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# On the semantics of dependencies: relative clauses and open clausal complements - extended abstract -

Philippe de Groote

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Philippe de Groote. On the semantics of dependencies: relative clauses and open clausal complements - extended abstract -. Logic and Engineering of Natural Language Semantics 20 (LENLS20), Nov 2023, Osaka, Japan. hal-04402969

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Distributed under a Creative Commons Attribution 4.0 International License
On the semantics of dependencies:
relative clauses and open clausal complements

— extended abstract —

Philippe de Groote

LORIA, UMR 7503, Université de Lorraine, CNRS, Inria,
54000 Nancy, France

# 1 Introduction

Dependency grammars, which stem from a long linguistic tradition [12, 7], have seen a resurgence of interest in recent years, especially due to the Universal Dependency project [9]. Their principle is to make explicit the dependency relations existing between the words of a sentence, these relations being identified by the syntactic functions they represent.

Dependency parsing offers an interesting alternative to constituent parsing. In particular, parsing an agrammatical sentence does not result in a failure, but in a partial dependency structure that still contains some information. For this reason, dependency parsing is considered to be more robust than constituent parsing.

Formal compositional semantics, in the tradition of Montague [8], is based on a homomorphism between syntactic structure and semantic representation. As a result, it relies heavily on the notion of constituent, which does not appear explicitly in dependency structures. For this reason, providing a dependency grammar with a Montagovian semantic interpretation is not straightforward.

Nonetheless, in recent years, several ways of adapting Montague's semantics to the case of dependency grammars have been proposed in the literature. In a series of papers, Haug and his co-authors show how to assign a formal interpretation to a dependency structure using glue semantics [6, 3, 5, 4]. Other authors, including ourself, rely on a compositionality principle close to Montague's by exploiting a functional representation of dependency structures [10,
11, 2]. In this paper, we continue the latter line of thought. In particular, we aim to show that the basic principles that we have stated in [2] allow one to deal with more advanced syntactic phenomena. We consider two cases: the relative clauses (which depend on the *acl:relcl* relation) and the open clausal complements (which depend on the *xcomp* relation).

## 2 Relative clauses and wh-extraction

In this section, we propose a solution to the problem of the semantic treatment of wh-extraction. We deal with relative clauses as a paradigmatic example.

First, consider the following sentence together with its dependency tree:

<figure><img src="image_3.png" /></figure>

Following the approach that we have advocated in [2], the above sentence may be assigned a meaning by interpreting the following term:

root (nsubj (obj PRAISES AMANDINE) (det MAN A))

using the following semantics recipes:

A = SOME = λpq. ∃x. (px) ∧ (qx) : DET

MAN = λda. d (a man) : NP

AMANDINE = λdap. pa : NP

PRAISE = λp. ∃e. (praise e) ∧ (pe) : VP

det = λnd. λea. nda : NP → DET → NP

nsubj = λvn. λp. n SOME (λx. x) (λx. v (λe. (agent ex) ∧ (pe))) : VP → NP → VP

obj = λvn. λp. n SOME (λx. x) (λx. v (λe. (theme ex) ∧ (pe))) : VP → NP → VP

root = λv. v (λe. true) : VP → S

where:

DET = (e → t) → (e → t) → t

NP = ((e → t) → (e → t) → t) → ((e → t) → e → t) → (e → t) → t

VP = (v → t) → t

S = t
Now consider the following noun phrase, where the **acl:relc** relation indicates that a noun governs a verb that is the head of a relative clause:

<figure><img src="image_4.png" /></figure>

A usual way to give a semantic account of wh-movement is by means of λ-abstraction and, possibly, combinators to simulate it. This is the approach we follow. Accordingly, we posit:

$$WNP = e \to NP$$

$$WVP = e \to VP$$

This allows our semantic lexicon to be extended as follows:

$$\mathbf{WHO} = \lambda x \mathbf{d}ap. \mathbf{p} \mathbf{x} : \mathbf{WNP}$$

$$\mathbf{acl:relcl} = \lambda n \mathbf{v}da. \mathbf{n} d (\lambda p. \mathbf{a} (\lambda x. (\mathbf{p} \mathbf{x}) \wedge (\mathbf{v} \mathbf{x} (\lambda e. \mathbf{true})))) : \mathbf{NP} \to \mathbf{WVP} \to \mathbf{NP}$$

But then, the expression below gives rise to a type mismatch:

nsubj PRAISES WHO

Indeed, (nsubj PRAISES) is of type NP → VP while WHO is of type WNP. One way out of this typing problem is to coerce the types of the dependency relations using combinators similar to the B and C combinators of combinatory logic [1]:

$$\mathbf{warg1} = \lambda r \mathbf{v}nx. r (v \mathbf{x}) \mathbf{n} : (\mathbf{VP} \to \mathbf{NP} \to \mathbf{VP}) \to \mathbf{WVP} \to \mathbf{NP} \to \mathbf{WVP}$$

$$\mathbf{warg2} = \lambda r \mathbf{v}nx. r v (n \mathbf{x}) : (\mathbf{VP} \to \mathbf{NP} \to \mathbf{VP}) \to \mathbf{VP} \to \mathbf{WNP} \to \mathbf{WVP}$$

Then, the semantic interpretation of the noun phrase, *a man who* *praises amandine*, may be computed by evaluating the following expression:

root (det (acl:relcl MAN (warg2 nsubj (obj PRAISES AMANDINE) WHO)) A)

which reduces to the following formula:<fnref n="1" />

$$\lambda q. \exists x. (\mathbf{man} x) \wedge (\exists e. (\mathbf{praise} e) \wedge (\mathbf{theme} e a) \wedge (\mathbf{agent} e x) \wedge \mathbf{true}) \wedge (q x)$$

<footnote n="1">¹ When applied to a noun phrase, root is defined as root = λn.n SOME (λx.x).</footnote>
The solution we have sketched is consistent with the approach that we have advocated in [2]. In particular, it satisfies our coherence principle. Indeed, all the following expressions provide the same semantic interpretation:

root (det (acl:relcl MAN (warg2 nsubj (obj PRAISES AMANDINE) WHO))) A)

root (det (acl:relcl MAN (warg1 obj (warg2 nsubj PRAISES WHO) AMANDINE))) A)

root (acl:relcl (det MAN A) (warg2 nsubj (obj PRAISES AMANDINE) WHO)))

root (acl:relcl (det MAN A) (warg1 obj (warg2 nsubj PRAISES WHO) AMANDINE)))

## 3 Control verbs and open clausal complements

Open clausal complements are clauses that are missing their own subject and that are typically governed by a control verb or a raising verb.<fnref n="2" /> The dependency relation between a control verb and its open clausal complement is marked by the xcomp relation, like in the following example:

<figure><img src="image_5.png" /></figure>

The puzzle here is that the nsubj dependency relation must provide a subject not only for promises but also for praise.

The solution we propose is based on a new interpretation of verb phrases that is obtained by type-raising the type of events not on t but on e → t:

$$VP = (v \to e \to t) \to e \to t$$

This idea, the details and rationale of which would be too long to develop in this extended abstract, leads to the following interpretation of xcomp together with a revisited interpretation of nsubj to be used for control verbs:

$$xcomp = \lambda v cpx. v (\lambda ey. (\mathbf{topic} e (c (\lambda fz. \mathbf{agent} f z) y)) \wedge (pey)) x$$

$$nsubj = \lambda v npx. n \text{ SOME } (\lambda x. v) (\lambda ey. v (\lambda ez. (\mathbf{agent} ey) \wedge (pey)) y)$$

But this is not the end of the story, because it is not always the subject of the control verb that provides the open clausal complement with a semantic subject. It can also be the object, as the following sentence illustrates:

<footnote n="2">² An open clausal complement may also be the governee of an adjective.</footnote>
<figure><img src="image_6.png" /></figure>

It is even the case that a same control verb admits both construc-
tions:

Michel wants to praise Amandine

Michel wants Samuel to praise Amandine

Our solution can account for these phenomena by relying on a kind
of ad hoc polymorphism, which we discuss in the next section.

4 Ad hoc polymorphism and typing

The semantic developments we have outlined in the previous sec-
tions make it necessary to assign several different interpretations
to the same dependency relation. For instance, we have given two
interpretations to nsubj, and these two interpretations need to co-
exist. This need for some kind of polymorphism can be satisfied by
refining the typing system used at the syntax-semantics interface
level. In a full version of this paper, we will propose and discuss sev-
eral ways of doing so, explaining how they corresponds to encoding
notions of lexical semantics in the typing systems.

5 Conclusions

We have shown how dependency relations involving implicit seman-
tic arguments can be treated along the lines we developed in [2].
We argue that this helps to demonstrate of the feasibility and ad-
equacy of the formal theory of dependency semantics, whose basic
principles we laid out in [2].

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