A Topological Approach to Design Problems for Distributed Systems Using Category Theory: A Focus on Robotic Swarms

The mathematical formalization of design problems has received significant attention in recent years. A number of theoretical frameworks have been developed which express certain classes of design problems as mathematical optimization problems with constraints. Compositionality in complex systems has played a prominent role in structuring the reasoning behind these approaches, and category theory has been foundational in formalizing compositional structures. This work proposes an adaptation of the mathematical design problem to large distributed systems. A distributed system is modeled in terms of its local interactions, which are interpreted as a topological structure on the system and represented by a hypergraph. Category theory is used to match the structure of the design problem to the hypergraph of the system, and to formulate a general distributed design problem with global constraints. By applying concepts from algebraic topology, in particular homological algebra, the distributed design problem is brought into an explicit form which allows for the use of existing algorithms for optimization with consensus constraints. Compared to previous formalizations, this approach provides a finer structural view of systems with complex operational interdependencies, and has the potential to reduce the complexity of the design problem for large robotic swarms by exploiting the partial independences present in the system.