8,874 research outputs found

    Proof-theoretical studies on the bounded functional interpretation

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    Tese de doutoramento, Matemática (Álgebra Lógica e Fundamentos), 2009, Universidade de Lisboa, Faculdade de CiênciasThis dissertation studies the bounded functional interpretation of Ferreira and Oliva. The work follows two different directions. We start by focusing on the generalization of the bounded functional interpretation to second-order arithmetic (a.k.a. analysis). This is accomplished via bar recursion, a well-founded form of recursion. We carry out explicitly the bounded functional interpretation of the (non-intuitionistic) law of the double negation shift with bar recursive functionals of finite type. As a consequence, we show that full numerical comprehension has bounded functional interpretation in the classical case. In the other direction, we extend the bounded functional interpretation with new base types, representing an abstract class of normed spaces. Some studies on the representation of the real numbers are carried out, as it is useful to have a representation which meshes well with the notion of majorizability. A majorizability theorem holds. We carry out the extension of the bounded functional interpretation to new base types and prove a soundness theorem with characteristic principles similar to the numerical case. We also extend the classical direct bounded functional interpretation of Peano arithmetic to new base types and prove the corresponding soundness theorem. The characteristic principles are also similar to the ones in the numerical case. In the classical setting, these prove that linear operators are automatically bounded and that Cauchy sequences (with a modulus of Cauchyness) of elements of the new base type do converge. Relying on the characteristic principles (and on a special form of choice), a logical version of the Baire category theorem of functional analysis is proved. As a consequence, we also prove logical versions of the Banach-Steinhaus and the open mapping theorems.Programa Operacional Ciência e Inovação 2010 (POCI 2010) e FS

    A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators

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    First, we reconstruct Wim Veldman's result that Open Induction on Cantor space can be derived from Double-negation Shift and Markov's Principle. In doing this, we notice that one has to use a countable choice axiom in the proof and that Markov's Principle is replaceable by slightly strengthening the Double-negation Shift schema. We show that this strengthened version of Double-negation Shift can nonetheless be derived in a constructive intermediate logic based on delimited control operators, extended with axioms for higher-type Heyting Arithmetic. We formalize the argument and thus obtain a proof term that directly derives Open Induction on Cantor space by the shift and reset delimited control operators of Danvy and Filinski

    Stratified Negation in Limit Datalog Programs

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    There has recently been an increasing interest in declarative data analysis, where analytic tasks are specified using a logical language, and their implementation and optimisation are delegated to a general-purpose query engine. Existing declarative languages for data analysis can be formalised as variants of logic programming equipped with arithmetic function symbols and/or aggregation, and are typically undecidable. In prior work, the language of limit programs\mathit{limit\ programs} was proposed, which is sufficiently powerful to capture many analysis tasks and has decidable entailment problem. Rules in this language, however, do not allow for negation. In this paper, we study an extension of limit programs with stratified negation-as-failure. We show that the additional expressive power makes reasoning computationally more demanding, and provide tight data complexity bounds. We also identify a fragment with tractable data complexity and sufficient expressivity to capture many relevant tasks.Comment: 14 pages; full version of a paper accepted at IJCAI-1

    Perspectives for proof unwinding by programming languages techniques

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    In this chapter, we propose some future directions of work, potentially beneficial to Mathematics and its foundations, based on the recent import of methodology from the theory of programming languages into proof theory. This scientific essay, written for the audience of proof theorists as well as the working mathematician, is not a survey of the field, but rather a personal view of the author who hopes that it may inspire future and fellow researchers

    Sequentiality vs. Concurrency in Games and Logic

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    Connections between the sequentiality/concurrency distinction and the semantics of proofs are investigated, with particular reference to games and Linear Logic.Comment: 35 pages, appeared in Mathematical Structures in Computer Scienc
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