The operad of temporal wiring diagrams: formalizing a graphical language for discrete-time processes

Abstract

We investigate the hierarchical structure of processes using the mathematical theory of operads. Information or material enters a given process as a stream of inputs, and the process converts it to a stream of outputs. Output streams can then be supplied to other processes in an organized manner, and the resulting system of interconnected processes can itself be considered a macro process. To model the inherent structure in this kind of system, we define an operad $\mathcal{W}$ of black boxes and directed wiring diagrams, and we define a $\mathcal{W}$-algebra $\mathcal{P}$ of processes (which we call propagators, after Radul and Sussman). Previous operadic models of wiring diagrams use undirected wires without length, useful for modeling static systems of constraints, whereas we use directed wires with length, useful for modeling dynamic flows of information. We give multiple examples throughout to ground the ideas.