Two Useful Measures Of Word Order Complexity

This paper presents a class of dependency-based for-real grammars (FODG) which can be parametrized by two different but similar measures of non-projectivity. The measures allow to formulate constraints on the degree of word-order freedom in a language described by a FODG. We discuss the problem of the degree of word-order freedom which should be allowed ~, a FODG describing the (surface) syntax of Czech. 1 Introduction In [Kuboh,Holan:Pl~tek.1997] we have introduced a class of formal grammars. Robust Free.Order Dependency Grammars (RFODG's), in order to provide for a formal foundation to the way we are developing a grammar-checker for Czech. a natural language with a considerable level of word-order free-dora. The design of RFODG's was inspired by tile commutative CF-grammars (see [Huynh.83]), and several types of dependency based grammars (cf.. we have introduced different measures of incorrect-ness and of non-projectivity of a sentence. The measures of the non-projectivity create the focus of our interest in this paper. They are considered as the measures of word-order freedom. Considering this aim we work here with a bit simplified version of RFODG's. namely with Fre~-Order Dependency Grammars (FODG's). The measures of word-order freedom are used to formulate constraints which can be imposed on FODG's globally, or on their individual rules. Two types of syntactic structures, namely DR-trees (delete-rewrite-trees), and De-fre~s (dependency trees), are connected with FODG's. Any DR, tree can be transformed into a De-tree in an easy and uniform w~,. In [Kubofi,Holan.Pl/Ltek.1997] the measures of non-projectivity are introduced with the help of DR-trees only. Here we discuss one of them, called node-gaps complexity (Ng). It has some very interesting properties. CFI.'s are characterized by the complexity 0 of Ng. The N9 also characterizes the time complexity of the parser used by the above-n~'entioned grammar-checker. The sets of sentences with the Ny less than a fixed constant are parsable in a polynomial time. This led us to the idea to look for a fixed upper bound of Ng for all Czech sentences with a correct word order. In [Kubofi,Holan,Pbltek.1997] we even worked with the conjecture that such an upper bound can be set to 1. We will show in Section 5 that. it is theoretically impossible to find such an upper bound, and that even for practical purposes, e.g.. for grammar-checking. it should be set to a value considerably higher than 1. This is shown with the help of the measure dNg which is …