Abstract hypernormalisation, and normalisation-by-trace-evaluation for generative systems

Abstract: 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.