We present a typing system for the λ-calculus, with non-idempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λ-term is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound on the longest β-reduction sequence reducing a term to its normal form. We actually present these results in Klop’s extension of λ-calculus, where the bound that is read in the typing tree of a term is refined into an exact measure of the longest reduction sequence. This complexity result is, for longest reduction sequences, the counterpart of de Carvalho’s result for linear head-reduction sequences.

Abstract. We initiate a purely algebraic study of Ehrhard and Regnier’s resource λ-calculus, by introducing three equational classes of algebras: resource combinatory algebras, resource lambda-algebras and resource lambda-abstraction algebras. We establish the relations between them, laying down foundations for a model theory of resource λ-calculus. We also show that the ideal completion of a resource combinatory (resp. lambda-, lambda-abstraction) algebra induces a “classical ” combinatory (resp. lambda-, lambda-abstraction) algebra, and that any model of the classical λ-calculus raising from a resource lambda-algebra determines a λ-theory which equates all terms having the same Böhm tree. 1

Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.