```
date: 20221205115525
```

# Notacja Chomsky'ego:
$G=<V,\Sigma,P,\sigma>$, gdzie:
- V - symbole terminalne
- $\Sigma$ - metaalfabet - alfabet pomocniczy - symbole pomocnicze
- P - lista produkcji
- $\sigma$ - Głowa gramatyki


abc 
aaaabcccccc
aabbbc

## $\begin{gathered}a^{n}b^{m}c^{k} &n,m,k\geqslant1\end{gathered}$

$G_{1}=<V_1,\Sigma_1,P_1,\sigma_1>$
$V_1 = \{a,b,c\}$
$\Sigma_1=\{\alpha,\beta,\gamma\}$
$P_1=\begin{cases}\alpha \rightarrow a\alpha \\ \alpha \rightarrow a\beta \\ \beta \rightarrow b\beta \\ \beta \rightarrow b\gamma \\ \gamma \rightarrow c\gamma \\ \gamma \rightarrow c \end{cases}$
$\sigma_1 = \alpha$

## $\begin{gathered}a^{n}b^{m}c^{n} &n,m\geqslant1\end{gathered}$
$G_{2}=<V_2,\Sigma_2,P_2,\sigma_2>$
$V_2 = \{a,b,c\}$
$\Sigma_2=\{\alpha,\beta\}$
$P_2=\begin{cases}\alpha \rightarrow a\alpha c \\ \alpha \rightarrow a\beta c \\ \beta \rightarrow b \beta \\ \beta \rightarrow b \end{cases}$
$\sigma_2 = \alpha$

## $\begin{gathered}a^{n}b^{n}c^{n} &n\geqslant 1 \end{gathered}$
$G_{3}=<V_2,\Sigma_2,P_2,\sigma_2>$
$V_3 = \{a,b,c\}$
$\Sigma_3=\{\alpha,\beta\}$
$P_3=\begin{cases}\alpha \rightarrow abc \\ \alpha \rightarrow a\alpha \beta c \\ b\beta \rightarrow bb \\ c\beta \rightarrow \beta c \end{cases}$
$\gamma = \alpha$

## Przykłady
abbccc w $G_1$
$\alpha \rightarrow^2 a\beta \rightarrow^3 ab\beta \rightarrow^4 abb\gamma \rightarrow^5 abbc\gamma \rightarrow^5 abbcc\gamma \rightarrow^6 abbccc$

Drzewo gramatyki
```
abbccc
||||||
|||||γ
||||γ
|||γ
||β
|β
α
``` 

aaabbbccc $a^nb^nc^n$

$\alpha \rightarrow^2 a\alpha\beta c \rightarrow^2aa\alpha\beta c\beta c\rightarrow^1 aaabc\beta c\beta cc \rightarrow^{4,4} aaab\beta c\beta cc \rightarrow^4 aaab\beta\beta cc \rightarrow^3  aaabb\beta ccc\rightarrow^3 aaabbbccc$

# Notacja BNF:
$V={a,b,c}$
$\Sigma = \{<\alpha>,<\beta>\}$
$P = \{<\alpha>::= a<\alpha>c|a<\beta>c\ \\ <\beta>::= }$