In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful.
翻译:在概率一致性空间,即概率功能语言的省略模型,形态学是分析性的,因此很顺畅。我们探索了相应的衍生物的两个相关应用。首先,我们展示了衍生物如何在概率性 PCF (pPCF) 的低头降低中计算执行时间的预期值。接下来,我们将这些形态学的“本地”差异的一般概念应用于证明Lipschitz 特性的“当地”差异,从而可以反过来将PPCF 术语的观察距离与该模型自然具备的距离联系起来。这说明,按照差异性羊羔计算法的精神,将衍生物的概率性编程语言扩展到衍生物中,可能非常有意义。