From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps

Process Realizability

We develop a notion of realizability for Classical Linear Logic based on a concurrent process calculus.Comment: Appeared in Foundations of Secure Computation: Proceedings of the 1999 Marktoberdorf Summer School, F. L. Bauer and R. Steinbruggen, eds. (IOS Press) 2000, 167-18

Classical Logic with Mendler Induction

We investigate (co-)induction in Classical Logic under the propositions-as-types paradigm, considering propositional, second-order, and (co-)inductive types. Specifically, we introduce an extension of the Dual Calculus with a Mendler-style (co-)iterator that remains strongly normalizing under head reduction. We prove this using a non-constructive realizability argument.This is the author accepted manuscript. The final version is available from Springer via http://dx.doi.org/10.1007/978-3-319-27683-0_

On Tarski's fixed point theorem

A concept of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive and predicative version of Tarski's fixed point theorem is obtained.Comment: Proc. Amer. Math. Soc., to appea

On Extensions of AF2 with Monotone and Clausular (Co)inductive Definitions

This thesis discusses some extensions of second-order logic AF2 with primitive constructors representing least and greatest fixed points of monotone operators, which allow to define predicates by induction and coinduction. Though the expressive power of second-order logic has been well-known for a long time and suffices to define (co)inductive predicates by means of its (co)induction principles, it is more user-friendly to have a direct way of defining predicates inductively. Moreover recent applications in computer science oblige to consider also coinductive definitions useful for handling infinite objects, the most prominent example being the data type of streams or infinite lists. Main features of our approach are the use clauses in the (co)inductive definition mechanism, concept which simplifies the syntactic shape of the predicates, as well as the inclusion of not only (co)iteration but also primitive (co)recursion principles and in the case of coinductive definitions an inversion principle. For sake of generality we consider full monotone, and not only positive definitions, after all positivity is only used to ensure monotonicity. Working towards practical use of our systems we give them realizability interpretations where the systems of realizers are strongly normalizing extensions of the second-order polymorphic lambda calculus, system F in Curry-style, with (co)inductive types corresponding directly to the logical systems via the Curry-Howard correspondence. Such realizability interpretations are therefore not reductive: the definition of realizability for a (co)inductive definition is again a (co)inductive definition. As main application of realizability we extend the so-called programming-with-proofs paradigm of Krivine and Parigot to our logics, by means of which a correct program of the lambda calculus can be extracted from a proof in the logic

Quantitative Models and Implicit Complexity

We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by one of us), Light Affine Logic and Soft Affine Logic. The proofs are based on a common semantical framework which is merely instantiated in four different ways. The framework consists of an innovative modification of realizability which allows us to use resource-bounded computations as realisers as opposed to including all Turing computable functions as is usually the case in realizability constructions. For example, all realisers in the model for LFPL are polynomially bounded computations whence soundness holds by construction of the model. The work then lies in being able to interpret all the required constructs in the model. While being the first entirely semantical proof of polytime soundness for light logi cs, our proof also provides a notable simplification of the original already semantical proof of polytime soundness for LFPL. A new result made possible by the semantic framework is the addition of polymorphism and a modality to LFPL thus allowing for an internal definition of inductive datatypes.Comment: 29 page

Constructive set theory and Brouwerian principles

The paper furnishes realizability models of constructive Zermelo-Fraenkel set theory, CZF, which also validate Brouwerian principles such as the axiom of continuous choice (CC), the fan theorem (FT), and monotone bar induction (BIM), and thereby determines the proof-theoretic strength of CZF augmented by these principles. The upshot is that CZF+CC+FT possesses the same strength as CZF, or more precisely, that CZF+CC+FTis conservative over CZF for 02 statements of arithmetic, whereas the addition of a restricted version of bar induction to CZF (called decidable bar induction, BID) leads to greater proof-theoretic strength in that CZF+BID proves the consistency of CZF

Derived rules for predicative set theory: an application of sheaves

We show how one may establish proof-theoretic results for constructive Zermelo-Fraenkel set theory, such as the compactness rule for Cantor space and the Bar Induction rule for Baire space, by constructing sheaf models and using their preservation properties