The Umwelt of an Embodied Agent -- A Measure-Theoretic Definition

TL;DR

This paper formalizes Uexküll's Umwelt concept using measure theory, defining perspectives of an agent and external observer within a causal sensorimotor framework, and shows how world dynamics can be simplified while preserving the internal process.

Contribution

It introduces a measure-theoretic formalization of the Umwelt concept, linking agent perspectives with world dynamics through sigma-algebras and causal structures.

Findings

Defined two perspectives using sigma-algebras: external and agent.

Proposed the smaller sigma-algebra as the formal Umwelt.

Showed that world dynamics can be simplified without affecting the internal process.

Abstract

We consider a general model of the sensorimotor loop of an agent interacting with the world. This formalises Uexk\"ull's notion of a \emph{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 Uexk\"ull's \emph{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.