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Normal Forms and CutFree Proofs as Natural Transformations
 in : Logic From Computer Science, Mathematical Science Research Institute Publications 21
, 1992
"... What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax g ..."
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What equations can we guarantee that simple functional programs must satisfy, irrespective of their obvious defining equations? Equivalently, what nontrivial identifications must hold between lambda terms, thoughtof as encoding appropriate natural deduction proofs ? We show that the usual syntax guarantees that certain naturality equations from category theory are necessarily provable. At the same time, our categorical approach addresses an equational meaning of cutelimination and asymmetrical interpretations of cutfree proofs. This viewpoint is connected to Reynolds' relational interpretation of parametricity ([27], [2]), and to the KellyLambekMac LaneMints approach to coherence problems in category theory. 1 Introduction In the past several years, there has been renewed interest and research into the interconnections of proof theory, typed lambda calculus (as a functional programming paradigm) and category theory. Some of these connections can be surprisingly subtle. Here we a...
Course Notes in Typed Lambda Calculus
, 1998
"... this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those ..."
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this paper is clearly stated, after recalling how the logical connectives can be explained in term of the Sheffer connective: "We are led to the idea, which at first glance certainly appears extremely bold of attempting to eliminate by suitable reduction the remaining fundamental notions, those of proposition, propositional function, and variable, from those contexts in which we are dealing with completely arbitrary, logical general propositions . . . To examine this possibility more closely and to pursue it would be valuable not only from the methodological point of view that enjoins us to strive for the greatest possible conceptual uniformity but also from a certain philosophic, or if you wish, aesthetic point of view."
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"... Reproduction of all or part of this workis permitted for educational or research use on condition that this copyright notice isincluded in any copy. ..."
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Reproduction of all or part of this workis permitted for educational or research use on condition that this copyright notice isincluded in any copy.