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Cyclic lambda calculi

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Theoretical Aspects of Computer Software (TACS 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1281))

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Abstract

We precisely characterize a class of cyclic lambda-graphs, and then give a sound and complete axiomatization of the terms that represent a given graph. The equational axiom system is an extension of lambda calculus with the letrec construct. In contrast to current theories, which impose restrictions on where the rewriting can take place, our theory is very liberal, e.g., it allows rewriting under lambda-abstractions and on cycles. As shown previously, the reduction theory is non-confluent. We thus introduce an approximate notion of confluence. Using this notion we define the infinite normal form or Lévy-Longo tree of a cyclic term. We show that the infinite normal form defines a congruence on the set of terms. We relate our cyclic lambda calculus to the traditional lambda calculus and to the infinitary lambda calculus. Since most implementations of non-strict functional languages rely on sharing to avoid repeating computations, we develop a variant of our calculus that enforces the sharing of computations and show that the two calculi are observationally equivalent. For reasoning about strict languages we develop a call-by-value variant of the sharing calculus. We state the difference between strict and non-strict computations in terms of different garbage collection rules. We relate the call-by-value calculus to Moggi's computational lambda calculus and to Hasegawa's calculus.

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Authors and Affiliations

  1. Department of Computer & Information Sciences, University of Oregon, 97401, Eugene, OR, USA

    Zena M. Ariola & Stefan Blom

  2. Department of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081a, 1081 HV, Amsterdam

    Stefan Blom

Authors
  1. Zena M. Ariola
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  2. Stefan Blom
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Martín Abadi Takayasu Ito

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© 1997 Springer-Verlag Berlin Heidelberg

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Ariola, Z.M., Blom, S. (1997). Cyclic lambda calculi. In: Abadi, M., Ito, T. (eds) Theoretical Aspects of Computer Software. TACS 1997. Lecture Notes in Computer Science, vol 1281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0014548

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  • DOI: https://doi.org/10.1007/BFb0014548

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63388-4

  • Online ISBN: 978-3-540-69530-1

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