Primary auxiliary verbs

English has three primary auxiliary verbs: do, be, and have. All three take part in the formation of various grammatical constructions, but carry very little meaning themselves. For example, the primary auxiliary be is used to form the progressive, as in: Bill is dancing. However, it makes very little sense to ask what is means in this sentence. Instead, what is of interest is what is does, i.e. that it helps form a verb phrase which, as a whole, indicates that Bill's dancing is going on at this moment. The same reasoning applies to all the primary auxiliaries. They are auxiliaries in the true sense of being 'helpers' in conveying verbal meaning. 

Do

The most important use of the primary auxiliary do is to help form negative and (most) interrogative clauses (questions) when no other auxiliary is present in the verb phrase. This use of do is referred to as do-insertion (do-support, do-periphrasis). 
Negation with not
When a finite clause is negated by the negative adverb not, exactly one auxiliary must occur before the negation. Consider, for example, the following positive-negative pairs, where the verb phrases have been highlighted:
(1a) The solution may seem apparent. (1b) The solution may not seem apparent. (2a) The results should be compared to those of our previous experiment. (2b) The results should not be compared to those of our previous  experiment.
Negating a verb phrase that only consists of a main verb requires the insertion of a 'dummy' auxiliary, namely do, to conform with the rule for forming negative clauses in English. Compare the following positive-negative pairs:
(3a) France occupied Germany. (3b) France did not occupy Germany. (4a) Einstein discovered general relativity. (4b) Einstein did not discover general relativity.
As these examples show, the role of do is to act as the first auxiliary in the verb phrase, thus making it possible to place not immediately after it. Note: It is only when a clause is negated with not that do-insertion is required. Other negative elements like never, do not trigger do-insertion:
(5) France never occupied Germany. (6) Einstein never discovered general relativity.
Do in interrogative clauses
The formation of Yes/No-questions (i.e. ones where the expected answer is either Yes or No) in English also requires the presence of an auxiliary verb in the verb phrase. Such questions are formed by inverting the order of the grammatical subject of the clause and the first auxiliary verb of the verb phrase. For example:
(7a) The results should be compared to those of our previous experiment.
(7b) Should the results be compared to those of our previous experiment?
If you do not remember what subjects and verbs are, please follow this link:As (7a) and (7b) show, the difference between the first sentence and the second is that the subject, the results, appears in its normal initial position in the first one, whereas it has switched places with the first auxiliary verb, should, in the second. Again, if no other auxiliary is present, do-insertion occurs.
(8a) France occupied Germany. (8b) Did France occupy Germany? (9a) Einstein discovered general relativity. (9b) Did Einstein discover general relativity?
Besides Yes/No questions, English has another main type of interrogative clause, namely the so called wh-question. The term reflects the fact that this type of interrogative is introduced by an word like what, who, where, why, and how, or by a phrase containing an interrogative word, e.g. in which way, to whom, for what reason.  When interrogatives of this types occur as main clauses (i.e. are not part of another clause) the subject inverts with the first auxiliary, in a way similar to Yes/No questions.
(10a) The results should be compared to those of our previous experiment. (10b) To what should the results be compared?
          (alternatively: What should the results be compared to?)
Again, if no other auxiliary is present, do-insertion occurs:
(11a) France occupied Germany. (11b) What country did France occupy? (12a) Einstein discovered general relativity. (12b) What did Einstein discover?
Do-insertion in wh-questions is less general than in Yes/No questions. Thus it does not occur in:
  1. Interrogatives that are part of other clauses ('indirect questions'):
    The professor asked what country France occupied.
    (Not: *The professor asked what country did France occupy.)

  2. When the interrogative word or phrase functions as the subject of the clause.
    Who discovered general relativity?
    (Not: *Who did discover general relativity?)
Emphatic do
There is another use of do, which follows slightly different rules than the ones discussed above. This so called emphatic do occurs chiefly in speech, and is very rare in formal writing. It is mentioned here since non-native writers are not always familiar with its function and tend to use do where it is not called for. The following examples illustrate the use of do to make a contrast with a negative utterance or thought in the context.
(13) Mother: Lisa, why didn't you take out the garbage?         Lisa: But I did take out the garbage. (14) Please stop asking. I do love you.
By the way, if there is another auxiliary available, we put stress on this auxiliary instead of adding do, as in 15 and 16:
(15) I got the message. I will apologise. (16) You have to believe me. I am sorry.

Be

The primary auxiliary be takes part in the formation of progressive verb phrases, as well as the formation of passive verb phrases.
The progressive form
The progressive form consist of the auxiliary BE followed by the present participle (the ing-form). In the simplest case, a verb phrase in the progressive consists of just two verbs, as in the following example:
(17) The government was planning to cut its overseas aid budget. (18) Ohio is showing a small profit from its operations.
See the following for more complex examples and information on the use of the progressive:
The passive
A second use of the primary auxiliary BE is found in the formation of passive verb phrases. The passive consists of the auxiliary BE followed by the past participle (the second ed-form). In the simplest case, a passive verb phrase consist of just two verbs, as in the following example:
(19) Pluto was discovered in 1930. (20) The parents' bedroom is considered the most private room for outsiders.
See the following for more complex examples and information on the use of the passive:

In a Swedish indirect question, such as
(1) Jag vet inte vilka som kommer ikväll.
there is a som which has no English correspondent, i.e. there is no English word that is used to translate this som. What we say in English is simply
(2) I dont know who___will come tonight.
This is a very important contrastive difference for Swedes to pay attention to. Note that the Swedish som is not even translated if we expand the phrase including vilka to, for instance, vilka av hans många kusiner från Finland so that we get
(3) Jag vet inte vilka av hans många kusiner från Finland som kommer ikväll.
This would still be translated without anything corresponding to the Swedish som:
(4) I dont know which of his many cousins from Finland___ will come tonight.
Swedish has to have a dummy som, a simple placeholder, in these cases, while English cannot have one! Note that in Swedish indirect questions in which the question-word is the object of the subordinate clause, and not the subject as in the cases above, there is not supposed to be a som in Swedish either:
(5) Jag vet inte vem (som) han gav den till.
These cases are thus not complicated from a contrastive point of view. You can read up on subjects and objects if you follow this link:

Have

The primary auxiliary have is used for the formation of the perfect.
The perfect
The perfect consist of the auxiliary have followed by the past participle (the second ed-form).  In the simplest case, a verb phrase in the perfect consists of just two verbs, as in the following examples:
(21) Most multinational computer services companies have established a direct presence in Italy. (22) The first speaker gave a resumé of what chemistry had accomplished in just a few decades.
See the following for more complex examples and information on the use of the perfect:

Primary auxiliaries used as main verbs

All the primary auxiliaries can also be used as main verbs. This usage is illustrated in the following examples:
(23) Kurchatov's laboratory in Leningrad did groundbreaking work in nuclear physics. (24) Yellowstone is famous for its geysers. (25) Vacuum distillation has two main applications.
Do, and to a large extent have, fully behave as main verbs, e.g. by requiring do-insertion in the formation of interrogative and negative clauses:
(26) Does vacuum distillation have two main applications? (27) Kurchatov's laboratory in Leningrad did not do groundbreaking work in nuclear physics.
Be, on the other hand, retains some auxiliary properties, even when it is used as a main verb. Thus, interrogatives and negatives do not require the insertion of do:
(28) Is Yellowstone famous for its geysers? (29) Yellowstone is not famous for its geysers.
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