Reasoning about the set of trees licensed by a grammar

Here’s a very broad question I got recently, about the HPSG formalism. I am trying to understand this question better.

“…for formalisms like CFG, LCFRS, TAG, etc. […] there are characterizations of what kind of trees you can and can’t build with the grammar. For example if your LCFRS has fan-out 2, then you know that you can have discontinuous constituents with one gap, but no more than one. If your LCFRS is well-nested, you know that you can nest constituents in some ways but not others, etc. Has anyone attempted to give an abstract characterization of the set of trees that can be covered by an HPSG grammar?”

I am not sure I understand this question well enough. Could someone help me by either rephrasing it or pointing me to some specific literature?

Delph-in grammars have a context-free backbone so the constituents are simple: they just form a normal context-free tree. An HPSG grammar with a context-free backbone can still define a non-context-free language, because of the constraints encoded in the feature structures.

There are variants of HPSG where constraints are defined on something other than a context-free backbone (e.g. linearisation-based HPSG). But in general, the trees are only one part of the analysis, so there’s no obvious need to have fancy kinds of tree with discontinuous constituents (as in LCFRS).

Perhaps a different kind of “abstract characterization” is meant. If so, perhaps the person asking you could clarify on this discussion.