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Commit 72ad4b11 authored by Theo von Arx's avatar Theo von Arx
Browse files

Add Fix-point iteration

parent 97609f67
...@@ -147,15 +147,27 @@ Determine the set of all direct successor states of a given set of states $Q$ by ...@@ -147,15 +147,27 @@ Determine the set of all direct successor states of a given set of states $Q$ by
\subsubsection{Fixed-Point Iteration} \subsubsection{Fixed-Point Iteration}
TODO Starte mit dem Initialzustand und bestimme die Menge der innert einem oder mehreren Schritten
erreichbaren Zustände.
\begin{align}
Q_0 &= \{q_0\}\\
Q_{i+1} &= Q_i \cup \{ q' | ∃ q \text{with } \psi_Q(q) \cdot \psi_\delta(q,q')\}\\
\shortintertext{oder}
\psi_{Q_{i+1}}(q') &= \psi_{Q_i}(q) + (∃q : \psi_{Q_i}(q) \cdot \psi_\delta (q, q'))\\
\intertext{wiederhole den Iteratiosschritt solange, bis gilt:}
Q_{i+1} &= Q_i =: \hat{Q}
\end{align}
Dann beschreibt $\hat{Q}$ die Menge aller erreichbaren Zustände.
\subsubsection{Comparison of Finite Automatas} \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 $u$: 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 $u$:
\begin{align} \begin{align}
\intertext{transition function of $A$:} & ψ_r^A (x_A', x_A , 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 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 states:} & ψ_X (x_A , x_B) \\
\intertext{joint function of reachable output:} \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)\\ 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))
\end{align} \end{align}
\begin{fsatz} \begin{fsatz}
......
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