A rough attempt at characterizing Vim's Normal Mode grammar that I made while writing this Vim series. See the bibliography there for more context.
Notation: | means or, [] means optional, * means zero or more, <> means literal class (e.g. <number> could expand to 1, 2, etc.), ... means there are more instances not enumerated here (these would need to be fleshed out to get a complete grammar).
It could be interesting to use a grammar like this one to organize an interactive Vim cheatsheet.
normal-command = motion-form
| verb-phrase
| input-expression
| prepositional-verb-phrase
| pronoun-verb-form
| macro-expression
| special-form
motion-form = [quantifier] motion
verb-phrase = [quantifier] verb-form noun-form
| [quantifier] grammaticalized-verb-form
input-expression = [quantifier] input-verb
prepositional-verb-phrase = prepositional-destination-form verb-phrase
| prepositional-source-form pronoun-verb-form
pronoun-verb-form = p | P | gp | gP
macro-expression = [quantifier] built-in-macro
special-form = quantifier %
| q direct-object
| @ indirect-object
| m mark
| R | Q | : | @ | % | v | V | C-v | u | C-r | U | g- | g+ | q: | q/ | q?
motion = h j k l w W b B e E 0 $ ^ _ | gE gi gj g0 g$ gg G / ? n N f F t T , ; 'mark `mark ...
quantifier = <number>
verb-form = verb
| verb verb-clause*
noun-form = [quantifier] noun
input-verb = r i a o O
prepositional-destination-form = preposition indirect-object
prepositional-source-form = preposition direct-object
built-in-macro = x X s C S D I A J gJ ~ & .
direct-object = register | append-register
mark = <lowercase letter> | <uppercase letter> | < | > | ' | ` | ...
verb = c d y g~ gu gU gq = !
verb-clause = adverb | quantifier
noun = motion | text-object
grammaticalized-verb-form = dd | cc | yy | ...
preposition = "
indirect-object = register
register = <lowercase letter> | <number> | " | : | _ | . | ...
append-register = <uppercase letter>
adverb = v | V | C-v
text-object = modifier object
modifier = i a
object = w s p ( ) { } [ ] < > " ' ` gn gN