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Extracting constructive content from classical logic via controllike reductions
 In Bezem and Groote [12
, 1993
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A semantic view of classical proofs  typetheoretic, categorical, and denotational characterizations (Extended Abstract)
 IN PROCEEDINGS OF LICS '96
, 1996
"... Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, class ..."
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Cited by 30 (2 self)
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Classical logic is one of the best examples of a mathematical theory that is truly useful to computer science. Hardware and software engineers apply the theory routinely. Yet from a foundational standpoint, there are aspects of classical logic that are problematic. Unlike intuitionistic logic, classical logic is often held to be nonconstructive, and so, is said to admit no proof semantics. To draw an analogy in the proofsas programs paradigm, it is as if we understand well the theory of manipulation between equivalent specifications (which we do), but have comparatively little foundational insight of the process of transforming one program to another that implements the same specification. This extended abstract outlines a semantic theory of classical proofs based on a variant of Parigot's λµcalculus [24], but presented here as a type theory. After reviewing the conceptual problems in the area and the potential benefits of such a theory, we sketch the key steps of our approach in ...
Dependent choices, ‘quote’ and the clock
 Th. Comp. Sc
, 2003
"... When using the CurryHoward correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an inst ..."
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Cited by 28 (10 self)
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When using the CurryHoward correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem: to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "substitution method (due to W. Ackermann), that for every theorem A (A prenex) of firstorder Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
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Cited by 8 (3 self)
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A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
A Strong Normalization Result for Classical Logic
 Annals of Pure and Applied Logic
, 1995
"... In this paper we give a strong normalization proof for a set of reduction rules for classical logic. These reductions, more general then the ones usually considered in literature, are inspired to the reductions of Felleisen's lambda calculus with continuations. 1 Introduction Recently, in the logic ..."
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Cited by 7 (0 self)
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In this paper we give a strong normalization proof for a set of reduction rules for classical logic. These reductions, more general then the ones usually considered in literature, are inspired to the reductions of Felleisen's lambda calculus with continuations. 1 Introduction Recently, in the logic and theoretical computer science community, there has been an ever growing interest in the computational features of classical logic. The problem on which research is beginning to focus now is not the theoretical possibility of having constructive content present in classical proofs, established in old and well known results, but the practical applicability of such results. It was Kreisel in [12] who first pinpointed the presence of constructive content in classical proofs by proving the equality of the sets of \Sigma 0 1 sentences provable respectively in intuitionistic and classical logic. Friedman in [7] showed how to get the computational content of a classical proof of a \Sigma 0 1 ...
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how ..."
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Cited by 5 (2 self)
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
JACQUES HERBRAND: LIFE, LOGIC, AND AUTOMATED DEDUCTION
"... The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1 ..."
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The lives of mathematical prodigies who passed away very early after groundbreaking work invoke a fascination for later generations: The early death of Niels Henrik Abel (1802–1829) from ill health after a sled trip to visit his fiancé for Christmas; the obscure circumstances of Evariste Galois ’ (1811–1832) duel; the deaths of consumption of Gotthold Eisenstein (1823–1852) (who sometimes lectured his few students from his bedside) and of Gustav Roch (1839–1866) in Venice; the drowning of the topologist Pavel Samuilovich Urysohn (1898–1924) on vacation; the burial of Raymond Paley (1907–1933) in an avalanche at Deception Pass in the Rocky Mountains; as well as the fatal imprisonment of Gerhard Gentzen (1909–1945) in Prague1 — these are tales most scholars of logic and mathematics have heard in their student days. Jacques Herbrand, a young prodigy admitted to the École Normale Supérieure as the best student of the year1925, when he was17, died only six years later in a mountaineering accident in La Bérarde (Isère) in France. He left a legacy in logic and mathematics that is outstanding.
"Classical" programmingwithproofs in $\lambda^{Sym}_{PA}$: an analysis of nonconfluence
"... . Sym PA is a natural deduction system for Peano Arithmetic that was developed in order to provide a basis for the programmingwith proofs paradigm in a classical logic setting. In the paper we analyze one of its main features: nonconfluence. After looking at which rules can cause nonconfluence ..."
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. Sym PA is a natural deduction system for Peano Arithmetic that was developed in order to provide a basis for the programmingwith proofs paradigm in a classical logic setting. In the paper we analyze one of its main features: nonconfluence. After looking at which rules can cause nonconfluence, we develop in the system a formal proof for a formula that can be seen as a simple but meaningful program specification. The computational behaviour of the corresponding term will be analysed by interpreting it as a (higherorder communicating) process formed by distinct subprocesses which cooperate in different ways, producing different results, according to the reduction strategy used. We also show how to restrict the system in order to get confluence without loosing its computational features. The restricted system enables us to argue for the expressive power of symmetric and nondeterministic calculi like Sym PA . 1 Introduction The possibility of extracting constructive content f...
unknown title
, 2005
"... Le contenu computationnel des preuves: Nocounterexample interpretation et spécification des théorèmes de l’arithmétique mémoire sous la direction de JeanLouis Krivine ..."
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Le contenu computationnel des preuves: Nocounterexample interpretation et spécification des théorèmes de l’arithmétique mémoire sous la direction de JeanLouis Krivine