# Definition:Formal Grammar/Bottom-Up

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## Definition

Let $\LL$ be a formal language whose alphabet is $\AA$.

A **bottom-up grammar** for $\LL$ is a formal grammar whose rules of formation allow the user to build well-formed formulas from primitive symbols, in the following way:

- Letters are well-formed formulas.
- A collection of specified collations of well-formed formulas, possibly labeled with additional signs, are also well-formed formulas.

In certain use cases, the first clause is adjusted to allow for more complex situations, for example in the bottom-up specification of predicate logic

### Extremal Clause

The **extremal clause** of a bottom-up grammar is the final rule which excludes all collations other than those specified in the formation rules from being well-formed formulas.