==== How Can We Define Languages in Computer Science? ====

Several scientific methods have been developed to precisely define the syntactic rules of languages.

=== Backus-Naur Form (BNF): ===
A **meta-language** used to describe the syntax of languages.

| ''<name>''  | meta-symbol |
| ''::=''     | definition |
| ''|''       | alternative |
| ''{expression}'' | repetition (minimum and maximum repetitions can be specified using subscripts) |
| ''GOTO''    | terminal symbol (in quotes for clarity, can also use apostrophes instead) |

=== 1. Example: Syntax of License Plates ===

Let’s start with a few typical examples and try to generalize:

''ABC-1234, GHT-234, HSD-333, AI-BB-654''

Syntax definition:
$$
<license\_plate>::=<newType> | <oldType> \\
<oldType>::=\{<letter>\}_3^3 − \{<number>\}_3^3 \\
<newType>::=\{<letter>\}_2^2 − \{<letter>\}_2^2 − \{<number>\}_3^3 \\
<letter>::=A|B|C...|Z \\
<number>::=0|1|2|3|4|5|6|7|8|9 \\
$$

=== 2. Example: Syntax of Phone Calls in Hungary ===

Let’s list a few examples and try to generalize:

''062012345, +36301234567, 0680460046''

Syntax definition:
$$
<phone call>::= \{<prefix>\}_0^1 <city> <customer> \\
<prefix>::=\{+\}_0^1 36|06 \\
<city>::=\{<number>\}_1^2\\
<customer>::=\{<number>\}_6^7\\
<number>::=0|1|2|3|4|5|6|7|8|9 \\
$$

=== 3. Example: How can we describe the BNF formula using itself? ===

\[
\langle \text{BN formula} \rangle ::= \langle \text{rule} \rangle 
\]
\[
\langle \text{rule} \rangle ::= \langle \text{identifier} \rangle "::=" \langle \text{expression} \rangle
\]
\[
\langle \text{identifier} \rangle ::= "\langle" \langle \text{letter} \rangle \mid \left\lbrace \langle \text{letter} \rangle \mid \langle \text{digit} \rangle \right\rbrace "\rangle"
\]
\[
\langle \text{expression} \rangle ::= \langle \text{term} \rangle \left\lbrace "|" \langle \text{term} \rangle \right\rbrace
\]
\[
\langle \text{term} \rangle ::= \langle \text{factor} \rangle \left\lbrace \langle \text{factor} \rangle \right\rbrace
\]
\[
\langle \text{factor} \rangle ::= \langle \text{identifier} \rangle \mid \langle \text{terminal\_symbol} \rangle
\]
\[
\langle \text{terminal\_symbol} \rangle ::= \left\lbrace \langle \text{character} \rangle \right\rbrace
\]
\[
\langle \text{letter} \rangle ::= \langle \text{uppercase} \rangle \mid \langle \text{lowercase} \rangle
\]
\[
\langle \text{uppercase} \rangle ::= A \mid B \mid C \dots Z
\]
\[
\langle \text{lowercase} \rangle ::= a \mid b \mid c \dots z
\]
\[
\langle \text{digit} \rangle ::= 0 \mid 1 \mid 2 \mid 3 \mid 4 \mid 5 \mid 6 \mid 7 \mid 8 \mid 9
\]