Denis Béchet

A Complete Axiomatisation for the Inclusion of Series-Parallel Partial Orders

Series-parallel orders are defined as the least class of partial orders containing the one-element order and closed by ordinal sum and disjoint union. From this inductive definition, it is almost immediate that any series-parallel order may be represented by an algebraic expression, which is unique up to the associativity of ordinal sum and to the associativivity and commutativity of disjoint union. In this paper, we introduce a rewrite system acting on these algebraic expressions that axiomatises completely the sub-ordering relation for the class of series-parallel orders.

Dependency structure grammars

In this paper, we define Dependency Structure Grammars (DSG), which are rewriting rule grammars generating sentences together with their dependency structures, are more expressive than CF-grammars and non-equivalent to mildly context-sensitive grammars. We show that DSG are weakly equivalent to Categorial Dependency Grammars (CDG) recently introduced in [6,3]. In particular, these dependency grammars naturally express long distance dependencies and enjoy good mathematical properties.

Two models of learning iterated dependencies

We study the learnability problem in the family of Categorial Dependency Grammars (CDG), a class of categorial grammars defining unlimited dependency structures. CDG satisfying a reasonable condition on iterated (i.e., repeatable and optional) dependencies are shown to be incrementally learnable in the limit.

Parsing Pregroup Grammars and Lambek Calculus Using Partial Composition

The paper presents a way to transform pregroup grammars into contextfree grammars using functional composition. The same technique can also be used for the proof-nets of multiplicative cyclic linear logic and for Lambek calculus allowing empty premises.

Fully lexicalized pregroup grammars

Pregroup grammars are a context-free grammar formalism introduced as a simplification of Lambek calculus. This formalism is interesting for several reasons: the syntactical properties of words are specified by a set of types like the other type-based grammar formalisms; as a logical model, compositionality is easy ; a polytime parsing algorithm exists. However, this formalism is not completely lexicalized because each pre-group grammar is based on the free pregroup built from a set of primitive types together with a partial order, and this order is not lexical information. In fact, only the pregroup grammars that are based on primitive types with an order that is equality can be seen as fully lexicalized. We show here how we can transform, using a morphism on types, a particular pregroup grammar into another pregroup grammar that uses the equality as the order on primitive types. This transformation is at most quadratic in size (linear for a fixed set of primitive types), it preserves the parse structures of sentences and the number of types assigned to a word.

k-Valued Non-Associative Lambek Grammars are Learnable from Function-Argument Structures

This paper is concerned with learning categorial grammars in the model of Gold. We show that rigid and k-valued non-associative Lambek grammars are learnable from function-argument structured sentences. In fact, function-argument structures are natural syntactical decompositions of sentences in sub-components with the indication of the head of each sub-component. This result is interesting and surprising because for every k, the class of k-valued NL grammars has innite elasticity and one could think that it is not learnable, which is not true. Moreover, these classes are very close to unlearnable classes like k-valued associative Lambek grammars learned from function-argument sentences or k-valued non-associative Lambek calculus grammars learned from well-bracketed list of words or from strings. Thus, the k-valued non-associative Lambek grammars learned from function-argument sentences is at the frontier between learnable and unlearnable classes of languages.

Minimality of the correctness criterion for multiplicative proof nets

Almost a decade ago, Girard invented linear logic with the notion of a proof-net. Proof-nets are special graphs built from formulas, links and boxes. However, not all nets are proof-nets. First, they must be well constructed (we say that such graphs are proof-structures). Second, a proof-net is a proof-structure that corresponds to a sequential proof. It must satisfy a correctness criterion. One may wonder what this static criterion means for cut-elimination. We prove that every incorrect proof-structure (without cut) can be put in an environment where reductions lead to two kinds of basically wrong configurations: deadlocks and disconnected proof-structures. Thus, this proof says that there does not exist a bigger class of proof-structures than proof-nets where normalization does not lead to obviously bad configurations.

k -Valued non-associative Lambek grammars are learnable from generalized functor-argument structures

This paper is concerned with learning categorial grammars from positive examples in the model of Gold. Functor-argument structures (written FA) are usual syntactical decompositions of sentences in sub-components distinguishing the functional parts from the argument parts defined in the case of classical categorial grammars also known as AB-grammars. In the case of nonassociative type-logical grammars, we propose a similar notion that we call generalized functor-argument structures and we show that these structures capture the essence of non-associative Lambek (NL) calculus without product.We show that (i) rigid and k-valued non-associative Lambek (NL without product) grammars are learnable from generalized functor-argument structured sentences.We also define subclasses of k-valued grammars in terms of arity. We first show that (ii) for each k and each bound on arity the class of FA-arity bounded k-valued NL languages of FA structures is finite and (iii) that FA-arity bounded k-valued NL grammars are learnable both from strings and from FA structures as a corollary.Result (i) is obtained from (ii); this learnability result (i) is interesting and surprising when compared to other results: in fact we also show that (iv) this class has infinite elasticity. Moreover, these classes are very close to classes like rigid associative Lambek grammars learned from natural deduction structured sentences (that are different and much richer than FA or generalized FA) or to k-valued non-associative Lambek grammars unlearnable from strings or even from bracketed strings. Thus, the class of k-valued non-associative Lambek grammars learned from generalized functor-argument sentences is at the frontier between learnable and unlearnable classes of languages.

k-valued non-associative lambek grammars (without product) form a strict hierarchy of languages

The notion of k-valued categorial grammars where a word is associated to at most k types is often used in the field of lexicalized grammars as a fruitful constraint for obtaining several properties like the existence of learning algorithms. This principle is relevant only when the classes of k-valued grammars correspond to a real hierarchy of languages. This paper establishes the relevance of this notion for two related grammatical systems. In the first part, the classes of k-valued non-associative Lambek (NL) grammars without product is proved to define a strict hierarchy of languages. The second part introduces the notion of generalized functor argument for non-associative Lambek (NL∅) calculus without product but allowing empty antecedent and establishes also that the classes of k-valued (NL∅) grammars without product form a strict hierarchy of languages.

Universal Interaction Systems with Only Two Agents

In the framework of interaction nets [6], Yves Lafont has proved [8] that every interaction system can be simulated by a system composed of 3 symbols named γ, δ and Ɛ. One may wonder if it is possible to find a similar universal system with less symbols. In this paper, we show a way to simulate every interaction system with a specific interaction system constituted of only 2 symbols. By transitivity, we prove that we can find a universal interaction system with only 2 agents. Moreover, we show how to find such a system where agents have no more than 3 auxiliary ports.