The weak algebraic λ-calculus
Résumé
We introduce a variant of the algebraic λ-calculus, which extends the ordinary λ-calculus with the possibility of forming linear combinations of terms, with coefficients taken in a fixed semiring. The only difference from the original version is that we drop the identity 0M = 0, i.e., multiplication by the scalar 0 does not yield the empty sum in general: terms thus form a weak module instead of a module. We then study how this minor relaxing of axioms yields a simpler and more robust rewriting theory. In particular, we show that the obtained reduction is still confluent; however, contrasting with the original version, the induced equivalence is conservative over β-equivalence, without any assumption on the semiring. We moreover characterize normalization properties via a system of intersection type systems, and provide a normalization strategy, again without assumption on scalars. Finally, we show that the Taylor expansion of algebraic λ-terms can be adapted to this setting.
Domaines
Informatique [cs]Origine | Fichiers produits par l'(les) auteur(s) |
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