Definition:Bottom-Up Form of Top-Down Grammar

Definition

Let $\LL$ be a formal language.

Let $\LL$ be given by a top-down formal grammar $\TT$.

The bottom-up form of $\TT$ is the formal grammar $\BB$ defined by declaring that:

Each letter of $\LL$ is a $\BB$-WFF

and, for each rule of formation $\mathbf R$ of $\TT$ of the form:

A metasymbol may be replaced by the collation $\phi$ with metasymbols $\phi_1, \ldots, \phi_n$

declaring that $\BB$ has the rule of formation $\mathbf R_\BB$:

If $\phi_1, \ldots, \phi_n$ are $\BB$-WFFs, so is $\phi$.