Execution Time of λ-Terms via Denotational Semantics and Intersection Types

This paper presents a work whose aim is to obtain information on execution time of λ-terms by semantic means. By execution time, we mean the number of steps in a computational model. As in [Ehrhard and Regnier 2006], the computational model considered in this paper will be Krivine’s machine, a more realistic model than β-reduction. Indeed, Krivine’s machine implements (weak) head linear reduction: in one step, we can do at most one substitution. In this paper, we consider two variants of this machine : the first one (Definition 2.4) computes the head-normal form of any λ-term (if it exists) and the second one (Definition 2.11) computes the normal form of any λ-term (if it exists). The fundamental idea of denotational semantics is that types should be interpreted as the objects of a category C and terms should be interpreted as arrows in C in such a way that if a term t reduces to a term t, then they are interpreted by the same arrow. By the Curry-Howard isomorphism, a simply typed λ-term is a proof in intuitionistic logic and the β-reduction of a λ-term corresponds to the cut-elimination of a proof. Now, the intuitionistic fragment of linear logic [Girard 1987] is a refinement of intuitionistic logic. This means that when we have a categorical structure (C, . . .) for interpreting intuitionistic linear logic, we can derive a category K that is a denotational semantics of intuitionistic logic, and thus a denotational semantics of λ-calculus. Linear logic has various denotational semantics; one of these is the multiset based relational model in the category Rel of sets and relations with the comonad associated to the finite multisets functor (see [Tortora de Falco 2000] for interpretations