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info4/kapitel-3/PDA-Kellerautomaten.ipynb
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%% Cell type:markdown id: tags:
# PDA (Push Down Automata - Kellerautomaten)
%% Cell type:markdown id: tags:
Ein __(nichtdeterministischer) Kellerautomat__
(kurz PDA für __push-down automaton__) ist ein $6$-Tupel
$M = (\Sigma, \Gamma, Z, \delta , z_0, \#)$, wobei
* $\Sigma$ das Eingabe-Alphabet ist,
* $\Gamma$ das Kelleralphabet,
* $Z$ eine endliche Menge von Zuständen,
* $\delta : Z \times (\Sigma \cup \{\lambda\}) \times \Gamma
\rightarrow \mathfrak{P}_e(Z \times \Gamma^{\ast})$ die
Überführungsfunktion,
* $z_0 \in Z$ der Startzustand,
* $\# \in \Gamma$ das Bottom-Symbol im Keller.
Anmerkung: $\mathfrak{P}_e(Z \times \Gamma^{\ast})$ ist die Menge aller
__endlichen__ Teilmengen von $Z \times \Gamma^{\ast}$.
%% Cell type:code id: tags:
``` prob
::load
MACHINE PDA
/* B Modell eines PDA */
SETS
Z = {z0,z1}; // die Zustände des Automaten, z0 ist der Startzustand
SYMBOLE={a,b, A, BOT, lambda} /* BOT = # = Bottom-Symbol im Keller*/
DEFINITIONS
Σ == {a,b}; // das Eingabe-Alphabet
Γ == {A,BOT}; // das Kelleralphabet
ANIMATION_FUNCTION_DEFAULT == {(1,1,z)};
ANIMATION_FUNCTION == {2}*α \/ {3}*(γ);
ANIMATION_FUNCTION1 == {(1,0,"z: "),(2,0,"α:"),(3,0,"γ:")};
ANIMATION_STR_JUSTIFY_LEFTx == TRUE;
SET_PREF_PRETTY_PRINT_SEQUENCES == TRUE
CONSTANTS δ
PROPERTIES
/* A PDA für {a^mb^m| m>=1} ; Beispiel von Info 4 (Folie 95ff) */
/* Der PDA für {a^m b^m| m>=1} ; Beispiel von Info 4 (Folie 95ff) */
δ = { (z0,a,BOT) ↦ (z0,[A,BOT]),
(z0,a,A) ↦ (z0,[A,A]),
(z0,b,A) ↦ (z1,[]),
(z1,lambda,BOT) ↦ (z1,[]),
(z1,b,A) ↦ (z1,[]) }
// Anmerkung: δ ist hier als Relation anstatt als Funktion zu Mengen definiert
// Deshalb entspricht δ[{(z,a,g)}] in der B Maschine δ(z,a,g) aus dem Skript
VARIABLES
z, α, γ // Konfiguration in dem sich der PDA befindet
INVARIANT
z ∈ Z ∧ // der aktuelle Zustand
α ∈ seq(Σ) ∧ // der noch zu lesende Teil des Eingabeworts
γ ∈ seq(Γ) // aktuelle Kellerinhalt
INITIALISATION
z := z0 ||
γ := [BOT] || // Initialisierung des Stapels
α := [a,a,b,b] // das Eingabewort
OPERATIONS
// die Operationen Schritt und LambdaSchritt modellieren
// Schritte in der Ableitungsrelation
Schritt(z‘,s) = PRE α ≠ ∅ ∧ γ ≠ ∅ ∧
z‘↦s ∈ δ[{(z,first(α),first(γ))}] THEN
z := z‘ || // in den neuen Zustand wechseln
α := tail(α) || // das erste Symbol auf der Eingabe löschen
γ := s^tail(γ) // s auf den Stapel packen
END;
LambdaSchritt(z‘,s) = PRE γ ≠ ∅ ∧
z‘↦s ∈ δ[{(z,lambda,first(γ))}] THEN
z := z‘ || // in den neuen Zustand wechseln
γ := s^tail(γ) // s auf den Stapel packen
END;
Akzeptieren = PRE γ = ∅ ∧ α = ∅ THEN
/* Wir akzeptieren wenn Eingabe und Stapel leer sind */
skip END
END
```
%% Output
Loaded machine: PDA
%% Cell type:code id: tags:
``` prob
:constants
```
%% Output
Machine constants set up using operation 0: $setup_constants()
%% Cell type:code id: tags:
``` prob
:init
```
%% Output
Machine initialised using operation 1: $initialise_machine()
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z: </td>
<td style="padding:10px">z0</td>
<td style="padding:0px"></td>
<td style="padding:0px"></td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">α:</td>
<td style="padding:10px">a</td>
<td style="padding:10px">a</td>
<td style="padding:10px">b</td>
<td style="padding:10px">b</td>
</tr>
<tr>
<td style="padding:10px">γ:</td>
<td style="padding:10px">BOT</td>
<td style="padding:0px"></td>
<td style="padding:0px"></td>
<td style="padding:0px"></td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:browse
```
%% Output
Machine: PDA
Sets: Z, SYMBOLE
Constants: δ
Variables: z, α, γ
Operations:
Schritt(z0,{(1|->A),(2|->BOT)})
%% Cell type:code id: tags:
``` prob
:exec Schritt
```
%% Output
Executed operation: Schritt(z0,{(1|->A),(2|->BOT)})
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z: </td>
<td style="padding:10px">z0</td>
<td style="padding:0px"></td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">α:</td>
<td style="padding:10px">a</td>
<td style="padding:10px">b</td>
<td style="padding:10px">b</td>
</tr>
<tr>
<td style="padding:10px">γ:</td>
<td style="padding:10px">A</td>
<td style="padding:10px">BOT</td>
<td style="padding:0px"></td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
(z,α,γ)
```
%% Output
$(\mathit{z0}\mapsto\{(1\mapsto \mathit{a}),(2\mapsto \mathit{b}),(3\mapsto \mathit{b})\}\mapsto\{(1\mapsto \mathit{A}),(2\mapsto \mathit{BOT})\})$
(z0↦{(1↦a),(2↦b),(3↦b)}↦{(1↦A),(2↦BOT)})
%% Cell type:code id: tags:
``` prob
:browse
```
%% Output
Machine: PDA
Sets: Z, SYMBOLE
Constants: δ
Variables: z, α, γ
Operations:
Schritt(z0,{(1|->A),(2|->A)})
%% Cell type:code id: tags:
``` prob
:exec Schritt
```
%% Output
Executed operation: Schritt(z0,{(1|->A),(2|->A)})
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z: </td>
<td style="padding:10px">z0</td>
<td style="padding:0px"></td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">α:</td>
<td style="padding:10px">b</td>
<td style="padding:10px">b</td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">γ:</td>
<td style="padding:10px">A</td>
<td style="padding:10px">A</td>
<td style="padding:10px">BOT</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:browse
```
%% Output
Machine: PDA
Sets: Z, SYMBOLE
Constants: δ
Variables: z, α, γ
Operations:
Schritt(z1,{})
%% Cell type:code id: tags:
``` prob
:exec Schritt
```
%% Output
Executed operation: Schritt(z1,{})
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z: </td>
<td style="padding:10px">z1</td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">α:</td>
<td style="padding:10px">b</td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">γ:</td>
<td style="padding:10px">A</td>
<td style="padding:10px">BOT</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:browse
```
%% Output
Machine: PDA
Sets: Z, SYMBOLE
Constants: δ
Variables: z, α, γ
Operations:
Schritt(z1,{})
%% Cell type:code id: tags:
``` prob
:exec Schritt
```
%% Output
Executed operation: Schritt(z1,{})
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z: </td>
<td style="padding:10px">z1</td>
</tr>
<tr>
<td style="padding:10px">α:</td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">γ:</td>
<td style="padding:10px">BOT</td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:browse
```
%% Output
Machine: PDA
Sets: Z, SYMBOLE
Constants: δ
Variables: z, α, γ
Operations:
LambdaSchritt(z1,{})
%% Cell type:code id: tags:
``` prob
:exec LambdaSchritt
```
%% Output
Executed operation: LambdaSchritt(z1,{})
%% Cell type:code id: tags:
``` prob
:show
```
%% Output
<table style="font-family:monospace"><tbody>
<tr>
<td style="padding:10px">z: </td>
<td style="padding:10px">z1</td>
</tr>
<tr>
<td style="padding:10px">α:</td>
<td style="padding:0px"></td>
</tr>
<tr>
<td style="padding:10px">γ:</td>
<td style="padding:0px"></td>
</tr>
</tbody></table>
<Animation function visualisation>
%% Cell type:code id: tags:
``` prob
:browse
```
%% Output
Machine: PDA
Sets: Z, SYMBOLE
Constants: δ
Variables: z, α, γ
Operations:
Akzeptieren()
%% Cell type:code id: tags:
``` prob
:exec Akzeptieren
```
%% Output
Executed operation: Akzeptieren()
%% Cell type:markdown id: tags:
Das Eingabewort ``aabb`` wurde akzeptiert.
%% Cell type:code id: tags:
``` prob
```
......
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