Quantitative semantics of the lambda calculus: Some generalisations of the relational model

We present an overview of some recent work on the quantitative semantics of the λ-calculus. Our starting point is the fundamental degenerate model of linear logic, the relational model MRel. We show that three quantitative semantics of the simply-typed λ-calculus are equivalent: the relational semantics, HO/N game semantics, and the Taylor expansion semantics. We then consider two recent generalisations of the relational model: first, R-weighted relational models where R is a complete commutative semiring, as studied by Laird et al.; secondly, generalised species of structures, as introduced by Fiore et al. In each case, we briefly discuss some applications to quantitative analysis of higher-order programs.