TY - GEN

T1 - The Girard-Reynolds Isomorphism

AU - Wadler, Philip

PY - 2001

Y1 - 2001

N2 - The second-order polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in second-order intuitionistic propositional logic, P2, can be represented in F2. Reynolds additionally proved an abstraction theorem: for a suitable notion of logical relation, every term in F2 takes related arguments into related results. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embeddingof F2 into P2, and that the Reynolds embeddingfollo wed by the Girard projection is the identity. The Girard projection discards all first-order quantifiers, so it seems unreasonable to expect that the Girard projection followed by the Reynolds embeddingshould also be the identity. However, we show that in the presence of Reynolds’s parametricity property that this is indeed the case, for propositions correspondingto inductive definitions of naturals, products, sums, and fixpoint types.

AB - The second-order polymorphic lambda calculus, F2, was independently discovered by Girard and Reynolds. Girard additionally proved a representation theorem: every function on natural numbers that can be proved total in second-order intuitionistic propositional logic, P2, can be represented in F2. Reynolds additionally proved an abstraction theorem: for a suitable notion of logical relation, every term in F2 takes related arguments into related results. We observe that the essence of Girard’s result is a projection from P2 into F2, and that the essence of Reynolds’s result is an embeddingof F2 into P2, and that the Reynolds embeddingfollo wed by the Girard projection is the identity. The Girard projection discards all first-order quantifiers, so it seems unreasonable to expect that the Girard projection followed by the Reynolds embeddingshould also be the identity. However, we show that in the presence of Reynolds’s parametricity property that this is indeed the case, for propositions correspondingto inductive definitions of naturals, products, sums, and fixpoint types.

U2 - 10.1007/3-540-45500-0_24

DO - 10.1007/3-540-45500-0_24

M3 - Conference contribution

SN - 978-3-540-42736-0

T3 - Lecture Notes in Computer Science

SP - 468

EP - 491

BT - Theoretical Aspects of Computer Software

PB - Springer

ER -