Normal Forms for the Algebraic Lambda-Calculus

We study the problem of defining normal forms of terms for the algebraic λ-calculus, an extension of the pure λ-calculus where linear combinations of terms are first-class entities: the set of terms is enriched with a structure of vector space, or module, over a fixed semiring. Towards a solution to the problem, we propose a variant of the original reduction notion of terms which avoids annoying behaviours affecting the original version, but we find it not even locally confluent. Finally, we consider reduction of linear combinations of terms over the semiring of polynomials with non-negative integer coefficients: terms coefficients are replaced by indeterminates and then, after reduction has taken placed, restored back to their original value by an evaluation function. Such a special setting permits us to talk about normal forms of terms and, via an evaluation function, to define such notion for any semiring.