Dependent Directed Wiring Diagrams for Composing Instantaneous Systems

Directed wiring diagrams can be used as a composition pattern for composing input/output systems such as Moore machines. In a Moore machine, the input parametrizes an internal state and the internal state defines the output. Because the value of the output is shielded from the input by the internal state, Moore machines can compose by connecting the output of any machine to the input of any other machine. These connections are defined by the trace wires in a directed wiring diagram. Unlike Moore machines, Mealy machines allow the output to be directly and instantaneously affected by the input. In order to compose such machines via directed wiring diagrams, it is necessary to avoid cycles between trace wires in the wiring digram and dependencies of outputs on inputs. To capture these patterns of composition, we introduce an operad of dependent directed wiring diagrams. We then define an algebra of Mealy machines on this operad and an algebra of stock and flow diagrams in which the values of auxiliary variables are parameterized by inputs. Finally, we give a semantics for this algebra of stock and flow diagrams by giving a morphism of algebras from stock and flow diagrams into Mealy machines.