Explicit substitutions

@article{Abadi1989ExplicitS,
  title={Explicit substitutions},
  author={Mart{\'i}n Abadi and Luca Cardelli and Pierre-Louis Curien and Jean-Jacques L{\'e}vy},
  journal={Journal of Functional Programming},
  year={1989},
  volume={1},
  pages={375 - 416}
}
The λ&sgr;-calculus is a refinement of the λ-calculus where substitutions are manipulated explicitly. The λ&sgr;-calculus provides a setting for studying the theory of substitutions, with pleasant mathematical properties. It is also a useful bridge between the classical λ-calculus and concrete implementations. 

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References

SHOWING 1-10 OF 50 REFERENCES

Proof of termination of the rewriting system subst on CCL

Un résultat de complétude pour les substitutions explicites

We establish a completeness result of λσ-calculus with respect to λ-calculus. Specifically we construct an interpretation of λσ-calculus, which is such that a source term is typable iff its

The lambda calculus - its syntax and semantics

  • H. Barendregt
  • Mathematics
    Studies in logic and the foundations of mathematics
  • 1985

On laziness and optimality in lambda interpreters: tools for specification and analysis

TLDR
It is shown that ACCL has properties not possessed by Curien's original combinators that make it particularly appropriate as the basis for implementation and analysis of a wide range of reduction schemes using shared environments, closures, or λ-terms.

Combinatory logic

the equations derivable from the axioms of the theory. When this two-step algebraization has been carried out it is possible to reduce problems of validity in a first-order language to questions

Equations and rewrite rules: a survey

Typeful Programming

  • L. Cardelli
  • Computer Science
    Formal Description of Programming Concepts
  • 1989
TLDR
It is shown how typeful programming is best supported by sophisticated type systems, and how these systems can help in clarifying programming issues and in adding power and regularity to languages.

Categorical Combinators