Taylor Expansion, β-Reduction and Normalization

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. 1998 ACM Subject Classification F.3.2 Semantics of Programming Languages