68 lines
3.4 KiB
Markdown
68 lines
3.4 KiB
Markdown
**Př. 1**: Vytvořte gramatiku, která bude nad abecedou $\{0, 1\}$ generovat řetězec obsahující lichý počet 0 a sudý počet 1.
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- $\{0, 1\} \quad w \dots \text{lichý počet } 0 \text{, sudý počet } 1$
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- 4 stavy
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**Př. 2**:
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- $S \to abA \mid bS \mid aa \mid A$
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- $A \to abA$
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- $B \to aS \mid baA \mid a$
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a) Najděte $G'$ typu G3R takovou, že $L(G') = L(G)$.
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+ $S \to bS | aS_{1} | aS_{2} \mid aS | bB_{1} | aB_{2} \mid aA_{1}$
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+ $A \to aA_{1} \mid aS | bB_{1} | aB_{2}$
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+ $B \to aS | bB_{1} | aB_{2}$
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- $S_{1} \to bA$
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- $B_{1} \to aA$
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- $A_{1} \to bA$
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- $S_{2} \to aS_{3}$
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- $S_{3} \to e$
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- $B_{2} \to e$
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b) Vytvořte tabulku popisující nedeterministický konečný automat A takový, že platí $L(A) = L(G') = L(G)$.
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| | a | b |
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| ------------------ | ----------------------------------- | -------------- |
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| $\to S$ | $\{S, S_{1}, S_{2}, A_{1}, B_{2}\}$ | $\{S, B_{1}\}$ |
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| $S_{1}$ | - | $\{A\}$ |
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| $S_{2}$ | $\{S_{3}\}$ | - |
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| $\leftarrow S_{3}$ | - | - |
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| $A$ | $\{S, A_{1}, B_{2}\}$ | $\{B_{1}\}$ |
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| $A_{1}$ | - | $\{A\}$ |
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| $B$ | $\{S, B_{2}\}$ | $\{B_{1}\}$ |
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| $B_{1}$ | $\{A\}$ | - |
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| $\leftarrow B_{2}$ | - | - |
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c) Vytvořte tabulku popisující deterministický konečný automat A' takový, že platí $L(A') = L(G') = L(G)$.
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| | a | b |
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| -------------------------------------------------------- | ---------------------------------------------- | --------------------- |
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| $\to S$ (0) | $\{S, S_{1}, S_{2}, A_{1}, B_{2}\}$ (1) | $\{S, B_{1}\}$ (2) |
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| $\leftarrow \{S, S_{1}, S_{2}, A_{1}, B_{2}\}$ (1) | $\{S, S_{1}, S_{2}, S_{3}, A_{1}, B_{2}\}$ (3) | $\{S, A, B_{1}\}$ (4) |
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| $\{S, B_{1}\}$ (2) | $\{S, S_{1}, S_{2}, A, A_{1}, B_{2}\}$ (5) | $\{S, B_{1}\}$ (2) |
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| $\leftarrow\{S, S_{1}, S_{2}, S_{3}, A_{1}, B_{2}\}$ (3) | $\{S, S_{1}, S_{2}, S_{3}, A_{1}, B_{2}\}$ (3) | $\{S, A, B_{1}\}$ (4) |
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| $\{S, A, B_{1}\}$ (4) | $\{S, S_{1}, S_{2}, A, A_{1}, B_{2}\}$ (5) | $\{S, B_{1}\}$ (2) |
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| $\leftarrow\{S, S_{1}, S_{2}, A, A_{1}, B_{2}\}$ (5) | $\{S, S_{1}, S_{2}, S_{3}, A_{1}, B_{2}\}$ (3) | $\{S, A, B_{1}\}$ (4) |
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**Př. 3**: Sestrojte NKA $A$, kde platí $L(A) = L(G_{1})^R \cup L(G_{2})$.
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- $G_{1}$
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- $S \to aS | bbA$
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- $A \to aaA | B$
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- $B \to bbB | e$
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- $G_{2}$
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- $S \to Aba | Ab | B$
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- $A \to Aaa | B$
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- $B \to Bbb | e$
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- G3L -> reverze -> G3P -> NKA -> reverze -> NKA
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Plán
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1) $A_{1} \qquad L(A_{1}) = L(G_{1})$
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2) $A_{1}^R \qquad L(A_{1}^R) = L(A_{1})^R = L(G_{1})^R$
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3) $G_{2}^R \qquad L(G_{2}^R) = L(G_{2})^R$
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4) $A_{2}^R \qquad L(A_{2}^R) = L(G_{2}^R) = L(G_{2})^R$
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5) $A_{2} \qquad A_{2} = (A_{2}^R)^R \quad L(A_{2}) = \dots = L(G_{2})$
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6) $A \qquad L(A) = L(A_{1}^R) \cup L(A_{2}) = L(G_{1})^R \cup L(G_{2})$
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$G_{2}^R$
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- $S \to abA | bA | B$
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- $A \to aaA | B$
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- $B \to bbB | e$ |