Deduction from past form of " have (got) to " [duplicate]

I'd like to ask if there is any form of " have (got) to " in order to make a deduction from the past equivalent to " must have + past participle" ?

For example:

They hadn’t eaten all day. They must have been hungry.

Is there any structures as follows:

They hadn’t eaten all day. They have to have been hungry.

or

They hadn’t eaten all day. They had to have been hungry.

Solution 1:

The last one is definitely acceptable: the point you're talking about is in the past. Imagine someone looking for a fugitive, and wondering if they were the one who was reported to have stolen a sandwich off a park bench the day before. You're putting yourself back in time to yesterday, when the sandwich was stolen, and then talking about the past relative to that point.

"They hadn’t eaten all day. They have to have been hungry.", to me, seems like an ungrammatical version of 3), which isn't technically correct because you've jumped from the past to the present and then back to the past again. I think it's an example of the sort of mistake which is quite technical and minor, and so would probably be forgiven or possibly just treated as acceptable informal language.

I think that when people would use this latter construction, they're using the present tense to refer to their own thought processes and deductions, which obviously are occurring within their minds in the present. Within their mind, the past and present coincide at the same time, hence the confusion.

Solution 2:

Have (got) to, along with its contractions hafta and gotta, is called a Periphrastic Modal Idiom.

Periphrastic means 'using a phrase consisting of several words (instead of an inflection)'.

Modal refers to modal auxiliary verbs, which all have several types of meaning.
In particular, all modals have at least two senses, the Deontic sense and the Epistemic sense.

Since modal auxiliaries are defective verbs (they lack verbal forms, like gerunds and participles),
there are many constructions they can't be used in. No past tense for must, for instance:

• You must be at your desk by 8 am tomorrow. ~ This must be what they do.

but not

• *You musted be at your desk by 8 am in those days ~ *This musted be what they did.

Instead, one uses a periphrastic modal, which does not have these limitations:

• You had to be at your desk by 8 am in those days ~ This had to be what they did.

It's a bit unusual to use have (got) to in the epistemic sense, but it's quite possible

• This has (got) to/This must be the biggest pile of burlap bags in the world.
• After working all day, you must/have (got) to be hungry.

Modals falute a bit higher than their paraphrases, so they are sometimes felt as more formal.