Prepositions come specified from the lexicon with their assign-case feature.
Case assignment by verbs has two parts: assignment of case to the object(s) and assignment of case to the subject. Assignment of case to the object is simpler. English verbs always assign accusative case to their NP objects (direct or indirect). Hence this is built into the tree and not put into the lexical entry of each individual verb.
Assignment of case to the subject involves the following two equations.
This is a two step process - the final case assigned to the subject
depends upon the assign-case
feature of the
verb as well as whether an auxiliary verb adjoins in.
Finite verbs like sings have nom as the value of their assign-case
feature. Non-finite verbs have
none as the value of their assign-case
feature. So if no auxiliary adjoins in, the only subject they can have
is PRO which is the only NP with none as the value its case