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Permutability of Proofs in Intuitionistic Sequent Calculi
, 1996
"... We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the sa ..."
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Cited by 23 (4 self)
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We prove a folklore theorem, that two derivations in a cutfree sequent calculus for intuitionistic propositional logic (based on Kleene's G3) are interpermutable (using a set of basic "permutation reduction rules" derived from Kleene's work in 1952) iff they determine the same natural deduction. The basic rules form a confluent and weakly normalising rewriting system. We refer to Schwichtenberg's proof elsewhere that a modification of this system is strongly normalising. Key words: intuitionistic logic, proof theory, natural deduction, sequent calculus. 1 Introduction There is a folklore theorem that two intuitionistic sequent calculus derivations are "really the same" iff they are interpermutable, using permutations as described by Kleene in [13]. Our purpose here is to make precise and prove such a "permutability theorem". Prawitz [18] showed how intuitionistic sequent calculus derivations determine natural deductions, via a mapping ' from LJ to NJ (here we consider only ...
On the unusual effectiveness of Logic in computer science
 Bulletin of Symbolic Logic
"... Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effecti ..."
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Cited by 7 (0 self)
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Effectiveness of Mathematics in the Natural Sciences [Wig60]. This paper can be construed as an examination and affirmation of Galileo’s tenet that “The book of nature is written in the language of mathematics”. To this effect, Wigner presented a large number of examples that demonstrate the effectiveness of
Symmetry and Interactivity in Programming
 Bulletin of Symbolic Logic
, 2001
"... We recall some of the early occurrences of the notions of interactivity and symmetry in the operational and denotational semantics of programming languages. We suggest some connections with ludics. ..."
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Cited by 5 (1 self)
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We recall some of the early occurrences of the notions of interactivity and symmetry in the operational and denotational semantics of programming languages. We suggest some connections with ludics.
An Introduction to Polymorphic Lambda Calculus
 Logical Foundations of Functional Programming
, 1994
"... Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or secondorder) typed lambda calculus was invented by JeanYves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that ..."
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Cited by 4 (0 self)
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Introduction to the Polymorphic Lambda Calculus John C. Reynolds Carnegie Mellon University December 23, 1994 The polymorphic (or secondorder) typed lambda calculus was invented by JeanYves Girard in 1971 [11, 10], and independently reinvented by myself in 1974 [24]. It is extraordinary that essentially the same programming language was formulated independently by the two of us, especially since we were led to the language by entirely different motivations. In my own case, I was seeking to extend conventional typed programming languages to permit the definition of "polymorphic" procedures that could accept arguments of a variety of types. I started with the ordinary typed lambda calculus and added the ability to pass types as parameters (an idea that was "in the air" at the time, e.g. [4]). For example, as in the ordinary typed lambda calculus one can write f int!int : x int : f(f (x)) to denote the "doubling" function for the type int, which accepts a function from integers
Scope Ambiguities through the mirror
 John Bejamins Publishing Company. vol
"... In this paper we look at the interpretation of Quantifier Phrases from the perspective of Symmetric Categorial Grammar. We show how the apparent mismatch between the syntactic and semantic behaviour of these expressions can be resolved in a typelogical system equipped with two Merge relations: one f ..."
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Cited by 2 (2 self)
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In this paper we look at the interpretation of Quantifier Phrases from the perspective of Symmetric Categorial Grammar. We show how the apparent mismatch between the syntactic and semantic behaviour of these expressions can be resolved in a typelogical system equipped with two Merge relations: one for syntactic units, and one for the evaluation contexts of the semantic values associated with these syntactic units. Keywords:
Soundness and Principal Contexts for a Shallow Polymorphic Type System based on Classical Logic
"... In this paper we investigate how to adapt the wellknown notion of MLstyle polymorphism (shallow polymorphism) to a term calculus based on a CurryHoward correspondence with classical sequent calculus, namely, theX icalculus. We show that the intuitive approach is unsound, and pinpoint the precise ..."
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Cited by 1 (0 self)
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In this paper we investigate how to adapt the wellknown notion of MLstyle polymorphism (shallow polymorphism) to a term calculus based on a CurryHoward correspondence with classical sequent calculus, namely, theX icalculus. We show that the intuitive approach is unsound, and pinpoint the precise nature of the problem. We define a suitably refined type system, and prove its soundness. We then define a notion of principal contexts for the type system, and provide an algorithm to compute these, which is proved to be sound and complete with respect to the type system. In the process, we formalise and prove correctness of generic unification, which generalises Robinson’s unification to shallowpolymorphic types. Key words: CurryHoward, classical logic, generic unification, principal types, cut elimination 1.