Poking around the LIABG ("Linguistics Institute of Ancient and Biblical Greek") site, I happened upon the proceedings from the 2006 symposium. This was nice, because while I was invited I wasn't able to make it (was married the previous month).

In the proceedings, I skimmed the article "What's Up in Syntax" and ran across the following, which I found helpful:

In traditional analysis predication involves a grammatical subject about which a predication is made. In the sentence All dogs have tails. The noun phrase All Dogs is the subject and the possession of tails is predicated of dogs with ‘have’ serving the predicating function. An alternative view of predication asserts that this relation is between a predictor and one or more arguments, in which case both All dogs and tails are viewed as arguments or terms with have (the predicator) functioning to form a relation of predication between the two. Complements are generally taken to be elements required by the predicator whereas adjuncts are nonobligatory elements, providing circumstantial information. Head terms or controlling terms govern the terms to which they stand in relation, as in ‘The head is the noun in the NP’ (Van Valin and LaPolla, 1997: 67). The marking of head terms typologically continues to be debated among linguists (see Zwicky, 1985; Hudson, 1987). Head terms receive various levels of modification in which a dependant terms modify the head through qualification (a limiting relation), specification (a classifying or identifying relation) or definition (further defining) (see Halliday, 1976: 59-66). A head term with its modifiers forms a group. Groups can be nominal, verbal, adverbial or conjunctive (Halliday, 2004: 310; Morley, 2004: 74-83). Coordination relations, realized through the conjunction system, are used to relate groups or modifiers within groups. (2006 LIABG Proceedings, p. 11)

If you've worked at all with the OpenText.org analysis in Logos Bible Software, the above may help a bit in understanding the analysis (which uses these labels). For example, if you've ever wondered what the difference between a complement and adjunct is, the above gives you a rule of thumb: Complements are required, Adjuncts are non-essential.

If you dig this stuff, or if you want to know more, I'd recommend reading the introduction to Jeffrey T. Reed's A Discourse Analysis of Philippians (amazon.com) (also part of the "Studies in New Testament Greek" collection, hopefully available sometime in the future for Logos Bible Software (on pre-pub at the time of post composition))

Bonus: A blog post I wrote in 2006 is footnoted in one of the articles of the proceedings (p. 151 of the proceedings). It was news to me when I learned of it. Interestingly, that blog post had to do with αλλα. Let's just say I've come a long way in understanding αλλα in the past two years. Maybe I'll write about the footnote in a future post.