# Definition:Collation

## Definition

A collation is a structured alignment with certain placeholders that underpins the construction of formal languages.

These placeholders may be replaced by elements of an alphabet $\mathcal A$ under consideration.

A collation in $\mathcal A$ is one where all placeholders are replaced by symbols from $\mathcal A$.

For example, if we take $\square$ to denote a placeholder, then $\square\square\square\square\square$ represents the collation "a word of length $5$".

We can see that then the word "sheep" is an instance of the collation "a word of length $5$" in the English alphabet, as is "axiom".

Typical examples of collations encountered in mathematics are words or structured graphics like labeled trees.

### Collation System

A key feature of collations is the presence of methods to collate a number of collations into a new one.

A collection of collations, together with a collection of such collation methods may be called a collation system.

For example, words and the method of concatenation.

Let $\mathcal C$ be a collation system, and let $\mathcal A$ be an alphabet.

Suppose that for any two collations from $\mathcal C$, $C$ and $C'$, in the alphabet $\mathcal A$, it holds that:

If $C$ and $C'$ are indistinguishable, then $C = C'$.

Then $\mathcal C$ has the unique readability property for $\mathcal A$.

## Examples

The concept of collation being a very fundamental and abstract one, it is helpful to discuss some examples.

• Any word in natural language is a collation in the standard alphabet;
• Any number is a collation in the alphabet of digits;
• Any sentence is a collation in the alphabet of all words;
• Any sentence is a collation in the alphabet of letters and punctuation marks;
• Any movie is a collation in the alphabet of stills;
• Any Lego construction is a collation in the alphabet of Lego bricks;
• Any labeled tree is a collation.

We see that unique readability is typically ensured by:

• Position on the paper (or any other 2D carrier);
• Position in time;
• Position in the real world (or any other 3D environment).