We study a class of quantitative models for higher-order computation: Lafont categories with (infinite) biproducts. Each of these has a complete “internal semiring” and can be enriched over its modules. We describe a semantics of nondeterministic PCF weighted over this semiring in which fixed points are obtained from the bifree algebra over its exponential structure. By characterizing them concretely as infinite sums of approximants indexed over nested finite multisets, we prove computational adequacy.

We can construct examples of our semantics by weighting existing models such as categories of games over a complete semiring. This transition from qualitative to quantitative semantics is characterized as a “change of base” of enriched categories arising from a monoidal functor from coherence spaces to modules over a complete semiring. For example, the game semantics of Idealized Algol is coherence space enriched and thus gives rise to to a weighted model, which is fully abstract.

我们研究了一类用于高阶计算的定量模型：具有（无限）双积的Lafont类别。每个模块都有一个完整的“内部半环”，并且可以在其模块上进行扩充。我们描述了在此半环上加权的不确定PCF的语义，其中从双自由代数的指数结构中获得固定点。通过将它们具体化为在嵌套有限多集上索引的近似值的无限和，我们证明了计算的充分性。

我们可以通过对现有模型（例如游戏类别）在整个半环上进行加权来构造语义示例。从定性语义到定量语义的这种转变的特征是，丰富的类别的“基础变化”是由于单等函子从相干空间到完整半环上的模块而引起的。例如，理想化Algol的游戏语义丰富了相干空间，因此产生了一个完全抽象的加权模型。