Observable Semantics for Characterising Consistency Between Heterogeneous Models

The design of complex cyber-physical systems increasingly relies on heterogeneous, multi-domain models, each capturing different system aspects. As these models evolve independently, inconsistencies may arise. Such inconsistencies remain difficult to detect due to gaps between the modelling domains. We propose to bridge this gap by introducing a formal notion of observables, inspired by physics, as measurable system properties, such that every model can constrain the possible values of an observable. We define a semantic framework where observables induce consistency relations and show that any such relation can be expressed using suitable observable semantics. To capture realistic engineering scenarios, we extend the framework with meta-model spanning and compound observable semantics, enabling the modular construction of complex constraints. We also provide encodings of these observable semantics back into the original framework, preserving the original results and showing that extensions remain expressively equivalent to initial ones. Finally, we discuss some practical implications of our framework, namely how observables can support cross-domain communication, separation of concerns, and extensibility, allowing consistency requirements to evolve as part of the modelling process.