... @@ -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} ... ...