Endofunctors modelling higher-order behaviours

TL;DR

This paper introduces a novel, syntax-agnostic method for modeling higher-order system behaviors using final coalgebras of circularly defined behavioural functors, shifting from syntax-based to semantics-based approaches.

Contribution

It provides a general construction for defining circular behavioural functors, enabling syntax-independent modeling of higher-order behaviors.

Findings

Successfully models higher-order behaviors as final coalgebras

Solves circularity in defining behavioral functors

Distinguishes between syntactic and semantic higher-order approaches

Abstract

In this paper we show how the abstract behaviours of higher-order systems can be modelled as final coalgebras of suitable behavioural functors. These functors have the challenging peculiarity to be circularly defined with their own final coalgebra. Our main contribution is a general construction for defining these functors, solving this circularity which is the essence of higher-order behaviours. This characterisation is syntax agnostic. To achieve this property, we shift from term passing to behaviour passing: in the former higher-order is expressed by passing around syntactic objects (such as terms or processes) as representations of behaviours whereas the former ditches the syntactic encoding altogether and works directly with behaviours i.e. semantic objects. From this perspective, the former can be seen as syntactic higher-order whereas the later as semantic higher-order.