Glue Semantics

Origin. Mary Dalrymple, John Lamping and Vijay Saraswat, "LFG Semantics via Constraints" (EACL 1993); Fernando Pereira joined the group for "Linear Logic for Meaning Assembly" (1995), which is where the linear-logic formulation is set out in full. LFG interface. Resource-sensitive composition. Foundation of LFG semantics.

Models. Syntax provides resources. Linear logic assembles meaning. Meaning constructors. Multiple attachments. Flexible composition.

Formalism.

Meaning constructors: f : G. f: meaning (λ-term). G: glue formula (linear logic). Resource contribution.

Glue logic: Linear logic fragment. ⊸: linear implication (consume-produce). ⊗: tensor (both required). Resource-sensitive.

Assembly: Collect meaning constructors from words. Prove goal type from resources. Proof = meaning derivation. Multiple proofs = ambiguity.

Example: "John sleeps." John: j : g_john sleeps: λx.sleep(x) : g_subj ⊸ g_sentence Prove: g_sentence. Result: sleep(j).

Modifiers: Contribute resource and consume modified. Adjective: λPλx.P(x) ∧ red(x) : (g ⊸ g) ⊸ (g ⊸ g). Flexible attachment. Scope through resource handling.

Quantifier scope: Multiple derivations. Each proof = one reading. Underspecification natural. No scope mechanism needed.

LFG interface: F-structure provides resources. σ-projection: f-structure to semantics. Glue assembles. Modular architecture.

Symbols.

Symbol	Unicode	Name	Meaning
λ	U+03BB	Lambda	Abstraction in the meaning term
σ	U+03C3	Sigma	The σ-projection from f-structure to semantic structure
∧	U+2227	Logical and	Conjunction inside a meaning term
⊗	U+2297	Circled times	Tensor: both resources required
⊸	U+22B8	Multimap	Linear implication: consume and produce
!	—	Exponential	Licenses reuse; restores contraction and weakening
f : G	—	Meaning constructor	A meaning term paired with a glue formula
g	—	Semantic resource	An atomic resource such as g_subj
P, x	—	Variables	Predicate and individual variables in meaning terms
j	—	Individual constant	The denotation of "John"
LFG	—	Lexical-Functional Grammar	The host architecture
MILL	—	Multiplicative intuitionistic linear logic	The glue fragment
f-structure	—	Functional structure	LFG's syntactic level supplying resources

Metatheory. The glue logic is the multiplicative implicational fragment of intuitionistic linear logic, and the Curry–Howard correspondence for that fragment is what makes the architecture work: a proof of the goal resource is a linear λ-term, so the meaning is read off the derivation and no separate composition rule is required. Linearity is not decoration — every premise is used exactly once, which turns LFG's Completeness and Coherence conditions into proof-theoretic facts rather than stipulations. Scope ambiguity is derivational: distinct proofs of the same goal from the same premises are distinct readings, so n scope-bearing elements yield their permutations with no quantifier-raising operation and no type shifting. The work moves into proof search — provability in the multiplicative fragment is NP-complete (Kanovich 1992; Lincoln, Mitchell, Scedrov, and Shankar 1992), so enumerating readings means enumerating proofs of an NP-complete problem. Asudeh (2012) extended the resource logic to resumption and anaphora using manager resources, and Asudeh and Crouch showed the assembly logic is independent of LFG and transfers to other syntactic frameworks.

Applies to. LFG semantics. Compositional semantics. Scope. Modification.

Limitations. Resource sensitivity cuts both ways. A meaning that must be reused, as in anaphora, or discarded, as with expletives, violates linearity outright, and the repairs — manager resources, or the exponential ! — reintroduce contraction and weakening and so surrender the property that motivated the system. Proof search is the cost of derivational ambiguity: provability in the multiplicative fragment is NP-complete, and distinct proofs are not distinct meanings, so spurious ambiguity has to be quotiented away by normal forms the logic does not itself supply. Because every proof of the goal counts as a reading, the system generates all scope permutations and overgenerates wherever scope is constrained — islands and clause-boundedness must be imposed on the f-structure rather than derived. And the empirical content sits in the lexicon: nothing in the logic fixes which meaning constructor a word contributes.