Page 282 - Artificial Intelligence for the Internet of Everything

260 Artificial Intelligence for the Internet of Everything

controlled power source, a compressor, and a fan, and the heater as a boiler
together with a collection of radiators. Each of these component decompo-
sitions could be represented graphically, and by substituting those decompo-
sitions into Fig. 13.1B we would get a new decomposition of the HVAC
system at a finer level of granularity.
This composition is required to satisfy an important technical condition
called associativity, which concerns threefold decompositions. Notationally,
c
we represent an architecture as a many—one relationship d : C 1 ,…,C n !S
or, more compactly, d : C i ! S. This associates the interior interfaces C i
with the exterior interface S.
Now, suppose that we further decompose each component C i into sub-
components K ij , which are linked by a family of subarchitectures
c i : K ij ! C i . Graphically, the system boundary of c i matches one of the com-
ponent boundaries in d; composition in the operad corresponds to pasting
the former into the latter, yielding a decomposition of S into the subsub-
components K ij :

c i d ¼ðc 1 ,…,c n Þ d : K ij ¼ K 11 ,…,K nm ! S

Now, given a third layer of decompositions b ij : L ijk ! K ij we can ostensibly
relate S to L ijk in two different ways; associativity demands that these should
coincide:

ðb ij c i Þ d ¼ b ij ðc i dÞ
In other words, we could either combine the descriptions of the first and
second layers, and then insert the third layer, or vice versa combine the
second-layer components with the third layer individually and then incor-
porate the results into the first layer. One of the reasons diagrammatic syntax
is so useful for operads is that this condition is automatically satisfied by
graphical substitution.
Once we have an operad to provide the syntax for our architectures, we
can use it to structure the semantics of system models. First, we need a
semantic context like dynamics, probability, or computation. The breadth
of semantics for operads is a second element supporting the breadth of engi-
neering applications. To give an idea of how this works, we will represent
contracts on the system using logical semantics.
To provide logical semantics for an operad we must first associate each
object with a set of possible states or instances. For us, this state space is
defined by the list of boundary parameters and their types; the state space