Definition:Collation/Unique Readability
Definition
Let $\CC$ be a collation system.
Let $\AA$ be an alphabet.
Suppose that for any two collations from $\CC$, $C$ and $C'$, in the alphabet $\AA$, it holds that:
- If $C$ and $C'$ are indistinguishable, then $C = C'$.
Then $\CC$ has the unique readability property for $\AA$.
Examples
It is clear that any word in natural language is uniquely readable: we are able to distinguish words if and only if they have a different letter at some position.
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Some non-examples can help to illustrate the concept and its usefulness.
Non-examples come in two sorts: Ones where two placeholders cannot be distinguished, and ones where two symbols substituted for a placeholder cannot be distinguished.
As to the first, consider a coin with two identical sides. We consider a placeholder on each side, suitable for one letter.
Then if we write $A$ on one side and $B$ on the other, this cannot be distinguished from writing $B$ on the one side and $A$ on the other.
So since we cannot distinguish the two placeholders, there is no way of knowing whether we tried to express $AB$ or $BA$.
As to the second, consider writing down words of natural language without spaces.
Then we would have to guess whether $\mathsf {Iamnowhere}$ should be read as $\mathsf {I \, am \, now \, here}$ or as $\mathsf {I \, am \, nowhere}$.
Sources
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text {II}.4$ Polish Notation