Modal perfects in English are verb constructions that combine a modal verb with the perfect tense to express various ideas related to past possibilities, obligations, deductions, and events that could have happened or should have been completed in the past. The two main modal verbs used in modal perfect constructions are “could have” and “should have.”

1. Could Have: This modal perfect construction is used to express past possibilities that did not happen. For example: “He could have won the race if he had trained harder.” Here, the speaker is suggesting the possibility of winning the race in the past, but it did not happen due to lack of training.

2. Should Have: This modal perfect construction is used to express past obligations or duties that were not fulfilled. For example: “She should have finished her homework before watching TV.” In this case, the speaker is indicating that there was an obligation to finish homework before watching TV, but it was not completed.

Modal perfects are formed by combining the modal verb (could, should) with the perfect infinitive of the main verb. The perfect infinitive is formed using “have” + past participle of the main verb. For example: “I should have studied more for the exam.”

In summary, modal perfects in English are used to discuss hypothetical past events, missed obligations, or deductions about the past. They are a useful way to convey nuanced meanings related to past possibilities and requirements.