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z7~mOSKJAScq2>{nraG)(!(Xv~=RBsVUxp|xZO~?m742S2;zCD?m?h!z1L`nT?|gf}9xjw&|QNLm&K@}gCI3(*kAH`%pIYaHj$tI|?W z;5v&krethIrrGM{EZWeQ6=`68fY;4GKC=5aNQCEn|4L5apPn=Szx?zUTX5L%$@mav RC%6=XytIl`rNmpm{{bZ(#})tp literal 0 HcmV?d00001 diff --git a/Writerside/in.tree b/Writerside/in.tree index 323ad2e..5f6b467 100644 --- a/Writerside/in.tree +++ b/Writerside/in.tree @@ -126,6 +126,8 @@ + + diff --git a/Writerside/topics/04/Theoretische Informatik/Hausaufgaben/ti_hausaufgabe6.md b/Writerside/topics/04/Theoretische Informatik/Hausaufgaben/ti_hausaufgabe6.md new file mode 100644 index 0000000..d4247d9 --- /dev/null +++ b/Writerside/topics/04/Theoretische Informatik/Hausaufgaben/ti_hausaufgabe6.md @@ -0,0 +1,146 @@ +# Übungsblatt 6 +> Wenzel Schwan (1125033), Paul Kneidl (1125219), David Schirrmeister (1125746), Michelle Klein (1126422) + +## Übung 1 +Betrachten Sie folgendes algorithmisches Problem: + +**Bestimmung des Nerode-Index** +- **Eingabe**: ein (vollständiger) NEA $N$ +- **Ausgabe**: der Nerode-Index der Sprache $L(N)$ + +Beschreiben Sie eine algorithmische Lösung dieses Problems (stichpunktartig oder als Pseudocode). +Nutzen Sie dazu das in der Vorlesung gesammelte Wissen und begründen Sie damit die Durchführbarkeit und Korrektheit Ihrer Lösung. + +**Lösung:** + +- Potenzautomat bilden +- DEA minimieren +- Zustände zählen + +## Übung 2 +Betrachten Sie den Automaten $A$ aus Abbildung 1 über dem Alphabet $\Sigma = \{0,1\}$. +![image_864.png](image_864.png) + +### 2(a) +Wenden Sie das in den Vorlesungsfolien beschriebene Verfahren zur Minimierung des Automaten $A$ an, um die Zustands-Äquivalenzklassen des Automaten $A$ zu bestimmen. +Stellen Sie Ihr Vorgehen nachvollziehbar dar (orientieren Sie sich dazu an dem finalen Beispiel des Foliensatzes 4). + +- Äquivalenzklassen finden + - {a,b,c,d,h,f}, {e,g} + - Zeuge: ε + - {a,c,d,f}, {b,h}, {e,g} + - Zeuge: 1 + - {a,d,c}, {f}, {b,h}, {e,g} + - Zeuge: 01 + - {a,c}, {d}, {f}, {b,h}, {e,g} + - Zeuge: 11 + - {a,c}, {d}, {f}, {b}, {h}, {e,g} + - Zeuge: 001 + +- Zustandsübergänge + - $≡_{a,c}$ + - 1 → $≡_{d}$ + - 0 → $≡_{b}$ + - $≡_{d}$ + - 0,1 → $≡_{b}$ + - $≡_{b}$ + - 0 → $≡_{a,c}$ + - 1 → $≡_{e,g}$ + - $≡_{e,g}$ + - 0 → $≡_{f}$ + - 1 → $≡_{h}$ + - $≡_{h}$ + - 0 → $≡_{f}$ + - 1 → $≡_{e,g}$ + - $≡_{f}$ + - 0, 1 → $≡_{f}$ + +### 2(b) +Konstruieren Sie den Quotientenautomaten zu $A$ mit Hilfe der von Ihnen bestimmten Zustands-Äquivalenzklassen (in der Graphdarstellung). + +```plantuml +@startuml +scale 0.50 + +left to right direction +skinparam dpi 150 + +skinparam state { + BackgroundColor #FFFACD + BorderColor black + FontName Helvetica + RoundCorner 30 + Shadowing false + LineThickness 0 +} + + +state eg##[bold] + +[*] --> ac +ac --> d: 1 +ac --> b: 0 +d --> b: 0,1 +b --> ac: 0 +b --> eg: 1 +eg --> h: 1 +eg --> f: 0 +h --> eg: 1 +h --> f: 0 +f --> f: 0,1 + + +@enduml +``` + +### 2(c) +Welche Sprache akzeptiert der Automat $A$? + + + +## Übung 3 +Betrachten Sie die beiden deterministischen endlichen Automaten $A_1$ und $A_2$ aus Abbildung 2. +Lesen Sie Kapitel 2.3.3 über das Äquivalenzproblem für endliche Automaten aus dem Skript. +Nutzen Sie das dort beschriebene Vorgehen, um nachvollziehbar zu überprüfen, ob die Automaten $A_1$ und $A_2$ die gleiche Sprache akzeptieren. + +![image_865.png](image_865.png) + + + +## Übung 4 +### 4(a) +Geben Sie die folgende Grammatik $G_1$ in formaler Tupel-Darstellung an. +Leiten Sie außerdem 5 Wörter der Sprache $L(G_1)$ ab und beschreiben Sie anschließend $L(G_1)$ in formaler Mengenschreibweise. + +Grammatik $G_1$: +- $S \Rightarrow aSa\ |\ bSb\ |\ X$ +- $X \Rightarrow 0\ |\ 1$ + +### 4(b) +Geben Sie die folgende Grammatik $G_2$ in formaler Tupel-Darstellung an. +Leiten Sie außerdem 5 Wörter der Sprache $L(G_2)$ ab und beschreiben Sie anschließend $L(G_2)$ in formaler Mengenschreibweise. + +Grammatik $G_2$: +- $S \Rightarrow A00A$ +- $A \Rightarrow 0A\ |\ 1A\ |\ 0\ |\ 1$ + +### 4(c) +Geben Sie alle möglichen Ableitungen mit Regeln aus der Grammatik $G_2$ für das Wort $10001$ an. + + +## Übung 5 +Betrachten Sie folgende Grammatik $G$: +- $S \Rightarrow x\ |\ yS$ +- $xy \Rightarrow yx$ +- $yx \Rightarrow xy$ + +### 5(a) +Können Sie mit $G$ das Wort $xxy$ ableiten? +Können Sie mit $G$ das Wort $xyy$ ableiten? +Begründen Sie jeweils Ihre Antwort. + +### 5(b) +Beschreiben Sie die von $G$ erzeugte Sprache in formaler Mengenschreibweise. + +### 5(c) +Gibt es eine Grammatik $G'$ mit $L(G') = L(G)$, sodass $G'$ weniger als vier Regeln hat?