# 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.

### References

- Ay N, Ghazi-Zahedi K (2014) On the causal structure of the sensorimotor loop. In: Prokopenko M (ed) Guided self-organization: inception. Springer, New YorkGoogle Scholar
- Bauer H (1996) Probability theory. Walter de GruyterGoogle Scholar
- Bogachev VI (2007) Measure theory, vol I. Springer, New YorkCrossRefGoogle Scholar
- Bogachev VI (2007) Measure theory, vol II. Springer, New YorkCrossRefGoogle Scholar
- Boylan ES (1971) Equiconvergence of martingales. Ann Math Stat 42:552–559CrossRefGoogle Scholar
- Karr AF (1975) Weak convergence of a sequence of Markov chains. Z. Wahrscheinlichkeitstheorie verw. Gebiete 33:41–48CrossRefGoogle Scholar
- Neveu J (1972) Note on the tightness of the metric on the set of complete sub \(\sigma \)-algebras of a probability space. Ann Math Stat 43:1369–1371CrossRefGoogle Scholar
- Pfante O, Ay N (2015) Operator-theoretic identification of closed sub-systems of dynamical systems. Discontinuity Nonlinearity Complex 1:91–109CrossRefGoogle Scholar
- Pfeifer R, Bongard J (2007) How the body shapes the way we think. MIT Press, CambridgeGoogle Scholar
- Pfante O, Bertschinger N, Olbrich E, Ay N, Jost J (2014) Comparison between different methods of level identification. Adv Complex Syst 17:1450007CrossRefGoogle Scholar
- Pearl J (2000) Causality: models. Reasoning and inference. Cambridge University Press, CambridgeGoogle Scholar
- Rogge L (1974) Uniform inequalities for conditional expectations. Ann Prob 2:486–489CrossRefGoogle Scholar
- Tishby N, Polani D (2010) Information theory of decisions and actions. In: Cutsuridis V, Hussain A, Taylor J (eds) Perception-action cycle: models, architecture and hardware, pp 601–636Google Scholar
- Von Uexküll J (1926) Theoretical biology. Harcourt, Brace & Co., New YorkGoogle Scholar
- Von Uexküll J (1934) A stroll through the worlds of animals and men. In: Lashley K (ed) Instructive behavior. International University PressGoogle Scholar
- Von Uexküll J (2014) Umwelt und Innenwelt der Tiere. In: Mildenberger F, Herrmann B (eds) Klassische Texte der Wissenschaft. Springer, New YorkGoogle Scholar
- Zahedi K, Ay N, Der R (2010) Higher coordination with less control: a result of information maximization in the sensori-motor loop. Adapt Behav 18:338–355CrossRefGoogle Scholar