Underlying representation

In Functional Grammar and Functional Discourse Grammar, underlying representations (URs) are the formalised semantico-pragmatic structures that motivate the surface structure of linguistic expressions.

The generation of all URs is governed by a unified algorhitm, the generalised layered structure (GLS), which distinguishes two operations, restriction and application, the latter being comparable to merge in Chomsky's Minimalist Programme. Each layer is headed by a variable with a specific designation, the contents of which are restricted by the application of restrictors. The entire designation of the variable can be modified by one or more operators. Since restrictors also take the form of layers, the entire formalism is recurrent, and structures can be expanded by embedding them as restrictors in other layers, an operation known as upward layering.

History
The structure of URs is inspired by first-order predicate logic. Its original architecture (Dik 1978), which was designed for semantic representation, commanded slots for predicates and arguments with different formal properties. Hengeveld (1989) introduced the notions of GLS and upward layering, and introduced additional layers in subsequent years, each time less semantically and more interpersonally oriented. Keizer (1992) reconciled the hitherto problematic position of the predicate with this approach, observing that, in spite of its special relational properties, the denotation of predicates can be generated in much the same way as for their complements.

In multistratal FDG, the interpersonal layers introduced by Hengeveld and others were separated from the semantic layers they incorporated, in order to settle a long-lasting modularity debate. These layers (illocution being the most important of them) became the core of the interpersonal level, which co-existed orthogonally with the semantic level. GLS remained in vigour, however, as the sole available algorhitm to generate structures at either level.

Legend

 * round brackets indicate the scope of a layer
 * square brackets ([]) indicate restrictor slots
 * the colon represents the operation of restriction
 * α represents layer variables, subscripted for subsequent identification
 * ◊ represents restrictors, which themselves take the shape of layers
 * π represents operators
 * φ represents functions

Restriction
(παi: [◊i] (αi): [...] (αi): [◊n] (αi))


 * NB: Unlike in most other semantic theories, restriction forms subsets rather than set intersections. For this reason, the mutual order of restrictors is relevant for the designation of the layer. The first restrictor has the status of head, while subsequent restrictors serve as modifiers

Application (merge)
(παi: [(◊i) (◊j)undefined (◊k)φ] (αi))


 * NB: Multiple layers that occupy a single restrictor slot form an equipollent configuration, the constituents of which must merge before they can be applied to the layer as a restrictor. Functions indicate the qualitative relationship between the functor and its complement
 * NB: In equipollent configurations, the formalism requires the presence of a single functor, which assigns functions to all of its complements
 * NB: In pure structural terms, all restrictors (whether single-layer or equipollent) merge with the subsequent variable that designates the layer. Hence the formalism could be paraphrased as "layer αi, the designation of which is restricted by the application of (◊i) to αi, the designation of which is restricted by the application of (◊n)."'

Notational conventions

 * In actual analyses, the variable subscripts are substituted by numerics
 * In actual analyses in FDG, the primitives at the Interpersonal Level are usually rendered in capitals, and the primitives at the Representational Level in undercast
 * In recent publications, --◊-- is used to indicate the omission of the further internal structure of layers
 * The linear order of equipollent layers is irrelevant, but traditionally the functor precedes its complements
 * In order accommodate non-restrictive modification, as in a nice, red car (if the intersection of nice cars and red cars is intended, and not the subset of red cars that also happen to be nice), it has been proposed to use a comma rather than a colon  to separate the restrictors, to indicate the irrelevance of their mutual order of application