuitionistic propositional calculus considering all the standard (β and η) conversions.
- Atomic polymorphismPublication . Ferreira, Fernando; Ferreira, GildaIt has been known for six years that the restriction of Girard’s polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait’s method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the full intuitionistic propositional calculus translates into one or more βη-reduction steps in the restricted Girard system. As a consequence, we obtain a novel and perspicuous proof of the strong normalization property for the full intuitionistic propositional calculus. It is noticed that this novel proof bestows a crucial role to η-conversions.
- Atomic polymorphism and the existence propertyPublication . Ferreira, GildaWe present a purely proof-theoretic proof of the existence property for the full intuitionistic first-order predicate calculus, via natural deduction, in which commuting conversions are not needed. Such proof illustrates the potential of an atomic polymorphic system with only three generators of formulas – conditional and first and second-order universal quantifiers – as a tool for proof-theoretical studies.