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Commit 97609f67 authored by Theo von Arx's avatar Theo von Arx
Browse files

Improve Reachability of States

parent 45daac58
......@@ -109,6 +109,8 @@ Beim Zustandsübergang können Variablen neue Werte zugewiesen oder Events gener
Alle Events, Zustände und Aktionen sind global sichtbar.
State Charts werden in drei Phasen simuliert:
......@@ -117,6 +119,52 @@ State Charts werden in drei Phasen simuliert:
\item Übergänge werden durchgeführt, Variablen erhalten ihren neuen Wert.
\subsection{Reachability of States}
\paragraph*{Problem:} Is a state reachable by a sequence of state transitions?
\item Represent set of states and the transformation relation as OBDDs.
\item Use these representations to transform set of sets. Set corresponds to the set of states reachable after $i$ transitions.
\item Iterate the transformation until a fixed-point is reached, i.e., until the set of states does not change anymore (steady-state).
\subsubsection{Transformation of Sets of States}
Determine the set of all direct successor states of a given set of states $Q$ by means of the transformation function:
Q' &= \text{Suc}(Q, \delta) = \{ q' \vert ∃ q \text{ with } \psi_Q(q) \cdot \psi_\delta(q,q')\}\\
\psi_Q(q) = 1 &\Leftrightarrow q\in Q\\
\psi_\delta(q,q') = 1& \Leftrightarrow \text{There exists a transition between } q \text{ and } q'
\subsubsection{Fixed-Point Iteration}
\subsubsection{Comparison of Finite Automatas}
We define the following characteristic functions for two automatas $A$ and $B$ with states $x_A$, $x_B$ and outputs $y_A= w(x_A)$, $y_B= w(x_B)$ respectively and the shared input $u$:
\intertext{transition function of $A$:} & ψ_r^A (x_A', x_A , u)\\
\intertext{joint transition function:} ψ_f (x_A , x_A' , x_B , x_B' ) = & (∃u : ψ_r^A (x_A , x_A' , u) \cdot ψ_r^B (x_B , x_B' , u))\\
\intertext{joint function of reachable states:} & ψ_X (x_A , x_B) \\
\intertext{joint function of reachable output:}
ψ_Y (y_A , y_B) = &~(∃ x_A, x_B : \psi_X(x_A,x_B) \cdot \psi_w^A(x_A, y_A) \cdot \psi_w^B(x_B, y_B)\\
The automata are not equivalent iff the following term is true:
∃ y_A,y_B~:~ \psi_Y(y_A, y_B)\cdot (y_A \neq y_B)
Im Gegensatz zu hierarchischen State Maschinen (State Charts), Zustandsübergange in einem Petri-Netz sind asynchron. Die Reihenfolge der Übergänge ist zum Teil unkoordiniert. Sie ist gegeben durch eine Teilreihenfolge. \\
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