Definition:Formal Grammar

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

The formal grammar of $\LL$ comprises of rules of formation, which determine whether collations in $\AA$ belong to $\LL$ or not.

Roughly speaking, there are two types of formal grammar: top-down grammar and bottom-up grammar.

Top-Down Grammar

A top-down grammar for $\LL$ is a formal grammar which allows well-formed formulas to be built from a single metasymbol.

Such a grammar can be made explicit by declaring that:

From the words thus generated, those not containing any metasymbols are the well-formed formulas.

Bottom-Up Grammar

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:

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.

Also known as

The formal grammar may also be called syntax; however, a convenient viewpoint is to think of the formal grammar as explicating the syntax for the associated formal language.

Thus the formal grammar is a means to obtain a syntax for $\LL$, and multiple formal grammars may yield the same syntax.

Some sources call this merely a grammar, the term formal being taken for granted by the fact that a formal language is under discussion.