StanfordAlgebraAdditions

Revising the algebra

Here are some additional comments which may be useful for the algebra discussion:

Clarifications from presentation

Ambiguity of term `INDEX', `LTOP' etc

In DMRS, the index is a node in the DMRS graph, rather than a distinct instance variable. Since null semantics items have no nodes, they cannot have an index in that sense. However, the feature structure including the DMRS can have a feature INDEX, which could, for instance, link to a complement's index. What the DMRS treatment precludes, which is possible in MRS, is that the index is used as a sort of storage facility for properties such as tense, which can be accessed even if there are no syntactic arguments. This has consequences (predictions?) for the association of semantic properties with things like existential `it' - while it isn't completely ruled out, it would require an additional structure to be added.

Existing algebra is broken with intrinsic variable grammars

The algebra as originally defined says that the semantic head provides the slot and the hook (the index and ltop plus xarg or whatever). This worked for adjective-noun combination when we didn't use event variables on adjectives, because the hooks were equated. Once one does this with event variables on adjectives, the slot is on the adjective but the INDEX of the adjective-noun combination has to come from the noun, so we don't have a notion of semantic headedness any more. With scopal modification (adverbs like `probably'), the index comes from the modified verb and the ltop comes from the adverb, hence the hook isn't transmitted as a whole.

This could perhaps be fixed but points to a deeper issue: when we originally developed the algebra, we (at least me, and I think Alex) were influenced by previous approaches to want a notion of functor and argument, with the functor (aka semantic head) providing the slot and the hook. This is not tenable without some operation that does something drastic to the functor. Arguably this is analogous to some of the type-raising operations in conventional approaches. e.g., in lambda calculus, one can make an NP into a functor. But we always thought of this as a problem with those approaches, so this doesn't seem like a good path to go down.

Another issue is that the connection to the syntax is looser than I would like. The syntax and semantics cannot be completely isomorphic (again, this is a divergence from categorial grammars of various types) but we really would like the relationship to be regular with limited exceptions.

Modest proposal

1. We think of the algebra in terms of graphs (DAGs), since this is anyway what we're dealing with natively, and since it allows us to see the relationship of DMRS and other forms of MRS more clearly.
2. The TFS graphs really have the following type of structure:

   • [ SYN [ SYNROLE1 [1]

     • SYNROLE2 [2] ]

     • SEM < [ PRED

       • SEMROLE1 [1]

         SEMROLE [2] ] > ]

where [1], [2] indicates reentrancy. In what follows, for the purposes of the algebra, I'm going to try writing the semantic graphs with additional syntactic annotations. I expect this to go wrong, but it gives us the idea that syntax and semantics usually go together and we have to do something to disconnect them. (The hope is that what we do is less violent/unconstrained than with alternative approaches. )

1. In terms of the semantics,