Strong normalization for second order classical natural deduction

Parigot, Michel

doi:10.1109/lics.1993.287602

Cited by 85 publications

(74 citation statements)

References 6 publications

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“…Proof. As shown in the next section, the λ nC -tp calculus corresponds to Parigot's call-by-name λµ calculus, which is confluent and strongly normalizable (Py, 1998;Parigot, 1993b;Parigot, 1997). The λ vC -tp calculus corresponds to a subset of the λµ v calculus of Ong and Stewart (1997) which is strongly normalizing.…”

Section: Reduction Semanticsmentioning

confidence: 97%

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A proof-theoretic foundation of abortive continuations

Ariola

Herbelin

Sabry

2007

Higher-Order Symb Comput

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Abstract. We give an analysis of various classical axioms and characterize a notion of minimal classical logic that enforces Peirce's law without enforcing Ex Falso Quodlibet. We show that a "natural" implementation of this logic is Parigot's classical natural deduction. We then move on to the computational side and emphasize that Parigot's λµ corresponds to minimal classical logic. A continuation constant must be added to λµ to get full classical logic. The extended calculus is isomorphic to a syntactical restriction of Felleisen's theory of control that offers a more expressive reduction semantics. This isomorphic calculus is in correspondence with a refined version of Prawitz's natural deduction.

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Section: Reduction Semanticsmentioning

confidence: 97%

“…⊥ e is considered at the same level as Passivate). Parigot's normalization proof for minimal classical natural deduction (Parigot, 1993b;Parigot, 1997) applies also for full classical natural deduction. THEOREM 14 (Parigot).…”

Section: Minimal Classical Logicmentioning

confidence: 99%

A proof-theoretic foundation of abortive continuations

Ariola

Herbelin

Sabry

2007

Higher-Order Symb Comput

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“…We say that Red is a reducibility family if X ⊆ A for all A ∈ Red . There are essentially three kinds of reducibility families: Tait's saturated sets [17], Girard's reducibility candidates [7] and biorthogonals [13]. In this paper we focus on the last two.…”

Section: Example 21 ([18]mentioning

confidence: 99%

On the Values of Reducibility Candidates

Riba

2009

Lecture Notes in Computer Science

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Abstract. The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambda-calculus, to break strong normalization as well. Similarly, the straightforward elimination of implicit existential types breaks subject reduction. We propose elimination rules for union types and implicit existential quantification which use a form call-by-value issued from Girard's reducibility candidates. We show that these rules remedy the above mentioned difficulties, for strong normalization and, for the existential quantification, for subject reduction as well. Moreover, for extensions of the lambda-calculus based on intuitionistic logic, we show that the obtained existential quantification is equivalent to its usual impredicative encoding w.r.t. provability in realizability models built from reducibility candidates and biorthogonals.

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“…[18] and [20]) The F D2 type system has the following properties : 1) Type is preserved during reduction. 2) Typable λµ-terms are strongly normalizable.…”

Section: Pure and Typed λµ-Calculusmentioning

confidence: 99%

Mixed logic and storage operators

Nour¹

2000

Archive for Mathematical Logic

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In 1990 J-L. Krivine introduced the notion of storage operators. They are λ-terms which simulate call-by-value in the call-by-name strategy and they can be used in order to modelize assignment instructions. J-L. Krivine has shown that there is a very simple second order type in AF 2 type system for storage operators using Gődel translation of classical to intuitionistic logic. In order to modelize the control operators, J-L. Krivine has extended the system AF 2 to the classical logic. In his system the property of the unicity of integers representation is lost, but he has shown that storage operators typable in the system AF 2 can be used to find the values of classical integers. In this paper, we present a new classical type system based on a logical system called mixed logic. We prove that in this system we can characterize, by types, the storage operators and the control operators. We present also a similar result in the M. Parigot's λµ-calculus.

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Strong normalization for second order classical natural deduction

Cited by 85 publications

References 6 publications

A proof-theoretic foundation of abortive continuations

A proof-theoretic foundation of abortive continuations

On the Values of Reducibility Candidates

Mixed logic and storage operators

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