Perfect Tense
The Perfect Tense is a tense that in a way stands between the past and the present. It describes an action that was completed in the past, but the results of which remain in the present and have consequences for the present. It is mostly similar to the Present Perfect tense in English and can often be translated by it. (I.e. I have opened the door (and now it is open))
The Perfect Tense is formed using the stems of the fourth and the fifth Principal Parts. Perfect Active uses the fourth Principal Part and Perfect Middle\Passive uses the fifth Principal Part.
The regular way of forming the fourth Principal Part is as follows:
One takes the stem of the First Principal part. -- λυ
One takes the first consonant of the stem and adds it at the front of the stem, followed by epsilon. This is called reduplication. -- λελυ
One adds the kappa suffix to the end of the stem. -- λελυκ
After that, one has the complete stem, to which the endings can be added. The ending of the First person, singular, Perfect Indicative Active is α, so the Fourth Principal Part of λύω is λέλυκα.
A number of other Irregularities can occur, including:
If the verb has a prefix, the stem without a prefix is reduplicated and then the prefix is added. (So the Fourth Principal Part of ἀπολύω is ἀπολέλυκα)
Sometimes -κ- suffix may contract with other consonants. (For example, the Fourth Principal Part of γράφω is γέγραφα)
The final stem vowel of Contract Verbs might change, as in the formation of other Principal Parts. (The Fourth Principal Part of λαλέω is λελάληκα)
Other completely irregular verbs.
The best way to identify the fourth Principal Part is by noticing reduplication.
The forms of the Perfect Indicative Active are as follows:
The easiest way to form the fifth Principal Part is as follows. One takes the stem of the fourth Principal part (λελυκ). Then, one deletes the kappa suffix (λελυ). Then one can add the ending of the first Person, singular, Perfect Indicative Middle\Passive form, which is μαι. Hence, the fifth Principal Part of λύω is λέλυμαι.
The irregularities one can observe here are similar to the fourth Principal Part.
That means that, usually, one can recognize the fifth Principal Part by reduplication and the passive endings it uses.
Here are the forms of Perfect Indicative Middle\Passive. Note that Middle and Passive voices use the same forms in the Perfect Tense.