Using Perfect Infinitives with Modal Verbs

Form:

MODAL + HAVE + PAST PARTICIPLE 

Usage: 

• to refer to the past
• to refer to unreal situations 
• to show that the activity was different from what we wanted 
• to say how confident we are that something has happened

Express deduction, a logical conclusion, probability

• It is very cold; it must have snowed in the mountains.  
• If you can't find your car keys, you must have left them in the house.

Express negative deduction (impossibility in the past)

• She can't have passed such a difficult exam. 

Express past reference about something that was not carried out

• You could have done it. (You didn't do it)  
• She could have called the doctor early in the morning. (She didn't call the doctor)

Express the possibility that an action took place in the past

• The little girl may have lost the key. (It is possible that she lost the key.)  

Express a past possibility

• Our neighbours might have heard some noises when our car was stolen.  

Express an unnecessary action, which was, nevertheless, performed

• I needn't have knocked at the door since, in this way, I awoke the baby. (But I knocked)
• You needn't have bought the flowers.

Indicate that the past obligation was not fulfilled or carried out

•  You should have locked the door before leaving the house. (But you didn't lock it.)  

Express an unfulfilled duty or obligation

•  Paul ought to have waited until the lights were green before he crossed  the street. (But he didn't wait.)

The third conditional 

• I would have gone to university if my parents had had more money. (The speaker didn't go to university.)