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Compositional Models for Complex Systems 261

of the interface is the product of parameter spaces for each parameter of each
boundary of the interface. In a simplistic model, a window boundary might
define a state space Bool Temp Temp, where the first parameter defines
whether the window is open or closed, and the last two define the temper-
atures inside and outside the window, respectively. Sometimes, we may also
wish to include plausibility constraints on the state space; for example, we
might demand that when the window is open, then T in ¼ T out.
An operation is a decomposition of the system into components and
boundaries. This decomposition imposes constraints between the states of
component interfaces as well as the system boundaries. To define logical
semantics for an operation, we form the joint state space of all component
and system interfaces and identify which joint states can possibly be satisfied
given the architecture. In the example from Fig. 13.1B, the joint state space
might look like:
HVAC system Therm component
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
ðFloat sys Temp Þ ðFloat read Temp
sys therm Þ
AC component Heat component
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
ðBool AC Temp Þ ðBool heat Temp
AC heat Þ
Ctrl component
zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{
ðFloat set Float meas Bool cool Bool warm Þ
The specific architecture then defines a relation over this state space.
Usually, this relation just couples together variables that are linked by an
interaction in the diagram. Here, we would have:

ðF sys ¼ F set Þ^ðF meas ¼ F read Þ^ðB warm ¼ B heat Þ

^ðB cool ¼ B AC Þ^ðT sys ¼ T therm ¼ T AC ¼ T heat Þ
Some interactions, especially those governing interactions between three or
more boundaries, may require more sophistication. In the context of elec-
trical circuits, voltages across a bus are equal but currents add. In our exam-
ple, the temperature of the air is equal at all boundaries (assuming mixing),
but dynamical quantities like the energy input at an air boundary would
require more sophistication.
Recall that in addition to an architecture, the decomposition phase pro-
duces a contract for each component, which we will write with double
brackets ⟦C⟧ StateðCÞ. Analogously, we may write ⟦d⟧ for the constraint