Minimal Transitions

Static properties are only one half of the situation. The other half are dynamic actions. In our problem we can distinguish the following minimal actions:

$$'A' : S_{1} \longmapsto S_{2}$$ (4.11)

$$'B' : S_{1} \longmapsto S_{3}$$ (4.12)

$$'C' : S_{1} \longmapsto S_{4}$$ (4.13)

$$'Y_{1}' : S_{2}\longmapsto S_{1}$$ (4.14)

$$'Y_{2}' : S_{4}\longmapsto S_{1}$$ (4.15)

$$'Y_{3}' : S_{2}\longmapsto S_{5}$$ (4.16)

This shows that the description of the observable behavior has to presuppose some system actions $Y_{i}$ without knowing how these system actions are realized. They are only characterized by their effects.

Figure 4.2: Minimal sets of properties with transitions

Figure 4.3: Introducing states

What the figures 4.2 and 4.3 can demonstrate is the fact that states $Q_{i}$ are different from the sets of properties! Different states can have the same set of properties. Thus the states represent abstract objects. We can attach to one state $Q_{i}$ a certain set of properties. And therefore can the meaning of an action like pressing key 'A' be different depending in which 'context' this action occurs. In this sense does the sequence of some states connected by transitions represent a certain kind of 'history', a certain sequence of events. This history represents relations beyond that what is perceivable because perception is always attached to one concrete situation. But a sequence of states connected by transitions represent the connction of different perceivable situations, thus the structure given by objects and transitions is another 'level' of symbolic representation than that level where one represents pereceivable property sets effects of changes caused by perceivable actions. In this interpretation one should extend the mapping from states into property sets with a mapping from actions into transitions (see 4.18).

$$l : Q \longmapsto 2^{\Pi}$$ (4.17)

$$a : Q \times \Sigma^{*} \times Q \longmapsto ACTIONS$$ (4.18)