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Final semantics for untyped λ-calculus

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Typed Lambda Calculi and Applications (TLCA 1995)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 902))

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Abstract

Proof principles for reasoning about various semantics of untyped λ-calculus are discussed. The semantics are determined operationally by fixing a particular reduction strategy on λ-terms and a suitable set of values, and by taking the corresponding observational equivalence on terms. These principles arise naturally as co-induction principles, when the observational equivalences are shown to be induced by the unique mapping into a final F-coalgebra, for a suitable functor F. This is achieved either by induction on computation steps or exploiting the properties of some, computationally adequate, inverse limit denotational model. The final F-coalgebras cannot be given, in general, the structure of a “denotational” λ-model. Nevertheless the “final semantics” can count as compositional in that it induces a congruence. We utilize the intuitive categorical setting of hypersets and functions. The importance of the principles introduced in this paper lies in the fact that they often allow to factorize the complexity of proofs (of observational equivalence) by “straight” induction on computation steps, which are usually lengthy and error-prone.

Work supported by EEC Science contract MASK, HCM contract “Lambda Calcul Typé” and MURST 40% and 60% grants.

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Authors and Affiliations

  1. Dipartimento di Matematica e Informatica, Università di Udine, Italy

    Furio Honsell & Marina Lenisa

  2. Dipartimento di Informatica, Università di Pisa, Italy

    Marina Lenisa

Authors
  1. Furio Honsell
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  2. Marina Lenisa
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Mariangiola Dezani-Ciancaglini Gordon Plotkin

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© 1995 Springer-Verlag Berlin Heidelberg

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Honsell, F., Lenisa, M. (1995). Final semantics for untyped λ-calculus. In: Dezani-Ciancaglini, M., Plotkin, G. (eds) Typed Lambda Calculi and Applications. TLCA 1995. Lecture Notes in Computer Science, vol 902. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014057

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  • DOI: https://doi.org/10.1007/BFb0014057

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-59048-4

  • Online ISBN: 978-3-540-49178-1

  • eBook Packages: Springer Book Archive

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