Properties vs. States

Figure 4.1: Properties vs. States

To understand this translation one has to consider that this translations bridges two different worlds: the real world of concrete objects with properties and the abstract or symbolic world of formal models.

To describe a problem $P$ in the real world requires the description of real concrete objects by collections of individual properties -called Property Sets- and concrete actions -or events-, which are causing the change of at least one property of a property set. Thus if e.g. the interface of a PC has at a certain moment of time (always generated by some technical device called 'clock') a certain collection of properties $P_{1}$ (colored shapes on the screen, keys of a keybord, mouse buttons, etc.) and a user presses the left mouse button with the mouse arrow above a certain colored shape on the screen (action, event) $a_{1}$, then will this click eventually change the property set because e.g. the clicked shape will change it's color (or more). Then we have a new set of properties $P_{2}$. In this sense is the real world describable by property sets, which are static, and actions/ events, which are dynamic because they are causing minimal changes. The language which provides expressions to represent property sets we call Property Language $L_{PROP}$ and the language which describes actions we call Action Language $L_{ACT}$.

The problem with descriptions of property sets and actions is that they always can only describe concrete instances at concrete moments of time. The can not describe general relations between abstract concepts (categories,...).

For abstract relations and structures we can use Formal Models $M$ like automata. A typical finite automaton has the structure $\langle Q, I, F, \Sigma^{*}, \Xi^{*}, \Delta \rangle$. A formal automaton has no relation to the real world! With the aid of a formal automaton one can construct executions which can be represented as execution graphs or one can translate the description of an automaton in a transition graph showing the states of the automaton connected by transitions. Walking along the possible paths of a transition graph represents sequences of states and events in a certain order.

If one wants to use a formal automaton to interprete real world property sets and change causing events then one has to establish a mapping from the formal structures of the automaton into the real world descriptions (which usually are based on observations or even measurements). Examples of such mappings could be:

$$l : Q \longmapsto 2^{\Pi}$$ (4.2)

$$a : Q \times \Sigma^{*} \times Q \longmapsto ACTIONS$$ (4.3)

This can be read as: $l$ maps a state $q \in Q$ into a finite set of properties and $a$ maps a transition $\langle q, \sigma,q'\rangle$ onto a concrete action $a_{e}$ causing a change from one property set to another one. To be able to do this one has to extend the formal description of an automaton as follows $\langle Q, I, F, \Sigma^{*}, \Xi^{*}, \Delta, l, \Pi, a \rangle$.

The important point here is the following one: the same property set can be part of different states! This enables the possibility to represent within the formal model of an automaton the fact, that a certain property set can occure several times but at different positions in the sequence of actions. If someone wants to enter a door with an electronic key it can happen, that he/she enters several times the same key, but only then when he enters the key in the wanted order of a certain sequence of keys the door will open. With the language of properties and actions one can only describe the actual moment of pressing a certain key, but in an automaton one can represent a certain sequence of states and transitions (to be mapped onto actions) where a certain property set is associated/ attached to a certain state which can only be reached by passing certain other states with possibly other property sets attached.

Having this general idea in mind one can start a translation process.

(The following text has to be 'improved' based on the above view).