New Semantical Insights Into Call-by-Value λ-Calculus.

Despite the fact that call-by-value lambda-calculus was defined by Plotkin in 1977, we believe that its theory of program approximation is still at the beginning. A problem that is often encountered when studying its operational semantics is that, during the reduction of a lambda-term, some redexes remain stuck (waiting for a value). Recently, Carraro and Guerrieri proposed to endow this calculus with permutation rules, naturally arising in the context of linear logic proof-nets, that succeed in unblocking a certain number of such redexes. In the present paper we introduce a new class of models of call-by-value lambda-calculus, arising from non-idempotent intersection type systems. Beside satisfying the usual properties as soundness and adequacy, these models validate the permutation rules mentioned above as well as some reductions obtained by contracting suitable lambda I-redexes. Thanks to these (perhaps unexpected) features, we are able to demonstrate that every model living in this class satisfies an Approximation Theorem with respect to a refined notion of syntactic approximant. While this kind of results often require impredicative techniques like reducibility candidates, the quantitative information carried by type derivations in our system allows us to provide a combinatorial proof.