- The man could have been smiling
- All a man can do is smile
The Original Poster asks about the grammatical roles of can, do and is in example (2). For simplicity, I have extracted the clause we are interested in there from the larger example.
As all three of these words are verbs, it will come as no surprise that they are each heading their own verb phrase and may therefore be thought of as Heads or Predicators (a Predicator is just a special type of Head. A Head is just the main word which a larger phrase is built around, and which determines the structure of that phrase and where it occurs within larger phrases or clauses).
However, knowing that can, do and is in (2) are all Heads of verb phrases really only tells us a very little bit about their 'grammatical role' within the larger sentence. It tells us about their place in the structure of the immediate chunk of words that that verb occurs in and nothing else. And really, we could take any word of category 'X' and ask "What is the grammatical function of this word?" and the answer would nearly always be 'Head of an X phrase.' If you really want to know about the structural role played by a word, you need to also think about what job that phrase is doing in the larger sentence.
In example (1) you will see a string of four verbs in bold. Each of these verbs has a special relationship specifically with the verb that precedes or follows it. To start off with, the modal verb could heads the verb phrase:
- could have been smiling
It takes as its Complement another verb phrase headed by the auxiliary verb have:
- have been smiling
The verb have here takes as its complement another verb phrase headed by the verb been:
- been smiling
And of course within this verb phrase we see the auxiliary verb been taking a verb phrase headed by the present participle smiling (a one-word verb phrase in its own right).
So in (1) we see a chain or catenation of verbs and verb phrases where one verb takes a phrase headed by another verb, which takes a phrase headed by another verb, to give one large verb phrase within which successive verb phrases are nested:
- The man [could [have [been [smiling]]]].
In (6) above, the verbs could, have, been and smiling are all Heads of their own verb phrases. However, their arrangement and structure are very different indeed from those of the verbs can, do is and smile in (2).
Clauses such as the one in (2), reproduced here as example (7) are sometimes referred to as all-clefts:
- All (that) a man can do is smile.
All-clefts have three notable features: a noun phrase Subject consisting of the determiner all modified by an integrated relative clause; some form of the verb be used as a copula verb in its specifying sense; a phrase functioning as Predicative Complement of the verb be.
Notice then that in this sentence the verb can heads the verb phrase can do which appears inside the relative clause a man can do, which in turn appears within the subject noun phrase all (that) a man can do.
The following verb is is not a Complement of the verb can or the verb do but instead heads the verb phrase is smile which functions as the Predicate of the matrix clause (the main clause). It therefore occurs in a completely different constituent (a different chunk of words) from the preceding verbs. Structurally, it is arguably the most important word in the whole sentence.
Lastly we turn to the verb smile. Although this verb occurs as the Complement of the verb be here, the two verbs do not form part of an aspectual construction or a passive. Smile is not the Catenative Complement of is. In (1) we saw the verb phrase been smiling, where the -ing form smiling was mandatory because of the verb been. In other words, the verb be dictated the form of the verb. We could not have used a plain form of the verb, smile, for example:
- *could have been smile (ungrammatical)
In the verb phrase is smile, however, the form of the verb smile does not depend on the verb is. When be is used in its a specifying sense as a copular verb, it does not put restrictions on any verb it takes as a Complement. In such constructions it is the Subject, in fact, which determines the form of verb functioning as Predicative Complement. Consider:
- [All he is doing] is [smiling].
In the example:
- all a man can do is smile
... the verbs can, do and is are not part of the same verb phrase. They do not form a constituent. In fact can and do occur within a relative clause within the Subject, whereas is is the Head verb in the verb phrase forming the Predicate.
Within the relative clause in the subject, can is the Head of the Predicate. It takes a smaller verb phrase headed by do as its Complement.
Incidentally, the verb smile, which occurs as Complement of the verb is, is not a Catenative Complement, but a Predicative one. Thus although it occurs as a Complement of the verb, its form is actually dependent on the form of the Subject noun phrase, not the Head verb.