# An Adequate While-Language for Hybrid Computation

## TL;DR

This paper introduces a new language for hybrid computation that combines discrete and continuous dynamics, supported by a denotational semantics based on hybrid monads, enabling better analysis and description of complex natural and engineered systems.

## Contribution

It presents a novel hybrid computation language inspired by call-by-value, with a denotational semantics using hybrid and duration monads, advancing formal modeling of hybrid systems.

## Key findings

- Developed a hybrid monad supporting iteration for semantics.
- Introduced a lightweight duration monad as a simpler alternative.
- Provided a denotational semantics that is both adequate and computationally meaningful.

## Abstract

Hybrid computation combines discrete and continuous dynamics in the form of an entangled mixture inherently present both in various natural phenomena, and in applications ranging from control theory to microbiology. The emergent behaviours bear signs of both computational and physical processes, and thus present difficulties not only for analysis, but also for describing them adequately in a structural, well-founded way. Here, we introduce a language for hybrid computation, inspired by the fine-grain call-by-value paradigm, and equip it with a denotational and computationally adequate denotational semantics. Our denotational semantics crucially relies on a hybrid monad supporting an (Elgot) iteration operator, we developed elsewhere. As an intermediate step we introduce a more lightweight duration semantics furnished with analogous results and drawing on a new duration monad that we introduce as a lightweight counterpart to the hybrid monad.