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I asked this question one year ago. Nico gave an excellent answer, but I still have some confusion.

A: We saw a police helicopter overhead yesterday morning.
B: Really? They would have been looking for those bank robbers.

As Nico mentioned in his comments, in the original context, only "would have been looking for.." applies, because the use of "we saw..." (simple past) implies that the search is also in the past and doesn't extend to the present (at least in BrE). Whereas "will have been looking for.." suggests the search extends to the present or is relevant at present.

Besides, he pointed out that in the OP's example, B isn't pretty sure, B isn't expressing certainty. Note that B starts the sentence with "Really?". B is expressing a belief. B believes that the police had been looking for those bank robbers.

I suspect "They will have been looking for those bank robbers" applies to the context as well.

My main argument is taken from F. R. Palmer's Modality and the English Modals:

Generally the modality is in the present only, because the judgments are made in the act of speaking, epistemic modals being in this sense usually 'performative'. The modal verbs are not normally used, therefore, in past tense forms to refer to past judgments. Past tense forms are normally tentative with present time reference. It is, of course, possible to report past judgments, but this requires verbs such as THINK, BELIEVE, etc.

Based on this, I think "They will have been looking for those bank robbers yesterday" is a grammatical sentence, meaning "I'm sure they were looking for those bank robbers yesterday". Is my understanding correct?

This use doesn't seem to be very productive. I got a couple examples from the Internet:

Countless hearts will have been blown open yesterday; both in Ireland, and wherever in the world literature is loved.

In terms of the GC Contador, Uran and Aru will have been relatively happy with their rides yesterday. Contador's team in fact led at the intermediate time check at 9.9km but lost ground - and their leader briefly - and had to settle for second place on the stage.

I don't know if they are standard English or regional variations. Even the present perfect can sometimes be used together with expressions of finished time, but we know that's not standard English.

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    I cannot give an answer as comphrensive as your question, but i tend to agree with Nico. Using "will" and "yesterday" doesnt sound right.
    – Chu Wa Tim Tim
    Commented Oct 16, 2015 at 9:35
  • John Prescott excepted, we can reasonably assume British MPs have a perfectly good command of their native language. I see nothing particularly unusual about Undersecretary of State George Ward saying If the hon. Gentleman listened to my right hon. Friend the Minister of Health yesterday, he will have been assured that there is no danger. It's well-known that most Anglophones are quite careless about such tense usages in potentially true statements, whether past, present, or future.
    – FumbleFingers
    Commented Oct 25, 2015 at 14:17
  • John would have been mocked for a poor command of English, I suppose. :-) @FumbleFingers
    – Kinzle B
    Commented Oct 25, 2015 at 14:36
  • @Kinzle B: Doubtless. But he will have deserved it! :)
    – FumbleFingers
    Commented Oct 25, 2015 at 14:45
  • Come to think of it. I don't think B could say "Really? They would've been looking for those bank robbers.", because would is the past form of will here. @FumbleFingers
    – Kinzle B
    Commented Nov 5, 2015 at 16:28

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I have been reading quite a few books on English grammar and usage. Practical English Usage is one of my favourite. Among other things, Meaning and the English Verb, Modality and the English Modals, The Cambridge Grammar of the English Language, A Comprehensive Grammar of the English Language etc. are all first-rate reference books.

However, I still run into various unfamiliar uses when reading English novels, news, movie scripts, and so on. (For example, in https://english.stackexchange.com/questions/187936 FumbleFingers explained the meaning of "I should have known [something]" on which I couldn't find one bit relevant information in the books I have.) I can't help but wonder why even the most comprehensive grammar books couldn't cover all these uses. Therefore, I have to spend lots of time on the Internet in a search for any possible good explanations. Fortunately, I often stumble upon findings of great value.

The following excerpt is taken from one of my recent finds, Aspects of Modern English Usage for Advanced Students: A Comparison with French, by Paul Lambotte, John Potter, Harry A. Campbell. It addresses lots of usage problems in a detailed and innovative way. I think it's time for me to return the favor that ELL has done for me. :)

§ 506

1) 'Will have' is used to make an assumption about an action or situation in the past when the speaker considers something to be certain (modal certainty), which does not mean that it is certain, a fact (factual certainty), it generally refers to the recent past, when the speaker is most likely to be so assertive.

  • Peter has never been hardworking. He'll have dropped out of the scheme by now.

There is an assumption, not a logical conclusion, because you have no evidence on which to base a conclusion, only your general knowledge.

2) 'Will have' is not common at all to reter to the distant past and more especially to express a theory about the past. e.g. in historical studies, because it would make it appear too certain in the absence of adequate factual evidence. The standard phrase in this case is 'would have', which makes the theory more cautious, more tentative. (See § 509.)

§ 507

'Would' is used to indicate a little less certainty in the assumption than would be the case with 'will'. This past tense form of modal 'will', like that of 'may', with its conditional connotation, makes the judgment more tentative. The term 'conjecture' is meant here to refer to the tentative form of assumption.

About a building you cannot see properly in the distance, you could say: That building would be about ten storeys high.

This opinion is based on a rough estimate and you are not sure that it is correct. 'That building will be...' sounds more confident.

§ 509

'Would have' is the usual phrase to express a theory about the past, i.e. in historical accounts when there is hardly any doubt about it. It states what most probably happened, what things were most probably like etc. in the absence of evidence stating those things as facts.

‘Will have’ is uncommon in this case because it is too close to an assertion for what is, after all, only a theory.

(In Old Sturbridge Village, an American living museum where young craftsmen revive the eighteenth-century trades, explanations often run as follows :)

  • They would have baked their bread in this way...

Here are two examples from 'Chasing the Monsoon' by Alexander Frater (1990) :

  • One of da Gama's preoccupations would have been the safety of his fleet.

  • Each year, between 1852 and 1869, usually between the 6th and 7th day of June, he would have noted the appearance of the Monsoon Star.

The use of a simple past - 'was' and 'noted' - would imply that we actually know from e.g. his diary that it was indeed one of his preoccupations and that he noted the appearance of the Monsoon Star, but we do not know, we only assume that it was so.

§ 513

The speaker is looking at the past from the present. As the judgment is made in the present, the model is in the present tense, while the pastness of the action/situation is expressed by the perfect infinitive. (See § 508.)

'Will have' can only be used to make an assumption about the recent past (See § 506.)

'Would have', which expresses things in a more tentative way, is the usual way of expressing conjecture -and more particularly theories about the distant past. (See § 508,509.)

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