Lexical and Derivational Meaning in Vector-Based Models of Relativisation

Sadrzadeh et al (2013) present a compositional distributional analysis of relative clauses in English in terms of the Frobenius algebraic structure of finite dimensional vector spaces. The analysis relies on distinct type assignments and lexical recipes for subject vs object relativisation. The situation for Dutch is different: because of the verb final nature of Dutch, relative clauses are ambiguous between a subject vs object relativisation reading. Using an extended version of Lambek calculus, we present a compositional distributional framework that accounts for this derivational ambiguity, and that allows us to give a single meaning recipe for the relative pronoun reconciling the Frobenius semantics with the demands of Dutch derivational syntax.

💡 Research Summary

The paper investigates how to combine compositional distributional semantics with formal syntactic analysis for relative clauses, focusing on the particular challenges posed by Dutch. Earlier work by Sadrzadeh et al. (2013) showed that English relative clauses can be modelled using the Frobenius algebraic structure of finite‑dimensional vector spaces, but it required two distinct type assignments and two separate lexical recipes: one for subject‑relative clauses and another for object‑relative clauses. Dutch, however, is a verb‑final (SOV) language, and a single clause can be interpreted either as a subject‑relative or an object‑relative without any overt syntactic change. This creates a derivational ambiguity that cannot be handled by simply copying the English approach.

To address this, the authors extend the Lambek calculus with two controlled modal operators, ♦ and ✷, yielding the system NL⋄ (pronounced “N‑L‑diamond”). The modalities act like linear‑logic exponentials, allowing limited re‑ordering and restructuring of constituents while preserving the resource‑sensitive nature of the calculus. Two structural rules, αℓ⋄/σℓ⋄ (left) and αr⋄/σr⋄ (right), model the non‑local extraction patterns typical of SOV languages (for Dutch) and SVO languages (for English) respectively. The key insight is that a single type assignment for the Dutch relative pronoun “die”—(n \ n)/(♦✷np \ s)—is sufficient to capture both readings. The ♦✷np hypothesis can be discharged either as the subject argument of the verb or, after a σℓ⋄ re‑ordering step, as the object argument.

The semantic side of the framework maps NL⋄ types into a symmetric compact closed category (sCCC) whose concrete instantiation is the category of finite‑dimensional real vector spaces (FVect) and multilinear maps. The interpretation function ⌈·⌉ sends atomic types to vector spaces (e.g., ⌈np⌉ = N, ⌈s⌉ = S) and treats the modalities transparently (⌈♦A⌉ = ⌈✷A⌉ = ⌈A⌉). Complex types are interpreted via tensor product, and the usual left/right division operators become tensor‑product with a dual (A ⊗ B* or A* ⊗ B). The residuation rules of the calculus correspond to the familiar ε (inner product) and η (unit tensor) maps of compact closed categories; the Frobenius algebra provides copying (Δ), deletion (μ), and element‑wise multiplication (∘) operations needed for intersective semantics.

Crucially, the authors express all linear maps arising from proof steps using a generalized Kronecker delta. This notation compactly captures index renaming, contraction (inner products), and insertion of identity tensors, allowing the entire derivation to be reduced to a single tensor contraction pattern. For the Dutch example “mannen die vrouwen haten” (“men who women hate”), two distinct contraction patterns are derived:

1. Subject‑relative: the index of the ♦✷np hypothesis contracts with the subject index of the verb, yielding a vector v_subj ∈ N.
2. Object‑relative: the same hypothesis contracts with the object index, yielding v_obj ∈ N.

Both vectors are computed by the same lexical recipe for “die”, which is a linear map built from Frobenius copying and deletion combined with the appropriate Kronecker deltas. The derivations differ only in which indices are identified, reflecting the syntactic ambiguity while preserving a uniform semantic operation.

The paper also compares this vector‑based approach to traditional set‑theoretic semantics. In formal semantics, a restrictive relative clause denotes the intersection of the denotation of the head noun with the set of entities satisfying the clause; element‑wise multiplication in a distributional space naturally implements this intersection. The Frobenius algebra thus provides a categorical justification for the use of element‑wise multiplication as an intersective operator.

In summary, the authors present a unified compositional distributional model that simultaneously handles lexical meaning (via Frobenius algebra) and derivational ambiguity (via NL⋄). By assigning a single type to the relative pronoun and a single linear map to its meaning, the model elegantly captures both subject‑ and object‑relative readings in Dutch without resorting to ad‑hoc type switches. The framework bridges the gap between formal syntactic calculi and vector‑space semantics, and it suggests a path toward extending such integrated models to other languages and more complex syntactic phenomena.