Expressing discourse dynamics through
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Philippe de Groote.
At the beginning of the seventies,
Richard Montague originated a revolution in formal linguistics by
showing how standard tools from mathematical logic (proof-theory and
model-theory) could be used to assign semantics to natural language
utterances . His work, which is still a reference today, has given
rise to the research domain that is now known as Montague semantics.
From a technical point of view, Montague semantics is based on
higher-order logic and on the simply typed lambda-calculus.
Consequently, it suits quite well the categorial grammar approach whose
foundations are also type-theoretic . This has been pointed out at
the end of the eighties by van Benthem who showed how the two
formalisms could be connected using the Curry-Howard isomorphism .
This correspondence is the keystone of the syntax/semantics interface
of modern type-logical grammars. It also motivated the definition
of new grammatical formalisms such as Muskens' lambda-grammars  or
de Groote's abstract categorial grammars .
Montague semantics, in its current form, has its own limitations. In
particular, it is concerned with the meaning of single sentences, and
does not allow for the various dynamic phenomena that participate to
the interpretation of a discourse (interpretation context updating,
anaphora and co-reference resolution, ...). In order to make up
for this lack, new formalisms have been defined. Among these, the
most well-known are Kamp's discourse representation theory, and its
variants [6, 13].
Most of these formalisms are based, in one way or another, on a notion
of state change that is used to model the dynamic phenomena. As a
consequence, it is difficult to use the standard tools of mathematical
logic at the level of the discourse interpretation. In
particular, the simplicity of a syntax/semantics interface based on the
Curry-Howard isomorphism is lost.
The goal of this project is to lay new
theoretic foundations that will allow discourse dynamics to be taken
into account. Technically, it will consist in modelling the
notions of context and context change in a purely Montagovian way,
i.e., using only standard notions coming from type-theory and
lambda-calculus. A similar problem has been solved in the domain
of programming language denotational semantics, where the challenge was
to provide control operators (which present dynamic and
non-compositional features) with a mathematical semantics. The key
notion used to this end is the one of `continuation' .
Recently, de Groote has shown how this same notion of continuation may
be used to model the way quantifiers in natural language extend
dynamically their scope . The objectives of the project consist in
extending and generalising this work to several other dynamic phenomena
that play a part in the semantic interpretation of a discourse.
The project will follow an approach
based on the two following ideas:
- a sentence should be interpreted according to both its left and
- in order to get a compositional semantics, these two kind of
contexts must be abstracted over the meaning of the sentences.
In order to motivate these two
principles, consider the following piece of discourse:
A man enters the room. He switches
on the light.
On the one hand, the second sentence
refers to the first sentence through the pronominal anaphora. On
the other hand, the scope of the existential quantifier introduced by
the first sentence includes the second sentence. consequently, in
order to provide a single sentence with a self-contained meaning, both
its left and right contexts should be parameters of the interpretation.
Technically, Montague semantics is based on Church's simple type theory
, which provides a full hierarchy of functional types built upon two
atomic types: ι, the type of
individuals, and ο, the type
of propositions. In order to accommodate a notion of context, one
adds a third atomic type, namely, γ.
This type stands for the type of the left contexts. Then, if one
only considers discourses made of declarative sentences, a right
context is a piece of discourse that will be interpreted as a
proposition provided it is given its left context. Consequently, the
type of the right contexts must be γ→ο.
This type may be seen as the type of a continuation, which is indeed a
standard way of modeling right contexts in denotational semantics.
Now, let s
be the syntactic category of sentences, and t be the
syntactic category of texts (or discourses). As it has been said, one
intends to abstract over the meaning of a sentence its left and right
contexts. Consequently, one obtains the following semantic
interpretation for s and t:
[[s]] = [[t]] = γ→ (γ→ο)
This equation is the basis of a
Montagovian account of dynamics. The classical approach to
dynamics consits in interpreting a sentence as a context
modifier. Using the present notations, such an approach gives
rise to the following semantic interpretation:
Unfortunately, Equation (2) does not
allow the quantifiers to extend dynamically their scopes. In
fact, in , Montague is facing a similar problem: interpreting a noun
phrase as an individual (i.e., a term of type ι) does not allow a quantified
noun phrase to extend its scope over the complete sentence. His
solution consists in interpreting a noun phrase as a `type-raised'
individual, i.e., as a term of type (ι→ο) →ο. Similarly, Equation (1)
may be seen as the `type-raised' version of Equation (2). This is not
quite surprising because Montague's type raising is an operation that
may be explained in terms of continuations [1, 2].
 Barker, C. (2002). `Continuations
and the nature of quantification'. Natural Language Semantics 10, 211-242.
 Barker, C. (2004). `Continuations in Natural Language'. In H.
Thielecke (ed.), Proceedings of the
Fourth ACM SIGPLAN Continuations Workshop, 1-11.
 Church, A. (1940). `A formulation of the simple theory of
types'. Journal of Symbolic
 de Groote, P. (2001). `Towards abstract categorial grammars'.
In Association for Computational
Linguistics, 39th Annual Meeting and 10th Conference of the European
Chapter, Proceedings of the Conference, pp. 148-155.
 de Groote, P. (2006). `Towards a Montagovian account of
dynamics'. In Proceedings of
Semantics and Linguistic Theory XVI, CLC Publications, to appear.
 Kamp, H. and U. Reyle (1993). From
Discourse to Logic. Kluwer Academic Publishers, Dordrecht.
 Montague, R. (1973). `The proper treatment of quantification in
ordinary English'. In J. Hintikka, J. Moravcsik, and P. Suppes (eds.), Approaches to natural language:
proceedings of the 1970 Stanford workshop on Grammar and Semantics.
Reidel, Dordrecht. Reprinted: [8, pp. 247-270].
 Montague, R. (1974). Formal
Philosophy: selected papers of Richard Montague, edited and with an
introduction by Richmond Thomason. Yale University
 Moortgat, M. (1997). `Categorial type logics'. In J. van
Benthem and A. ter Meulen (Eds.), Handbook
of Logic and Language, Chapter 2, pp. 93-177. Elsevier.
 Muskens, R. (2001). `Lambda Grammars and the Syntax-Semantics
Interface'. In R. van Rooy and M. Stokhof (eds.), Proceedings of the Thirteenth Amsterdam
Colloquium, pp. 150-155.
 Stratchey, C. and C. Wadsworth (1974). `Continuations a
mathematical semantics for handling full jumps'. Technical Report
PRG-11, Oxford University, Computing Laboratory.
 J. van Benthem (1988). `The semantics of variety in
categorial grammar'. In W. Buszkowski, W. Marciszewski, and J.
van Benthem, editors, Categorial
Grammars. John Benjamins.
 van Eijck, J. and H. Kamp (1997). `Representing Discourse in
Context'. In J. van Benthem and A. ter Meulen (eds.), Handbook of Logic and Language.
Desired profile of candidate
Elementary knowledge of lambda-calculus
and type theory. Training in computational linguistics.