4.1.3 Orthogonal Components

Orthogonal components are a special kind of composite states.

Orthogonal components belonging to the same (orthogonal or non-orthogonal) composite state are separated by dashed lines across the round-corner box of the composite state. The current state of that composite state is the Cartesian product of the current states of all those orthogonal components. Substates may be defined inside each of the orthogonal components. If no substate is defined in it, the orthogonal component is a leaf state.

If a composite state has an orthogonal component as one of its children states, all its other children states must also be orthogonal components.

It is possible that an orthogonal component has orthogonal components as its children. Suppose M.A and M.B are orthogonal components of M, and M.B.C and M.B.D are orthogonal components of M.B. According to the definition of orthogonal components, the current state of M is equal to the Cartesian product of the current states of M.A and M.B, i.e., $S(M) = S(M.A) \times S(M.B)$ ($S(M)$ is the function to compute the current state(s) of $M$). Similarly, $S(M.B) = S(M.B.C) \times S(M.B.D)$. As a result, $S(M) = S(M.A) \times S(M.B.C) \times S(M.B.D)$.

The names of the orthogonal components are shown in rectangles inside them. According to the naming convention, orthogonal components of the same composite state should have different names.

Figure 4.3: An example of the graphical representation of orthogonal components

Figure 4.3 shows an example of orthogonal components. Composite state A has three orthogonal components defined in it: A.B, A.C and A.D, each of which has its inner structure.

Figure 4.4: Alternate graphical representation of orthogonal components in AToM$^3$

Alternately, AToM$^3$ shows the same example in a slightly different way (Figure 4.4).