INRIA-Lorraine, Nancy.

Calligramme

Philippe de Groote

• e-mail: degroote@loria.fr
• phone: + 33 3 54 95 84 04

Application is online.  If this link does not work or if you have any question, send an e-mail to 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 [7]. 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 [9]. 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 [12]. 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 [10] or de Groote's abstract categorial grammars [4].

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' [11].  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 [5]. 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:

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 [3], 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) →ο                 (1)

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:

[[s]] = [[t]] =  γ→γ                                 (2)

Unfortunately, Equation (2) does not allow the quantifiers to extend dynamically their scopes.  In fact, in [7], 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].

[1] Barker, C. (2002). `Continuations and the nature of quantification'.  Natural Language Semantics 10, 211-242.

[2] Barker, C. (2004). `Continuations in Natural Language'. In H. Thielecke (ed.), Proceedings of the Fourth ACM SIGPLAN Continuations Workshop, 1-11.

[3] Church, A. (1940).  `A formulation of the simple theory of types'.  Journal of Symbolic Logic, 5:56--68.

[4] 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.

[5] de Groote, P. (2006).  `Towards a Montagovian account of dynamics'.  In Proceedings of Semantics and Linguistic Theory XVI, CLC Publications, to appear.

[6] Kamp, H. and U. Reyle (1993). From Discourse to Logic.  Kluwer Academic Publishers, Dordrecht.

[7] 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].

[8] Montague, R. (1974). Formal Philosophy: selected papers of Richard Montague, edited and with an introduction by Richmond Thomason.   Yale University Press.

[9] 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.

[10] 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.

[11] Stratchey, C. and C. Wadsworth (1974). `Continuations a mathematical semantics for handling full jumps'. Technical Report PRG-11, Oxford University, Computing Laboratory.

[12] 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.

[13] 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.  Elsevier.

Elementary knowledge of lambda-calculus and type theory. Training in computational linguistics.