Embodied Functional Relations: A Formal Account Combining Abstract Logical Theory with Grounding in Simulation

Functional relations such as containment or support have proven difficult to formalize. Although previous efforts have attempted this using hybrids of several theories, from mereology to temporal logic, we find that such purely symbolic approaches do not account for the embodied nature of functional relations, i.e. that they are used by embodied agents to describe fragments of a physical world. We propose a formalism that combines descriptions of a high level of abstraction with generative models that can be used to instantiate or recognize arrangements of objects and trajectories conforming to qualitative descriptions. The formalism gives an account of how a qualitative description of a scene or arrangement of objects can be converted into a quantitative description amenable to simulation, and how simulation results can be qualitatively interpreted. We use this to describe functional relations between objects in terms of spatial arrangements, expectations on behavior, and counterfactual expectations for when one of the participants is absent. Our method is able to tackle important questions facing an agent operating in the world, such as what would happen if an arrangement of objects is created and why. This gives the agent a deeper understanding of functional relations, including what role background objects, not explicitly asserted to participate in a functional relation such as containment, play in enabling or hindering the relation from holding.