Taylor expansion, β-reduction and normalization

Lionel Vaux (1)

1 I2M - Institut de Mathématiques de Marseille

We introduce a notion of reduction on resource vectors, i.e. infinite linear combinations of resource λ-terms. The latter form the multilinear fragment of the differential λ-calculus introduced by Ehrhard and Regnier, and resource vectors are the target of the Taylor expansion of λ-terms. We show that the reduction of resource vectors contains the image, through Taylor expansion, of β-reduction in the algebraic λ-calculus, i.e. λ-calculus extended with weighted sums: in particular , Taylor expansion and normalization commute. We moreover exhibit a class of algebraic λ-terms, having a normalizable Taylor expansion, subsuming both arbitrary pure λ-terms, and normalizable algebraic λ-terms. For these, we prove the commutation of Taylor expansion and normalization in a more denotational sense, mimicking the Böhm tree construction.

Keywords: lambda-calculus; non-determinism; normalization; denotational semantics

Domains: Logic in Computer Science [cs.LO]; Logic [math.LO]

Lionel Vaux. Taylor expansion, β-reduction and normalization. 26th EACSL Annual Conference on Computer Science Logic (CSL 2017), Aug 2017, Stockholm, Sweden. pp.39:1--39:16, ⟨10.4230/LIPIcs.CSL.2017.39⟩. ⟨hal-01834329⟩