Subject Optionality
Verbs are conjugated to contain number and person information, everything a pronoun contains except gender, so objects can almost always be dropped.  However, there is a high degree of homonimity between conjugations (sono can be 'I am' or 'They are'), so pronouns are still used for clarification or emphasis.  Most of the work and explanation for subject optionality was done in the last lab, but I did change a few things, like fix the semantics per your suggestion.  I spent most of my time in this lab on the rest of the assignment.  Sample sentences follow.

Io dorm-o.  (I sleep)
Dorm-o. (I sleep)

At the end of this step, 29 of my sentences parsed.  See run 1.

Prepositional Phrase
Before I can do complete argument optionality, I needed to implement ditransitive verbs.  Before I could implement ditransitive verbs, I need to do prepositional phrases.  In Italian, prepositional phrases are composed of a preposition followed by a noun phrase.  To deal with this, I created a lexical type called preposition whose head is adposition and whose complement is a noun phrase.  If the noun phrase is a pronoun, then the pronoun must be in a special form.  To handle this, I created a new case for prepositions, leaving noun phrases underspecified.  I then specified that the complement for a preposition must be of this prepositional type.  The head of the prepositional phrase is the preposition.  For now, I have also specified that the prepositional phrase can only fill an argument position, not act as a modifier.  I might change that if I can get it to modify correctly.

To test it, I created a new transitive verb whose complement is head adp and whose complements are fulfilled.  This prepositional phrase takes the same spot as a noun phrase on regular transitive verbs.  I choose the irregular verb 'essere' to test this.

Questi libri sono per te.  (These books are for you)

Ditransitive Verbs
From what I have been able to see, almost all ditransitive verbs take either a prepositional or embedded phrase as its third argument.  I have only dealt with ditransitives that take prepositional phrases.  I created a verb type that takes a nominative as its first argument, an accusative as its second argument, and a prepositional phrase as its last argument.  A typical example would be:

I amici danno un libro ad una ragazza.
The friends give a book to a girl.

The verb 'danno' only takes prepositional phrases headed by either 'a' or 'ad'.  To deal with this, I should modify the verb so that it only takes prepositions of form to (after the English gloss) and add that form to 'a' and 'ad'.  However, there are other ditransitives that can take a variety of prepositions, so I have not modified this yet.  

At the end of this step, 37 of my sentences parsed.  See run 2.

Null Argument Optionality
Italian has both indefinite and definite null instantiation.  Some verbs can take either, while some can only take the definite null instantiation.  I handled this by inheriting from basic-head-opt-comp-phrase and by making two types of verbs and assigning each verb to one of the types.  I am not positive on all of the assignments, since I had to go on my gut feeling rather than a native speaker, but I found that a lot of verbs the can go either way I had created a transitive and intransitive type in the lexicon (not understanding argument optionality).  I fixed this, creating one type that handled both cases.  

Am-o questa ragazza (I love this girl)
Am-o (I love (definite only))
Maria la-cant-a (Maria sings it)
Maria cant-a (Maria sings (definite or null))

Since I had created the two types, adding this argument optionality didn't increase coverage, but actually decreased it.  I had originally analyzed the noun/verb inversion rule to only apply to verbs with no arguments, and so I created it as a lexical rule.  However, verbs with a null argument keep their argument until after it passes the head-opt-comp-phrase, disallowing the inversion rule from taking effect.  It worked fine with both the transitive and intransitive interpretation, but didn't reflect what was happening in the grammar.  Sentences like 'Mangi-o io' (eat I) don't parse, and sentences like 'Mangi-a Maria' (Eat Maria) only parse with the interpretation that something is eating Maria, rather than Maria is eating.  I have not taken the time to fix this yet.

Non-Null Argument Optionality
Italian argument optionality is very complex.  They work differently in different circumstances.  For one verb constructions, the arguments usually show up as pronouns in front of the verb.  If there are two arguments, the dative always appears before the accusative.  For two verb constructions, they could show up either before the initial verb or as postfixes to the infinitival form of the final verb, but the must show up together.  In positive imperatives they can show up at as postfixes to the conjugated verb.  How they show up in negative imperatives depends on the number and person of the subject.  I do not completely understand how they work in all circumstances.  For this lab, I have only implemented the non-imperative one verb construction.

I have three rules to deal with the complements.  My first rule looks for a verb whose first complement is accusative.  It then removes that complement, preserving all other complements, and adds an accusative prefix with the right PNG values.  This rule is used for both one complement transitives and ditransitives.  See the following:

La-am-o.  (I love it)
Mi-piac-e il film.  (The movie pleases me.)
Lo-do a Maria.  (I give it to Maria.)

The second rule looks for a verb that has two complements.  This deals with the case when the dative is dropped, but the accusative is not.   It removes the 2nd complement and adds the appropriate prefix.

Mi-dai il libro. (You give me the book)

The last rule looks for a verb that has already gone through the accusative rule, but still has one more complement.  This deals with the case with adding the dative prefix onto the accusative prefix.  It removes the remaining complement and adds the appropriate prefix.  This prefix is a different prefix than the one added when the dative is dropped, but not the accusative.

Ce-lo-dai.  (You give it to us.) 

There is still a bug in this.  The second rule can fire first and then the first rule, so that you get the dative prefix with the non-optional accusative argument and then the optional accusative argument.  This allows the wrong order of the complements, like 'lo-mi-dai'.  I could fix this by having the accusative rule look to make sure that the daughter hasn't gone through a dative rule, or have the dative rule change the verb so that it has to have the non-optional accusative.  We will see.  Since this case wasn't in my test suite, I still have no over-generation.

At the end of this phase, 75 sentences parsed.  See run 3.

Adjectives
An adjective can come before or after the noun it modifies and takes on the number and gender of the noun it is modifying.  I did this by removing the posthead constraint and adding lexical rules that add the appropriate ending and constrain the mod value to be a noun with the corresponding png values.  

I raggazzi studios-i legg-ono.  (The studious youth read.)
Tu sei una bell-a ragazza.  (You are a beautiful girl.)

The adjective could also be used with a copula.  In this case it must agree with the gender of the subject of the copula.  To do this, I created a verb copula that took an adjective as its complement and stipulated that the PNG value of the subject and the PNG value of the noun that the adjective is modifying be the same.

'Giosuè è saggi-o' (Joshua is wise.)

Since LKB makes no difference between 'lei' (she) and 'Lei' (you), I decided to make two 'lei's, one female and one underspecified.  Since 'Lei' acts like a third person pronoun, they have the same semantics.  I can therefore, get both of these grammatical parses:

'Lei è saggi-a'  (She/You are wise)
'Lei è saggi-o'  (You are wise)

Of course, there are some issues I won't be able to handle.

*'So che lei è saggi-o' (I know that she is wise)
'So che Lei è saggi-o' (I know that you are wise)

Adverbs
Adverbs were almost the same as posted on the class web site.  I just copied and pasted and fixed a few syntax issues.  I added a few words to the lexicon (since I didn't have any adverbs in my test suite.)  

Sempre dorm-i.  (You always sleep.)
Mangi-amo oggi.  (We eat today.)

At the end of the lab, 87 sentences parsed.  (65.9% of possible parses)  I have no overgeneration in the test suite, though I do have overgeneration outside of the test suite and I lost some of the sentences that I originally parsed.  There are some sentences that it marks as ungrammatical, but are actually grammatical. I still need to work out a few more issues with the semantics and get my adjectives to combine.  Imperatives don't work with negation or argument optionality.  It is only by pure luck and alternate interpretations that I don't get overgeneration with that.