Euclidean geometry has always been regarded by scientists as being a priori and objective. When we assume the role of an agent, however, the challenge of determining the optimum path should also take into account the agent’s capabilities, its embodiment, and in particular its cognitive effort.

The geometry of movement between states in a world is taken into account in a 2021 article led by Karen Archer (Department of Computer Science, University of Hertfordshire, Hatfield, United Kingdom) by factoring for information processing costs and the required spatial distances. As information costs grow more significant, this results in a geometry that deviates more and more from the initial geometry of the provided environment.

When projected onto 2- and 3-dimensional spaces, this “cognitive geometry” may be seen to have different distortions that represent the evolution of epistemic and information-saving techniques as well as pivot states. The analogies between traditional cost-based geometries and those induced by additional informational costs invite a generalization of the traditional notion of geodesics as cheapest routes towards the notion of *infodesics*, which approximates the usual geometric property that, when traveling from a start to a goal along a geodesic, not only the goal but all intermediate points are visited equally at the best cost from the start.

We considered distances induced by cost-only MDPs (Markov decision process) which were additionally endowed with an informational cost reflecting the complexity of decision-making. Geodesics generalise the intuition about geometry determined by directions and distances, representing optimal transitions between states. We proposed that the addition of informational criteria would characterise a cognitive geometry which additionally captures the difficulty of pursuing a particular trajectory.

The researchers discovered that free energy incorporates the cost of information processing while maintaining part of the spatial geometry’s structure through the value function. Although it will favor trajectories that pass through informationally efficient states on route to the objective, highlighting the best informationally efficient policy among otherwise equal policies will. As a result, these trajectories frequently take a “detour” through hubs that are more easily maneuverable in terms of pure distance.

We found considerable distortions which place boundary states more centrally in the space, with the boundaries acting as guides. Additionally, when considering infodesics, i.e. sets of intermediate states which are optimally reachable from the starting state, intermediate goals can be achieved en route to the final goal, thus defining classes of problems that are solved as a side effect of solving the main one.

The team defined the infodesic property as the triangle inequality becoming an equality, similar to how geodesics work. They had to loosen the requirements since this discrepancy is not always properly honored in their framework. Splitting a trajectory also absorbs the cost of switching the policies of the two segments into the split itself because of the informational nature of the free energy distance, which makes the violation of the triangle inequality rather severe.

A generalized but tighter triangle inequality can include the splitting cost for a more thorough explanation of the infodesic, cost that the team suggests will allow the imposing of a quasi-geometrical signature on the state space and provide the basis of a genuinely geometrical notion of task spaces that takes into account cognitive processing: a cognitive geometry.

This we understand to be a structure with optimal trajectories determined by either two states or by one state and a “direction” (i.e. policy) that is informed not only by the pure spatial geometry, but also by the cognitive costs that an agent needs to process when moving from task to task and how it has to informationally organise policies to achieve nearby or related tasks.

Such a notion would imply that even simply navigational decision issues may not be best addressed by the plain application of Euclidean or geodesic-based geometry. It would be intriguing to look into this distance further for that reason.

*A space of goals: the cognitive geometry of informationally bounded agents.* Karen Archer, Nicola Catenacci Volpi, Franziska Bröker, Daniel Polani.

Published: November 2021

DOI: https://doi.org/10.48550/arXiv.2111.03699