Tree-Adjoining Grammars as Abstract Categorial Grammars

categorial grammars are not intended as yet another grammatical formalism that would compete with other established formalisms. It should rather be seen as the kernel of a grammatical framework — in the spirit of (Ranta, 2002) — in which other existing grammatical models may be encoded. This paper illustrates this fact by showing how tree-adjoining grammars (Joshi and Schabes, 1997) may be embedded in abstract categorial grammars. This embedding exemplifies several features of the ACG framework: • The fact that the basic objects manipulated by an ACG are λ-terms allows higher-order operations to be defined. Typically, tree-adjunction is such a higher-order operation (Abrusci, Fouqueré and Vauzeilles, 1999; Joshi and Kulick, 1997; Mönnich, 1997). • The flexibility of the framework allows the embedding to be defined in two stages. A first ACG allows the tree langage of a given TAG to be generated. The abstract language of this first ACG corresponds to the derivation trees of the TAG. Then, a second ACG allows the corresponding string language to be extracted. The abstract language of this second ACG corresponds to the object language of the first one. 2. Abstract Categorial Grammars This section defines our notion of an abstract categorial grammar. We first introduce the notions of linear implicative types, higher-order linear signature, linear λ-terms built upon a higher-order linear signature, and lexicon. Let A be a set of atomic types. The set T (A) of linear implicative types built upon A is inductively defined as follows: 1. if a ∈ A, then a ∈ T (A); 2. if α, β ∈ T (A), then (α−◦ β) ∈ T (A). A higher-order linear signature consists of a triple Σ = 〈A,C, τ〉, where: 1. A is a finite set of atomic types; c © 2002 Philippe de Groote. Proceedings of the Sixth International Workshop on Tree Adjoining Grammar and Related Frameworks (TAG+6), pp. 101–106. Universitá di Venezia. 102 Proceedings of TAG+6 2. C is a finite set of constants; 3. τ : C → T (A) is a function that assigns to each constant in C a linear implicative type in T (A). Let X be a infinite countable set of λ-variables. The set Λ(Σ) of linear λ-terms built upon a higher-order linear signature Σ = 〈A,C, τ〉 is inductively defined as follows: 1. if c ∈ C, then c ∈ Λ(Σ); 2. if x ∈ X , then x ∈ Λ(Σ); 3. if x ∈ X , t ∈ Λ(Σ), and x occurs free in t exactly once, then (λx. t) ∈ Λ(Σ); 4. if t, u ∈ Λ(Σ), and the sets of free variables of t and u are disjoint, then (t u) ∈ Λ(Σ). Λ(Σ) is provided with the usual notion of capture avoiding substitution, α-conversion, and β-reduction (Barendregt, 1984). Given a higher-order linear signature Σ = 〈A,C, τ〉, each linear λ-term in Λ(Σ) may be assigned a linear implicative type in T (A). This type assignment obeys an inference system whose judgements are sequents of the following form: Γ −Σ t : α where: 1. Γ is a finite set of λ-variable typing declarations of the form ‘x : β’ (with x ∈ X and β ∈ T (A)), such that any λ-variable is declared at most once; 2. t ∈ Λ(Σ); 3. α ∈ T (A). The axioms and inference rules are the following: