Consider the sentence I will go to Belfast tomorrow. How do its surface and deep syntax tree differ in a dependency-based grammar?
There are two main differences:
- Deep syntax structures are dags,
- Only content words serve as nodes there.
The idea is that deep syntax structures should be as language independent as possible. In your sentence "will" is only a tense carrier and "to", though being a semantic preposition, functions as a case marker (illative). From a graph-theoretic perspective, to translate a surface tree into a deep dag is to remove all the nodes that represent function words from the graph. Thus at the level of deep syntax a sentence has the same structure in all languages unless there's a lexico-semantic mismatch (in which case the edge labels will be different).
In grammars with feature structures, deep structures are constraint-based. In LFG, for example, c-structures are phrase structure trees but f-structures are isomorphic to (deep) dependency dags (if one substitutes thematic roles for grammatical functions).
From a logical perspectives, deep syntax dags are "hierarchical logical forms". Lambda calculus could be used to incrementally assemble the logical form and there's a one-to-one relationship between logical forms and dags.
Also worth mentioning is information structure which is sometimes added to dags (though it's not part of syntax sensu stricto). It may be expressed as an ordering on nodes (as in FGD). Note that to fully account for syntax one needs a LP component (such as topological trees) which means that surface dependency trees are uninformative since they can be reconstructed from structures at the other levels. They can be view as a blend of topological trees and deep syntax dags. Surface trees tell us little but are structurally closer to the phonetic form of the sentece.
Update: A few definitions might be useful:
- Surface syntax trees are totally ordered (graph-theoretic) trees. There's a one-to-one mapping between their nodes and the words in the sentence. Since they're ordered, they can be tangled/nonprojective (nonprojectivity is a scale which includes planarity, well-nestedness, and full nonprojectivity).
- Deep syntax trees are dags that contain only content words. In case of pro-drop or if there are other covert elements, they can contain nodes that have no counterpart in the surface syntax tree (in Spanish "me gusta" the subject is covert). Dags are the same as feature structures (in fact, feature structures are merely a pictorial representation of dags). Deep syntax trees are called "tectogrammatical" in FGD. At this level the literal meaning (linguistic meaning, character) is captured. The representations are scope-free and there's no coercion.
- Logical forms are formulae in formal logics. Here things get more complicated since there are many different logics. In general a logical form in higher-order logic is structurally very similar to the corresponding deep syntax tree. In first-order logic, logical forms are flattened deep syntax trees. The meaning of words (that is, the chunks used to build up logical forms) come from the lexicon (sometimes they're called "semantic forms").
That depends of course on the particular dependency grammar (DG) that one chooses. There are many versions of DG, and they disagree about such matters in important ways. Lucien Tesnière, the father of modern DGs, did not distinguish between deep and surface syntax. He would have produced just one dependency analysis of the sentence, and this analysis would have lacked linear order.
An analysis in Meaning-Text Theory (MTT) might acknowledge seven different analyses of the sentence, since MTT acknowledges seven layers of representation.
Word Grammar would probably acknowledge just one analysis in terms of dependency arcs, and these arcs would appear above the actual sentence, which means there would be just one surface analysis of the sentence.
The type of DG that I prefer is monostratal in syntax, so there would be just one syntactic analysis. It looks like this:
If I am forced to produce a deep syntactic analysis of the sentence, there are various ways that I could do it. All of these ways would, though, abstract away from linear order in the tree. Here's one possibility:
This analysis lacks linear order. Note, however, that the function words are present in the tree, will and to. Some analyses of deep syntax might assume that function words are not present at that layer; they would then exclude all function words from the hierarchical analysis. Or other DGs might subordinate function words to content words in deep syntax, in which case they would subordinate will to go and to to Belfast.
For me, these questions about the nature of deep syntax cannot be answered. We cannot see, hear, or measure deep syntax in any emperically verifiable way. I therefore prefer to avoid the notion of deep syntax, remaining with surface syntax entirely. Surface syntax is much closer to what we can hear, see, and measure emperically. But I know a number of DG people who strongly disagree with my position in this area.
In sum then, there is no one answer to the question that would be accepted by the DG community in general. There are, rather, a number of competing answers. The one who poses the question should therefore decide for him- or herself which of the answers is best.