# The *Umwelt* of an embodied agent—a measure-theoretic definition

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## Abstract

We consider a general model of the sensorimotor loop of an agent interacting with the world. This formalises Uexküll’s notion of a *function-circle*. Here, we assume a particular causal structure, mechanistically described in terms of Markov kernels. In this generality, we define two \(\sigma \)-algebras of events in the world that describe two respective perspectives: (1) the perspective of an external observer, (2) the intrinsic perspective of the agent. Not all aspects of the world, seen from the external perspective, are accessible to the agent. This is expressed by the fact that the second \(\sigma \)-algebra is a subalgebra of the first one. We propose the smaller one as formalisation of Uexküll’s *Umwelt* concept. We show that, under continuity and compactness assumptions, the global dynamics of the world can be simplified without changing the internal process. This simplification can serve as a minimal world model that the system must have in order to be consistent with the internal process.

## Keywords

*Umwelt*

*Function-circle*Sensorimotor loop Embodied agent Intrinsic perspective External observer \(\sigma \)-Algebra

## Notes

### Acknowledgments

Nihat Ay is grateful for stimulating discussions with Keyan Ghazi-Zahedi and Guido Montúfar. The authors would like to thank Jürgen Jost for his helpful comments.

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