How would you compare LFG, HPSG, Minimalism, and Construction grammar?
So for starters I'd stress that of these four I'm most familiar with LFG and Minimalism, less familiar with HPSG, and much less familiar with Construction grammar.
Broadly, LFG and HPSG belong to the same "family" of frameworks, in contrast to the other two — they are both lexicalist frameworks which make use of feature structures (represented by AVMs) to express functional relationships between lexical items. LFG captures syntagmatic and dependency relationships through a traditional tree structure, while HPSG captures this information through ordered sets, but that's really a mere formal variation. I'm sure there would be more fundamental differences if I was more familiar with HPSG, and the LFG formal distinction between constituency-based and functional grammatical relations does nicely echo certain findings in the psycholinguistic literature, which I appreciate; but generally a lot of what I've seen of HPSG is very familiar from an LFG perspective.
Minimalism is more in the G&B tradition. Contemporary minimalism is typically non-Lexicalist, making it slightly better at handling languages like Greenlandic and slightly worse at handling Semitic language and other forms of nonconcatenative morphology. It just uses a tree structure, and effectively subsumes all grammatical relations under syntagmatic order and dependency relations, which means you have to have some kind of regular "scrambling" operation to explain away all the languages that don't do that. It's very formally tight, so if you only really care about language qua a purely formal system, it's a decent framework for that.
From what I've seen of it, Construction grammar seems to be a third strand that's doing a bit of it's own thing. I like the way it makes the paradigmatic/syntagmatic distinction very explicit, and it feels like it would play very well with frame approaches to morphology. Definitely looks like it's proponents have some important critiques of both derivational and functional frameworks, so well worth engaging with! From the handful of introductions I've been able to find, it does look like it might run into a bit of an issue of either missing out on generalisations that can be made about related structures, or relying on proposing "constructions" that are so abstract as to amount to little more than grammatical functions under any other name. I'd also love to hear if/how Construction grammar handles discourse-level constraints (given presumably we're not treating whole conversations as Constructions that need to be learned) and differing degrees of freeness in word order (most of the introductions I read seem to treat Constructions as having fixed order of elements, which is fine for English but seems like it's going to struggle with Warlpiri's post-auxiliary region or the German's Mittelfeld).
At the end of the day, none of these frameworks are going to perfectly capture "what human brains really do", so a lot the value of one framework over another is going to depend on what you value in your analysis, and what data each framework is about to satisfactorily account for.
