There is yet another way to finitely represent a regular set: by a grammar—which will naturally be called a regular grammar. To motivate the core idea behind grammars, consider, for example, the (inductive) definition of formulae (126.96.36.199). Moreover, to simplify matters, let us stay in the Boolean domain—that is, we will include only the connectives ¬ and ∨ but no quantifiers—and we will also adopt as atomic formulae the set of Boolean variables,121 generated by the symbol p with or without primes. Thus, the atomic formulae include
p, p′, p‴, p(n)
where p(n) indicates p with n primes
The alphabet over which we build these simplified well-formed (Boolean) formulae is
(,), ¬, ∨, p, p′, p″, p‴, . . .
In the inductive clauses of 188.8.131.52 we have included “if and are formulae, then so is ( ∨ )”. In words this says that
One way to get a “complicated” formula is to take two formulae, and join themvia a “∨”, adding outermost brackets after that.
This generates ...