Robotics, machine learning, inference, control, decision theory, system identification. There are many different views on how the information would flow from the environment into a robot, and the other way around. One example, Ralf Der is working on "homeokinesis". A robot is driven towards these regions in state space that cause large perceptual variations. To be able to do so, it necessarily must understand enough of the environment and hence won't just exhibit chaotic behavior.
Is there a branch of category theory that considers the fundamentals of a "self-learning" system? A description of a system which performs its own "system identification". It would be great if there is a formal approach to it, because many of the scientist working in this field (Oudeyer, Pfeiffer, Wolpert, Metta, Friston to name just a few) tend to explain their concepts in a narrative manner. It would be great to have a thorough conceptual review using monads, etc.
Challenges (from my laymen's perspective in increasing order of difficulty for a categorical approach):
- Describe Pearl's calculus of interventions.
- Describe principle of optimality using category theory (http://en.wikipedia.org/wiki/Principle_of_optimality) or the related notion of optimal substructure (http://en.wikipedia.org/wiki/Optimal_substructure).
- Describe the curse of dimensionality (http://en.wikipedia.org/wiki/Curse_of_dimensionality).
- Describe "embodied cognition". For example "Extending Dynamical Systems Theory to Model Embodied Cognition" (Hotton, Yoshima) is a tad more formal, but reaches in no way the rigorous accuracy of the definition of concepts as in category theory.
Pointers to people who only tried(!) to apply category theory to these challenging problems will be also appreciated.
Regarding "Describe Pearl's calculus of interventions" and other related ideas, you might find this thesis interesting:
Depending upon the level of speculation you are interested in considering, you might also like to look at this book: