The canonical SBCG construct hierarchy with list-valued GAP in RSRL #

[Sag10] [SCA+20b] [Sag12] [BMS01b] [Ric00] [Ric24] [PS01] [BC24a]

The single canonical RSRL signature for the Sign-Based Construction Grammar fragment, formalized on the RSRL feature-structure substrate (Syntax/HPSG/{Signature,Interpretation,Description}). HPSG is a model-theoretic (not generative-enumerative) framework [PS01]: a grammar is a signature plus principles, and grammaticality is satisfaction — membership in the class of structures that are well-formed in every component with respect to every principle (Models) — not derivation from a start symbol ([Ric24] Ch. 3, on the RSRL model theory of [Ric00]). The signature carries RSRL's member relation (the "R" in RSRL): list membership giving GAP genuine set semantics, so a filler discharges any member, not only GAP|FIRST (final section). One Signature (sig) carries, in one feature-structure language:

the construct type hierarchy with monotonic multiple inheritance ([SCA+20b] Figs. 6–7);
a GAP (SLASH) feature that is a set of loc objects ([BC24a] (9)): a list with gap amalgamation ([Sag10] §4, after [BMS01b]), the member relation for the order-independent set reading, and loc (CAT + INDEX) elements so the filler shares the gap's referential index — connectivity (ATB, parasitic gaps), [BC24a] (3)–(4); and
islands as [GAP ⟨⟩] constraints on the construction types themselves ([Sag10] §5.1).

A construction is a constraint τ ⇒ D ([Sag12] (44)) on a construct = [MTR, DTRS] ([Sag12] (46)); inheritance is the sort order plus the implication semantics — a construct whose sort sits below several supersorts satisfies all their principles, with no per-construction restatement. This is the monotonic (no-overriding) inheritance SBCG commits to ([Sag12] fn. 17; the "category restriction operation" of [SCA+20b]).

The hierarchy is the authoritative two-dimensional one of [SCA+20b] (Figs. 6–7):

construct backbone (Fig. 6): construct > {phrasal-cxt, lexical-cxt}; phrasal-cxt > {headed-cxt, clause}; headed-cxt > {subject-head-cxt, pred-head-comp-cxt, aux-initial-cxt, …} — with filler-head-cxt the further headed-cxt subtype of [Sag10] Fig. A2 (elided behind Fig. 6's "…");
clausal dimension (Fig. 7): clause > {core-cl, relative-cl}; core-cl > {declarative-cl, interrogative-cl, exclamative-cl}.

The cross-classification is a lower bound across the two dimensions — the mechanism [SCA+20b] Fig. 5 illustrates for subj-pred-cl (declarative-cl ⊓ subject-head-cxt). Here [Sag10]'s nonsubject wh-interrogative ns-wh-int-cl ((80), Fig. 8) is a subtype of both filler-head-cxt (headed dim.) and interrogative-cl (clausal dim.), so it inherits the filler-head constraints (head verbal, filler nonverbal, filler↔gap token identity, gap amalgamation) and the interrogative semantics (MTR SEM a question) from one sort assignment — the keystone theorem below.

The GAP feature and islands #

A sign's GAP value is a genuine HPSG list of loc objects — elist (empty) or nelist with FIRST (a loc = CAT + INDEX) / REST ([Sag10]'s ⟨…⟩ notation; [BC24a] (9): SLASH is a set of locals). The filler-head construction binds the head daughter's first gap (token identity between the filler and GAP|FIRST — sharing both CAT and INDEX) and amalgamates the rest onto the mother (MTR|GAP = HD-DTR|GAP|REST), so a clause with two undischarged gaps passes the second up ([Sag10] (53), (59)). Islands fall out of this: an island construction (topicalization (67), wh-exclamative (74)) additionally constrains its mother to [GAP ⟨⟩] (elist); a would-be second gap then makes the amalgamated mother GAP non-empty, contradicting [GAP ⟨⟩], so the construct is rejected — the model-theoretic content of "topicalization is an absolute extraction island", a theorem about Models, not a universal Subjacency. Weak islands (weak-island-cxt) are selectively permeable (NP passes, PP blocked), the NP/PP asymmetry of [SWB03] Ch. 15.

Scope #

This file is the substrate: the type hierarchy, the cross-classification keystone, all five filler-gap constructions ([Sag10] §5: topicalization, wh-exclamative, nonsubject wh-interrogative, wh-relative, the-clause), and the gap/island mechanism. The hierarchy also carries the aux-initial-cxt/interrogative-SAI sort nodes and the INV feature ([SCA+20b] Fig. 6); the inversion construction's own principle and worked constructs are paper-anchored in Studies/SagEtAl2020.lean, and the island/extraction taxonomy theorems in Studies/SagWasowBender2003 and Studies/Sag2010, which consume this substrate.

The relational binding signature bindingSig (Binding.lean) stays a separate example signature over the same RSRL framework — it has its own binding-specific sort hierarchy (anaphor/pronoun/index) and the locO relation; mathlib keeps many example structures over one framework. Both signatures now carry relations (member here, locO there).

GAP elements are loc objects (CAT + INDEX) — the INDEX is a single referential atom (it stands in for the full HPSG CONTENT|INDEX agreement bundle); n-ary DTRS, the WH/REL/IC/VFORM finer variation, and compositional SEM are deferred. Decidability stays inside Models over fixed finite interpretations (Kepser 2004: full RSRL model-checking is undecidable).

The canonical SBCG construct hierarchy with list-valued GAP in RSRL #

The GAP feature and islands #

Scope #

Sorts: categories, semantic types, GAP lists, signs, and the construct/clausal hierarchy #

Attributes and the signature #

Principles (constructions as τ ⇒ D) #

Multiple inheritance: ns-wh-int-cl is a lower bound across the two dimensions #

Worked constructs #

Construct families (parameterized worked-model builders) #

The keystone: cross-classification by inheritance #

The five filler-gap constructions ([Sag10] §5) #

Gap amalgamation and islands ([Sag10] §4–§5.1, after [BMS01b]) #

Islands as a property of the construction type ([Sag10] §5.1) #

Head-modifier constructs ([Sag10] §6) #

Inflectional lexical-rule constructs ([Sag12] (48)) #

Genuinely set-valued GAP via the member relation #

Filler-gap connectivity: the INDEX #

Principles (constructions as `τ ⇒ D`) #

Multiple inheritance: `ns-wh-int-cl` is a lower bound across the two dimensions #

Genuinely set-valued `GAP` via the `member` relation #

Filler-gap connectivity: the `INDEX` #