Definition:Polish Notation/Formal Definition

Definition
Let $\mathcal A$ be an alphabet.

Let each $s \in \mathcal A$ be assigned a natural number called its arity.

The elements of $\mathcal A$ with arity $0$ are its letters.

The other elements are its signs.

The formal grammar for Polish notation is given by the single bottom-up rule:


 * If $s$ has arity $n$ and $\phi_1, \ldots, \phi_n$ are well-formed formulas, then:


 * $s \phi_1 \cdots \phi_n$


 * is also a well-formed formula.

Notably, in the case where $s$ has arity $0$, this is a vacuous truth, so any letter $s$ of $\mathcal A$ constitutes a well-formed formula.