Jacobs' hypernormalisation is a construction on finitely supported discrete probability distributions, obtained by generalising certain patterns occurring in quantitative information theory. In this paper, we generalise Jacobs' notion in turn, by describing a notion of hypernormalisation in the abstract setting of a symmetric monoidal category endowed with a linear exponential monad -- a structure arising in the categorical semantics of linear logic. We show that Jacobs' hypernormalisation arises in this fashion from the finitely supported probability measure monad on the category of sets, which can be seen as a linear exponential monad with respect to a non-standard monoidal structure on sets which we term the convex monoidal structure. We give the construction of this monoidal structure in terms of a quantum-algebraic notion known as a tricocycloid. Besides the motivating example, and its natural generalisations to the continuous context, we give a range of other instances of our abstract hypernormalisation, which swap out the side-effect of probabilistic choice for other important side-effects such as non-deterministic choice, logical choice via tests in a Boolean algebra, and input from a stream of values. Finally, we exploit our framework to describe a normalisation-by-trace-evaluation process for behaviours of various kinds of coalgebraic generative systems, including labelled transition systems, probabilistic generative systems, and stream processors.
翻译:Jacobs的超常正常化是建立在有限支持的离散概率分布的构造上,通过对定量信息理论中出现的某些模式进行概括化而获得的。在本文中,我们逐个概括了Jacobs的概念,通过描述一个具有线性指数性月圆形的对称单成形分类的抽象设置中的超常化概念 -- -- 这是线性逻辑绝对语义中出现的一种结构。我们表明,Jacobs超常化是从这种方式中产生的,它来自一个有限支持的概率度量子类的摩登分布,这可以被视为一个线性指数性单项结构上非标准单项结构的线性单项元。我们把这一单项结构的构造放在一个称为量子-al-algibra概念的抽象结构的抽象设置中,这个结构是线性指数-直线性线性逻辑的直线性线性逻辑线性逻辑线性逻辑。我们给出了一系列其它的抽象超常态化例子,这种超常态的系统可以取代概率性选择的副效应,例如非非确定性选择、逻辑性单项单项单一单项结构结构结构,通过测试来构造选择。我们从一个直系的逻辑选择,通过一种直系的直系的直系选择, 我们的直系的基因变的基因结构的基因结构的基因结构, 来描述性结构的基因结构的正变变变, 以及我们的基因变的基因变的基因结构的基因变式结构,我们的正常的基因变的基因变的基因变式的基因变的基因变式的基因化过程, 。