Hybrid realizability for intuitionistic and classical choice

International audienceIn intuitionistic realizability like Kleene's or Kreisel's, the axiom of choice is trivially realized. It is even provable in Martin-Löf's intu-itionistic type theory. In classical logic, however, even the weaker axiom of countable choice proves the existence of non-computable functions. This logical strength comes at the price of a complicated computational interpretation which involves strong recursion schemes like bar recursion. We take the best from both worlds and define a realizability model for arithmetic and the axiom of choice which encompasses both intuitionistic and classical reasoning. In this model two versions of the axiom of choice can co-exist in a single proof: intuitionistic choice and classical countable choice. We interpret intuitionistic choice efficiently, however its premise cannot come from classical reasoning. Conversely, our version of classical choice is valid in full classical logic, but it is restricted to the countable case and its realizer involves bar recursion. Having both versions allows us to obtain efficient extracted programs while keeping the provability strength of classical logic

Constructive Mathematics in Theory and Programming Practice

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics(BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focuses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH

Polarizing Double Negation Translations

Double-negation translations are used to encode and decode classical proofs in intuitionistic logic. We show that, in the cut-free fragment, we can simplify the translations and introduce fewer negations. To achieve this, we consider the polarization of the formul{\ae}{} and adapt those translation to the different connectives and quantifiers. We show that the embedding results still hold, using a customized version of the focused classical sequent calculus. We also prove the latter equivalent to more usual versions of the sequent calculus. This polarization process allows lighter embeddings, and sheds some light on the relationship between intuitionistic and classical connectives

Reverse mathematics and uniformity in proofs without excluded middle

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the \textit{Dialectica} interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis.Comment: Accepted, Notre Dame Journal of Formal Logi

The Fan Theorem, its strong negation, and the determinacy of games

IIn the context of a weak formal theory called Basic Intuitionistic Mathematics $\mathsf{BIM}$, we study Brouwer's Fan Theorem and a strong negation of the Fan Theorem, Kleene's Alternative (to the Fan Theorem). We prove that the Fan Theorem is equivalent to contrapositions of a number of intuitionistically accepted axioms of countable choice and that Kleene's Alternative is equivalent to strong negations of these statements. We also discuss finite and infinite games and introduce a constructively useful notion of determinacy. We prove that the Fan Theorem is equivalent to the Intuitionistic Determinacy Theorem, saying that every subset of Cantor space is, in our constructively meaningful sense, determinate, and show that Kleene's Alternative is equivalent to a strong negation of a special case of this theorem. We then consider a uniform intermediate value theorem and a compactness theorem for classical propositional logic, and prove that the Fan Theorem is equivalent to each of these theorems and that Kleene's Alternative is equivalent to strong negations of them. We end with a note on a possibly important statement, provable from principles accepted by Brouwer, that one might call a Strong Fan Theorem.Comment: arXiv admin note: text overlap with arXiv:1106.273

On Affine Logic and {\L}ukasiewicz Logic

The multi-valued logic of {\L}ukasiewicz is a substructural logic that has been widely studied and has many interesting properties. It is classical, in the sense that it admits the axiom schema of double negation, [DNE]. However, our understanding of {\L}ukasiewicz logic can be improved by separating its classical and intuitionistic aspects. The intuitionistic aspect of {\L}ukasiewicz logic is captured in an axiom schema, [CWC], which asserts the commutativity of a weak form of conjunction. This is equivalent to a very restricted form of contraction. We show how {\L}ukasiewicz Logic can be viewed both as an extension of classical affine logic with [CWC], or as an extension of what we call \emph{intuitionistic} {\L}ukasiewicz logic with [DNE], intuitionistic {\L}ukasiewicz logic being the extension of intuitionistic affine logic by the schema [CWC]. At first glance, intuitionistic affine logic seems very weak, but, in fact, [CWC] is surprisingly powerful, implying results such as intuitionistic analogues of De Morgan's laws. However the proofs can be very intricate. We present these results using derived connectives to clarify and motivate the proofs and give several applications. We give an analysis of the applicability to these logics of the well-known methods that use negation to translate classical logic into intuitionistic logic. The usual proofs of correctness for these translations make much use of contraction. Nonetheless, we show that all the usual negative translations are already correct for intuitionistic {\L}ukasiewicz logic, where only the limited amount of contraction given by [CWC] is allowed. This is in contrast with affine logic for which we show, by appeal to results on semantics proved in a companion paper, that both the Gentzen and the Glivenko translations fail.Comment: 28 page

Ecumenical modal logic

The discussion about how to put together Gentzen's systems for classical and intuitionistic logic in a single unified system is back in fashion. Indeed, recently Prawitz and others have been discussing the so called Ecumenical Systems, where connectives from these logics can co-exist in peace. In Prawitz' system, the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation, and the constant for the absurd, but they would each have their own existential quantifier, disjunction, and implication, with different meanings. Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties. In a recent work, Ecumenical sequent calculi and a nested system were presented, and some very interesting proof theoretical properties of the systems were established. In this work we extend Prawitz' Ecumenical idea to alethic K-modalities