Minimalist Grammars and Minimalist Categorial Grammars, definitions toward inclusion of generated languages

Stabler proposes an implementation of the Chomskyan Minimalist Program, [1] with Minimalist Grammars MG, [2]. This framework inherits a long linguistic tradition. But the semantic calculus is more easily added if one uses the Curry-Howard isomorphism. Minimalist Categorial Grammars MCG, based on an extension of the Lambek calculus, the mixed logic, were introduced to provide a theoreticallymotivated syntax-semantics interface, [3]. In this article, we give full definitions of MG with algebraic tree descriptions and of MCG, and take the first steps towards giving a proof of inclusion of their generated languages. The Minimalist Program MP, introduced by Chomsky, [1], unified more than fifty years of linguistic research in a theoretical way. MP postulates that a logical form and a sound could be derived from syntactic relations. Stabler, [2], proposes a framework for this program in a computational perspective with Minimalist Grammars MG. These grammars inherit a long tradition of generative linguistics. The most interesting contribution of these grammars is certainly that the derivation system is defined with only two rules: merge and move. The word Minimalist is introduced in this perspective of simplicity of the definitions of the framework. If the merge rule seems to be classic for this kind of treatment, the second rule, move, accounts for the main concepts of this theory and makes it possible to modify relations between elements in the derived structure. Even if the phonological calculus is already defined, the logical one is more complex to express. Recently, solutions were explored that exploited Curry’s distinction between tectogrammatical and phenogrammatical levels; for example, Lambda Grammars, [4], Abstract Categorial Grammars, [5], and Convergent Grammars [6]. First steps for a convergence between the Generative Theory and Categorial Grammars are due to S. Epstein, [7]. A full volume of Language and Computation proposes several articles in this perspective, [8], in particular [9], and Cornell’s works on links between Lambek calculus and Transformational Grammars, [10]. Formulations of Minimalist Grammars in a Type-Theoretic way have also been proposed in [11], [12], [13]. These frameworks were evolved in [14], [3], [15] for the syntax-semantics interface. Defining a syntax-semantics interface is complex. In his works, Stabler proposes to include this treatment directly in MG. But interactions between syntax and semantic properties occur at different levels of representation. One solution is to suppose that these two levels should be synchronized. Then, the Curry-Howard isomorphism could be invoked to build a logical representation of utterances. The Minimalist Categorial Grammars have been defined from this perspective: capture the same properties as MG and propose a synchronized semantic calculus. We will propose definitions of these grammars in this article. But do MG and MCG genrate the same language? In this article we take the first steps towrds showing that they do. The first section proposes new definitions of Minimalist Grammars based on an algebraic description of trees which allows to check properties of this framework, [3]. In the second section, we will focus on full definitions of Minimalist Categorial Grammars (especially the phonological calculus). We will give a short motivation for the syntax-semantics interface, but the complete presentation is delayed to a specific article with a complete example. These two parts should be viewed as a first step of the proof of mutual inclusion of languages between MG and MCG. This property is important because it enables us to reduce MG’s to MCG, and we have a well-defined syntax-semantics interface for MCG. 1 Minimalist Grammars Minimalist Grammars were introduced by Stabler [2] to encode the Minimalist Program of Chomsky, [1]. They capture linguistic relations between constituents and build trees close to classical Generative Analyses. These grammars are fully lexicalized, that is to say they are specified by their lexicon. They are quite different from the traditional definition of lexicalized because they allow the use of specific items which do not carry any phonological form. The use of theses items implies that MG represent more than syntactic relations and must be seen as a meta-calculus lead by the syntax. These grammars build trees with two rules:merge andmove which are trigged by features. This section presents all the definitions of MG in a formal way, using algebraic descriptions of trees. 1.1 Minimalist Tree Structures To provide formal descriptions of Minimalist Grammars, we differ from traditional definitions by using an algebraic description of trees: a sub-tree is defined by its context, as in [16] and [17]. For example, the figure on the left of the figure 1 shows two subtrees in a tree (t1 and t2) and their context (C1 and C2). Before we explain the relations in minimalist trees, we give the formal material used to define a tree by its context. Graded alphabets and trees: Trees are defined from a graded set. A graded set is made up of a support set, noted Σ, the alphabet of the tree, and a rank function, noted σ, which defines node arity (the graded terminology results from the rank function). In the following, we will use Σ to denote a graded (Σ, σ). The set of trees built on Σ, written TΣ , is the smallest set of strings (Σ ∪ {(; ); , }). A leaf of a tree is a node of arity 0, denoted by α instead of α(). For a tree t, if t = σ(t1, · · · , tk), the root node of t is written σ . Moreover, a set of variables X = {x1, x2, · · ·} is added for these trees. Xk is the set of k variables. These variables mark positions in trees. By using variables, we define a substitution rule: given a tree t ∈ TΣ(Xk) (i.e. a tree which contains instances of k variables x1, · · · , xk) and t1, · · · , tk, k trees in TΣ , the tree obtained by simultaneous substitution of each instance of x1 by t1, . . . , xk by tk is denoted by t[t1, · · · , tk]. The set of all subtrees of t is noted St. Thus, for a given tree t and a given node n of t, the subtree for which n is the root is denoted by t with this subtree replaced by a variable. Minimalist trees are produced by Minimalist Grammars and they are built on the graded alphabet {<,>,Σ}, whose ranks of < and > are 2 and 0 for strings of Σ. Minimalist Trees are binary ones whose nodes are labelled with < or >, and whose leaves contain strings of Σ. Relations between sub-trees We formalise relations for different positions of elements in St. Intuitively, these define the concept of be above, be on the right or on the left. A specific relation on minimalist trees is also defined: projection that introduces the concept of be the main element in a tree. In the following, we assume a given graded alphabet Σ. Proofs of principal properties and closure properties are all detailed in [3]. The first relation is the dominance which informally is the concept of be above. Definition 1 Let t ∈ TΣ, and C1, C2 ∈ St, C1 dominates C2 (written C1 ⊳ ∗ C2) if there exists C ′ ∈ St such that C1[C ] = C2. Figure 1 shows an example of dominance in a tree. One interesting property of this algebraic description of trees is that properties in sub-trees pass to tree. For example, in a given tree t, if there exists C1 and C2 such that C1 ⊳ ∗ C2, using a 1-context C, we could build a new tree t = C[t] (substitution in the position marked by the variable xx1 of t). Then, C[C1] and C[C2] exist (they are part of t) such that C[C1]⊳ C[C2]. Definition 2 Let t ∈ TΣ, C1, C2 ∈ St, C1 immediately precedes C2 (written C1 ≺ C2) if there exists C ∈ St such that: 1. C1 = C[σ(t1, . . . , tj , x1, tj+2, . . . , tk)] and 2. C2 = C[σ(t1, . . . , tj , tj+1, x1, . . . , tk)]. Precedence, written ≺, is the smallest relation defined by the following rules (transitivity rule, closure rule and relation between dominance and precedence relation): C1 ≺ ∼ C2 C2 ≺ ∼ C3 [trans] C1 ≺ ∼ C3 C1 ≺ C2 [∗] C1 ≺ ∼ C2 C1 ⊳ ∗ C2 [dom] C2 ≺ ∼ C1