Teoria da demonstração: abordagem lógico-computacional

Funder

Organizational Unit

Publications

A herbrandized functional interpretation of classical first-order logic

Publication . Ferreira, Fernando; Ferreira, Gilda

We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.

2017Journal article

Open access

Rasiowa–Harrop disjunction property

Publication . Ferreira, Gilda

We 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.

2017Journal article

Open access

η-conversions of IPC implemented in atomic F

Publication . Ferreira, Gilda

It 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.

2017Journal article

Open access

Analysis in weak systems

Publication . Fernandes, António M.; Ferreira, Fernando; Ferreira, Gilda

The 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.

2017Journal article

Open access

The computational content of atomic polymorphism

Publication . Ferreira, Gilda; Vasconcelos, Vasco T

We 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.

2019Journal article

Open access