Time complexity for deterministic string machines

Algorithms which learn environments represented by automata in the past have had complexity scaling with the number of states in the automaton, which can be exponentially large even for automata recognizing regular expressions with a small description length. We thus formalize a compositional language that can construct automata as transformations between certain types of category, representable as string diagrams, which better reflects the description complexity of various automata. We define complexity constraints on this framework by having them operate on categories enriched over filtered sets, and using these constraints, we prove elementary results on the runtime and expressivity of a subset of these transformations which operate deterministically on finite state spaces. These string diagrams, or "string machines," are themselves morphisms in a category, so it is possible for string machines to create other string machines in runtime to model computations which take more than constant memory. We prove sufficient conditions for polynomial runtime guarantees on these, which can help develop complexity constraints on string machines which also encapsulate runtime complexity.