add theo lecture

IN0011_Theoretische_Informatik/Endliche Automaten und Reguläre Ausdrücke.yaml

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+title: Endliche Automaten und reguläre Ausdrücke
+author: Hendrik Hübner
+id: 3709563985
+cards:
+  - type: latex_plus
+    front: Def. Konkatenation von Sprachen
+    back: |+
+      [$]AB = \{xy \mid x \in A \land y \in B\}[/$]
+
+  - type: latex_plus
+    front: Def. Kleene Stern bei Sprache A
+    back: |+
+      [$]A^* = \bigcup_{i=0}^{\infty} A^i[/$]
+  - type: latex_plus
+    front: Def. Kleene Plus bei Sprache A
+    back: |+
+      [$]A^+ = \bigcup_{i=1}^{\infty} A^i = AA^*[/$]
+
+  - type: latex_plus
+    front: [$]A \emptyset = ?[$]
+    back: |+
+      [$]\emptyset[/$]
+
+  - type: latex_plus
+    front: [$]A(B \cup C) = [/$]
+    back: |+
+      [$]AB \cup AC[/$]
+
+  - type: latex_plus
+    front: Def. Grammar
+    back: |+
+      [$]G = (V, \Sigma, P, S)[$]
+
+      [$]V[/$] is the set of non-terminal symbols.
+      [$]\Sigma[/$] is the set of terminal symbols.
+      [$]P \subseteq (V \cup \Sigma)^* \times (V \cup \Sigma)^*[/$] is the set of production rules.
+      [$]S[/$] is the start symbol.
+
+  - type: latex_plus
+    front: Def. Chomsky Hierarchy
+    back: |+
+      [$]
+      Type 0: All grammars
+      L = \{ w \in \Sigma^* \mid w \text{ can be derived from the start symbol using any production rule} \}
+      
+      Type 1: Context sensitive Grammars
+      L = \{ w \in \Sigma^* \mid \text{there exists a production } \alpha \rightarrow \beta \text{ such that} |\alpha| \leq |\beta| \text{ and} w = \alpha u \gamma \text{ where} u \in \Sigma^* \text{ and} \gamma \in \Sigma^* \}
+
+      Type 2: Context free Grammars
+      L = \{ w \in \Sigma^* \mid S \Rightarrow^* w \}
+
+      Type 3: Regular languages
+      L = \{ w \in \Sigma^* \mid w \text{ can be derived from the start symbol using production rules of the form} A \rightarrow aB \text{ or} A \rightarrow a \text{ where} A,B \in V \text{ and} a \in \Sigma \}
+
+      [/$]
+
+  - type: latex_plus
+    front: Def. DFA
+    back: |+
+      [$]DFA[/$] = (Q, \Sigma, \delta, q_0, F)[/$]
+      [$]Q[/$] is the set of states.
+      [$]\Sigma[/$] is the input alphabet.
+      [$]\delta: Q \times \Sigma \rightarrow Q[/$] is the transition function.
+      [$]q0 \in Q[/$] is the initial state.
+      [$]F \suseteq Q[/$] is the set of accepting (final) states.
+
+  - type: latex_plus
+    front: Def. NFA
+    back: |+
+      [$]NFA[/$] = (Q, \Sigma, \delta, q_0, F)[/$]
+      [$]Q[/$] is the set of states.
+      [$]\Sigma[/$] is the input alphabet.
+      [$]\delta: Q \times \Sigma \rightarrow \mathcal{P}(Q)[/$] is the transition function.
+      [$]q0 \in Q[/$] is the initial state.
+      [$]F \suseteq Q[/$] is the set of accepting (final) states.
+
+  - type: latex_plus
+    front: Def. [$]\hat{\delta}(q, w)[/$]
+    back: |+
+      The state which is reached with w from q
+
+      [$]\hat{\delta}(q, \epsilon) = q[/$]
+      [$]\hat{\delta}(q, aw) = \hat{\delta}(\delta(q, a), w)[/$]
+
+  - type: latex_plus
+    front: Welche gestalt haben Produktionen bei einer rechtslinearen Grammatik?
+    back: |+
+      [$]A \rightarrow a \mid aB [/$]
+
+  - type: latex_plus
+    front: Was gilt für rechtslineare Grammatiken?
+    back: |+
+      - Es existiert ein äquivalenter DFA (und umgekehrt)
+
+  - type: latex_plus
+    front: Def. [$]\bar{\delta}(S, a)[/$]
+    back: |+
+      The state which is reached with w from q
+
+      [$]\bar{\delta}(S, a) = \bigcup\limits{q \in S} \delta(q, a} [/$]
+
+
+  - type: latex_plus
+    front: Was ist die Potenzmengenkonstruktion?
+    back: |+
+      Sei N ein NFA
+      
+      Def. [$]DFA M = (\mathcal{P}(Q), \Sigma, \bar{\delta}, q_0, F_M)[/$]
+      Mit [$]F_M := {S \subseteq Q \mid S \cap F \neq \emptyset}[/$]
+      
+      Der enstehende DFA hat bis zu 2^|Q| zustände
+
+  - type: latex_plus
+    front: Def. pumping lemma
+    back: |+
+      [$]
+      A language $L$ is regular if and only if there exists a constant $p \geq 1$ such that for any string $s \in L$ with $|s| \geq p$, $s$ can be decomposed as $s = xyz$ satisfying the following conditions:
+
+      \begin{itemize}
+          \item $|xy| \leq p$: The length of the prefix $xy$ is at most $p$.
+          \item $|y| \geq 1$: The substring $y$ is non-empty.
+          \item For all $i \geq 0$, $xy^iz \in L$: Repeating the substring $y$ any number of times ($i \geq 0$) and concatenating it with $xz$ results in a string in $L$.
+      \end{itemize}
+      [/$]
+
+  - type: latex_plus
+    front: Was gilt für [$]\epsilon[/$]-NFAs?
+    back: |+
+      Es existiert immer ein äquivalenter NFA und damit auch DFA
+      
+
+  - type: latex_plus
+    front: Welche sechs erweiterungen von Regulären Ausdrücken für UNIX haben wir kennengelernt?
+    back: |+
+      . = beliebiges Zeichen des Alphabets
+      [a1 ... an] = beliebiges Zeichen aus {a1 ... an}
+      [^a1 ... an] = beliebiges Zeichen NICHT aus {a1 ... an}
+      w? = leeres Wort oder w
+      w+ = ww*
+      w{n} = www...ww (n mal)
+
+  - type: latex_plus
+    front: Satz von Kleene?
+    back: |+
+      Eine Sprache ist genau dann als regulärer ausdruck darstellbar, wenn sie regulär ist
+
+  - type: latex_plus
+    front: Def. Ardens lemma
+    back: |+
+      [$]
+        Sind \alpha, \beta und X reguläre ausdrücke mit \epsilon \notin L(\alpha), so gilt:
+        X \equiv \alpha X \mid \beta \Rightarrow X \equiv \alpha^*\beta
+      [/$]
+
+  - type: latex_plus
+    front: Zeitkomplexität umwandlung RE -> e-NFA?
+    back: n -> Q in O(n)
+
+  - type: latex_plus
+    front: Zeitkomplexität umwandlung NFA -> DFA?
+    back: n -> Q in O(2^n)
+
+  - type: latex_plus
+    front: Zeitkomplexität umwandlung NFA -> RE?
+    back: |+
+      n -> länge RE in O(3^n)
+
+  - type: latex_plus
+    front: Zeitkomplexität umwandlung e-NFA -> NFA?
+    back: Q -> Q
+
+  - type: latex_plus
+    front: Wann sind zwei Reguläre Ausdrücke oder Automaten äquivalent?
+    back: |+
+      [$] \alpha \equiv \beta \Leftrightarrow L(\alpha) = L(\beta) [/$]
+
+  - type: latex_plus
+    front: Def. Produktautomat
+    back: |+
+      Sind [$]M_1[/$] und [$]M_2[/$] DFAs so ist der Produktautomat wie folgt definiert:
+      [$]
+      (Q_1 \times Q_2, \Sigma, \delta, (s_1, s_2), F_1 \times F_2)
+      wobei, \delta((q_1, q_2), a) = (\delta_1(q_1, a), \delta_2(q_2, a))
+      
+      Der Produktautomat akzeptiert L(M_1) \cap L(M_2)
+      [/$]