Results 1  10
of
31
A callbyname lambdacalculus machine
 Higher Order and Symbolic Computation
"... We present, in this paper, a particularly simple lazy machine which runs programs written in λcalculus. It was introduced by the present writer more than twenty years ago. It has been, since, used and implemented ..."
Abstract

Cited by 39 (0 self)
 Add to MetaCart
(Show Context)
We present, in this paper, a particularly simple lazy machine which runs programs written in λcalculus. It was introduced by the present writer more than twenty years ago. It has been, since, used and implemented
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
Realizability algebras: a program to well order R
, 2010
"... When transforming mathematical proofs into programs, the main problem is naturally due to the axioms: indeed, it has been a long time since we know how to transform a proof in pure (i.e. without axioms) intuitionistic logic, even at second order [2, 7, 4]. ..."
Abstract

Cited by 13 (7 self)
 Add to MetaCart
(Show Context)
When transforming mathematical proofs into programs, the main problem is naturally due to the axioms: indeed, it has been a long time since we know how to transform a proof in pure (i.e. without axioms) intuitionistic logic, even at second order [2, 7, 4].
Syntax vs. semantics: a polarized approach
 Theoretical Computer Science
, 2005
"... We present a notion of sliced proofnets for the polarized fragment of Linear Logic and a corresponding game model. We show that the connection between them is very strong through an equivalence of categories (this contains soundness, full completeness and faithful completeness). An important topic ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
(Show Context)
We present a notion of sliced proofnets for the polarized fragment of Linear Logic and a corresponding game model. We show that the connection between them is very strong through an equivalence of categories (this contains soundness, full completeness and faithful completeness). An important topic in the recent developments of denotational semantics has been the quest for stronger and stronger connections between the syntactical systems and the denotational models. Work towards bringing the two notions closer has come from both sides, and can be seen as an attempt to solve the general question “what is a proof?”. Full abstraction and full completeness (see [1, 8]) results have been initiated with game semantics [1, 2, 15] and come with models containing only elements definable by the syntax. These results have been mainly obtained in the last ten years for fragments of linear logic (for example MLL with and without MIX [1, 14, 25, 5, 6], MALL [4], ILL [19], LLP [21],...) and for extensions of PCF (for example PCF [2, 15], µPCF [18], Idealized Algol [3],...). This full completeness property can be considered as a measurement of the precision of the semantics (whatever the syntax might be). On the other side, the syntactical settings for logical systems have evolved progressively: sequent
Disjunctive Normal Forms and Local Exceptions
, 2003
"... All classical λterms typable with disjunctive normal forms are shown to share a common computational behavior: they implement a local exception handling mechanism whose exact workings depend on the tautology. Equivalent and more efficient control combinators are described through a speci ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
All classical &lambda;terms typable with disjunctive normal forms are shown to share a common computational behavior: they implement a local exception handling mechanism whose exact workings depend on the tautology. Equivalent and more efficient control combinators are described through a specialized sequent calculus and shown to be correct.
Realizability : a machine for analysis and set theory
, 2006
"... In this tutorial, we introduce the CurryHoward (proofprogram) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this. ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
In this tutorial, we introduce the CurryHoward (proofprogram) correspondence which is usually restricted to intuitionistic logic. We explain how to extend this correspondence to the whole of mathematics and we build a simple suitable machine for this.
Proofs, programs, processes
"... Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming la ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
Abstract. We study a realisability interpretation for inductive and coinductive definitions and discuss its application to program extraction from proofs. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell. Programs extracted from proofs using coinduction can be understood as perpetual processes producing infinite streams of data. Typical applications of such processes are computations in exact real arithmetic. As an example we show how to extract a program computing the average of two real numbers w.r.t. to the binary signed digit representation. 1
Totality in arena games
, 2009
"... We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We tackle the problem of preservation of totality by composition in arena games. We first explain how this problem reduces to a finiteness theorem on what we call pointer structures, similar to the parity pointer functions of Harmer, Hyland & Melliès and the interaction sequences of Coquand. We discuss how this theorem relates to normalization of linear head reduction in simplytyped λcalculus, leading us to a semantic realizability proof à la Kleene of our theorem. We then present another proof of a more combinatorial nature. Finally, we discuss the exact class of strategies to which our theorems apply.
Realizability with constants
 Workshop on Formal Methods and Security
, 2003
"... ..."
(Show Context)
Getting results from programs extracted from classical proofs
, 2002
"... We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We present a new method to extract from a classical proof of ∀x(I[x] → ∃y(O[y] ∧ S[x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is verified on a non trivial example: a proof of Dickson’s lemma.