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Compositional Models for Complex Systems  253


                                   X
              choice but to set h(0) ¼ c . Once we have h(0), condition (ii) tells us where
              to send 1:

                                           ðiiÞ  X     X  X
                             hð1Þ¼ hðtð0jÞÞ ¼ t ðhð0ÞÞ ¼ t ðc Þ
              Similarly, the value at 1 entails the value at 2, and so on, so that in general we
              have no choice but to set

                                          n times
                                        zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{
                                        X    X  X
                                  hðnÞ¼ t ð…t ðt ðc X Þ…Þ
                 We can tell a similar story about controlled DDS. These are structures in
              which we have a set of commands A, and can issue a command to the system
              at each timestep. This corresponds to a signature in which we have a con-
              stant (initial state) c and several different transition functions t a , one for each
              a 2 A. A mapping must preserve the initial state as well as each of the
              controlled transitions.
                 The term algebra for controlled DDS is List(A), the set of lists with
              entries from A. The “initial state” is just the empty list hi, and the transition
              operations append elements of A to the front of the list. Just as above, a
              homomorphism h: List(A) ! X is completely determined by the con-
              trolled DDS constraints; for example,


                                  ha 0 ,a 1 ,a 2 i7!t a 0  ðt a 1  ðt a 2  ðc X ÞÞÞ

                 Rather than treating each element of the signature individually, we can
              wrap everything together into one component. First, we use the Cartesian
              product to replace the individual mappings t a : X ! X with a single map t :
              A   X ! X. We can also use a disjoint union to package together the tran-
              sition map with the constant, so that entire controlled DDS is represented by
              a single map α:1+(A   X)!X. Thus the construction FX ¼ 1+ (A   X)
              encodes the signature of the algebraic theory.
                 Similarly, the mapping condition for controlled DDS homomorphisms is
              expressed by a single square:

                                           1+(A×h)
                             1+(A × X)              1+(A × Y )
                                 α                        β
                                  X                     Y
                                             h
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