A Semantics for Hybrid Iteration

TL;DR

This paper develops a semantic framework for hybrid iteration using guarded traced categories and monads, enabling the modeling of hybrid computations with progressiveness and Zeno behavior.

Contribution

It introduces a modified hybrid monad with guarded iteration interpreted as progressiveness in time, unifying partial and total iteration theories for hybrid systems.

Findings

A new notion of guarded iteration for hybrid monads.

Semantic interpretation of hybrid computations with Zeno behavior.

Application to a simple hybrid programming language.

Abstract

The recently introduced notions of guarded traced (monoidal) category and guarded (pre-)iterative monad aim at unifying different instances of partial iteration whilst keeping in touch with the established theory of total iteration and preserving its merits. In this paper we use these notions and the corresponding stock of results to examine different types of iteration for hybrid computations. As a starting point we use an available notion of hybrid monad restricted to the category of sets, and modify it in order to obtain a suitable notion of guarded iteration with guardedness interpreted as progressiveness in time - we motivate this modification by our intention to capture Zeno behaviour in an arguably general and feasible way. We illustrate our results with a simple programming language for hybrid computations and interpret it over the developed semantic foundations.