Accepting phonology, the original generative problem was to model a string of tokens that are wordforms or morphs. However, the type-token distinction is abandoned with the commitment to competence modeling independent of performance. Token strings are demarcated by speech acts (clauses). A narrowly syntactic meaning of grammar is a generative production system (a set of combinatorial rules) that specifies soundly and completely the licensed set of strings. A state-machine generating strings.
This elevates the importance of not just word classes, but also the non-terminal categories or phrasal categories as the discrete tokens or alphabet for the generated strings.
Later, the alphabet of tokens can be replaced with feature-bundles. So the syntax rules can work over string of feature bundles.
An additional move in the mainstream generative grammar tradition (of Chomsky) is to allow tree-to-tree transformations. This is somewhat proof theoretic, rather than purely model theoretic. Non-reversible (thus not non-declarative) transformation rules somewhat correspond to proof steps. However, this was before the proofs-as-types approach (via the Curry-Howard isomorphism) brought out the equivalence of proof theory and model theory. For the critics in the declarative grammar traditions, the transformational move is a bad move that gives up the precision of modeling, and allows the theory to model anything rather than just reality. It also is the cultural foundation for a mainstream generative grammar to twiddle with perfecting the theory with internal concerns like minimalism, rather than sharpen the fit with ever broader data.
Graddol et al [1994] compare mainstream generative grammar with the systemic grammar alternative: "The framework that is used for analysing language has to be extravagant rather than economical. Where universal grammar seeks simplicity and economy, and draws on intuition [of competence] as its main data, systemic-functional grammar attempts to be comprehensive and gives much more emphasis to 'real' language that has been spoken or written."
Recently, declarative generative grammar approaches are converging with cognitive grammar, paying attention to performance and retaining the precision of generative models while taking on the early ambitions of systemic grammar to broadly model language in use.
Pollard in Convergent Grammar is also reclaiming a proof-theoretic approach, carried forward by Categorial Grammar, by bringing it in line with type-theoretic declarative grammars.
Jackendoff and Culicover are less concerned with building elaborate internal mechanisms for declarative grammar, although they are happy to inherit the detailed analysis of the generative tradition. They propose a flat model of syntax, returning to simple strings of tokens, with just enough structure in the representation to support the semantics-syntax interface and other interfaces in their parallel architecture.
HPSG is one of the more elaborate declarative grammar approaches, and while it remains compatible with the concerns of Jackendoff and Culicover, they make a series of moves that allow precise modeling of syntax and perhaps semantics as well. It would be nice if language specialists can model grammar in the flat simpler syntax, which the linguistic exploration environment generates the complex representations of HPSG in the background.
HPSG moves away from strings of feature bundles combined according to atomic phrasal categories. The features in a bundle allow recursive feature values, so each terminal category becomes itself a tree. The phrasal categories are also recursive feature structures, that duplicate (or in fact share) features from their constituents, notably heads pass certain features up to the phrase. By sharing rather than duplicating, the trees become directed graphs. This very expressive structure is tightly constrained by a set of type labels for nodes, with feature labels for edges. There is a type subsumption hierarchy as well.
Note that every utterance is modeled as a large graph where phrases are subgraphs, and wordforms are subgraphs in turn. These graphs are complete or sort-resolved [review Carpenter's distinctions], but the wordtypes or morphemes in the lexical unit are not complete, they are partial descriptions. These types are less specific than the token-level utterances and utterance fragments that they model, and they allow the complete structures to be generated from the partial feature descriptions in the lexicon.
HPSG was designed to work with the model-theoretic approach to modeling semantics of Montague Grammar (usually associated with Categorial Grammar for syntax). A related model-theoretic approach to modeling semantics is Minimal Recursion Semantics (and recently Robust MRS).
I am dissatisfied with the propositional level addressed by these model-theoretic approaches, but it is early days yet in semantic modeling. The traditional emphasis has been set-theoretic concerns like the scoping of quantifiers. I am interested in richer semantic representations than austere sets. Situation theory may fit the bill, but a lot of work needs to be done, especially at the semantics-syntax interface.
At this interface, I am interested in integrating the contributions of Construction Grammar and Jackendoff's parallel architecture. I am also interested in insights from lexical semantics (Levin, Pustejovsky).
A lexical unit, when we model it for some application, provides explicit and implicit information that is realized in the wordforms of an utterance. The prepositional level, including the conceptual structures of Jackendoff, is the more or less explicit level (some semantic elements in the conceptual or propositional structure may be implicit in the syntax, derivable from a fixed Event ontology). I believe that we can reuse some of this conceptual machinery for the implicit connoted information of individual word senses as well.
A lexical unit does not characterize a single unique meaning in usage, it subsumes a collection of related senses with the same wordform. Some of the meaning in usage may come from the construction, but perhaps a sense (learned from the shifting fads of usage, and construals passed on in a speaker's own utterances) brings in an additional circle of implicit concepts and conceptual relations.
For example, not all senses of "break" have an implicit participant of pieces, or the idea of sudden separation. But one fairly common sense does carry such information implicitly. What is explicit are the participant roles with arguments or complements, filled in the utterance by particular individuals of a noun (or referential index) type. What allows a hearer to zero in on one familiar sense or another, or to realize that this is a new sense in usage, is the outer circle of concepts that are implicit. So if one of the participants mentioned in an argument is a piece, then this sense or a related one is specified.
Where is the boundary between construal in performance, and precise type representation in competence? In the brain, there is certainly a fluid spectrum. For applications in computers, we would like to fix a particular discrete model. The account above can clarify some of the choices to fix.
