# Definition:Formal Language/Alphabet

## Definition

Let $\LL$ be a formal language.

The **alphabet** $\AA$ of $\LL$ is a set of symbols from which collations in $\LL$ may be constructed.

An **alphabet** consists of the following parts:

Depending on the specific nature of any particular formal language, these too may be subcategorized.

### Letter

A **letter** of a formal language is a more or less arbitrary symbol whose interpretation depends on the specific context.

In building a formal language, **letters** are considered to be the undefined terms of said language.

An important part of assigning semantics to a formal language is to provide an interpretation for its **letters**.

### Sign

A **sign** of a formal language $\LL$ is a symbol whose primary purpose is to structure the language.

In building a formal language, **signs** form the hooks allowing the formal grammar to define the well-formed formulae of the formal language.

Common examples of **signs** are parentheses, "(" and ")", and the comma, ",".

The logical connectives are also **signs**.

**Signs** form part of the alphabet of a formal language.

Unlike the letters, they must be the same for each signature for the language.

### Primitive Symbol

Let $\AA$ be the alphabet of a formal language $\LL$.

The symbols which comprise $\AA$ are called the **primitive symbols** of $\AA$.

It is usual, during the development of a formal system, to introduce further symbols in order to abbreviate what would otherwise be unwieldy constructions.

Hence the distinction between these newly-introduced symbols and the **primitive symbols**.

## Also denoted as

Some sources use $\Sigma$ to denote an arbitrary **alphabet**.

## Also see

- Results about
**alphabets**in the context of**Formal Language**can be found**here**.

## Sources

- 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (previous) ... (next): Chapter $1$: Preliminaries: $1.1$ Strings, Alphabets and Languages - 1988: Dominic Welsh:
*Codes and Cryptography*... (previous) ... (next): Notation: Alphabets and strings

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- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**alphabet** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**alphabet**