Chinese Numbers, MIX, Scrambling, and Range Concatenation Grammars

The notion of mild context-sensitivity was formulated in an at tempt to express the formal power which is both necessary and sufficient to define the syntax of natural languages. However, some linguistic phenomena such as Chinese numbers and German word scrambling lie beyond the realm of mildly contextsensitive formalisms. On the other hand, the class of range concatenation grammars provides added power w.r. t, mildly context-sensitive grammars while keeping a polynomial parse time behavior. In this report, we show that this increased power can be used to define the abovementioned linguistic phenomena with a polynomial parse time of a very low degree. 1 M o t i v a t i o n The notion of mild context-sensitivity originates in an attempt by [Joshi 85] to express the formal power needed to define the syntax of natural languages (NLs). We know that contextfree grammars (CFGs) are not adequate to define NLs since some phenomena are beyond their power (see [Shieber 85]). Popular incarnations of mildly context-sensitive (MCS) formalisms are tree adjoining grammars (TAGs) [Vijay-Shanker 87] and linear context-free rewriting (LCFR) systems [Vijay-Shanker, Weir, and Joshi 87]. However, there are some linguistic phenomena which are known to lie beyond MCS formalisms. Chinese numbers have been studied in [Radzinski 91] where it is shown that the set of these numbers is not a LCFR language and that it appears also not to be MCS since it violates the constant growth property. Scrambling is a word-order phenomenon which also lies beyond LCFR systems (see [Becket, Rambow, and Niv 92]). On the other hand, range concatenation grammar (RCG), presented in [Boullier 98a], is a syntactic formalism which is a variant of simple literal movement grammar (LMG), described in [Groenink 97], and which is also related to the framework of LFP developed by [Rounds 88]. In fact it may be considered to lie halfway between their respective string and integer versions; RCGs retain from the string version of LMGs or LFPs the notion of concatenation, applying it to ranges (couples of integers which denote occurrences of substrings in a source text) rather than strings, and from their integer version the ability to handle only (part of) the source text (this later feature being the key to tractability). RCGs can also be seen as definite clause grammars acting on a flat domain: its variables are bound to ranges. This formalism, which extends CFGs, aims at being a convincing challenger as a syntactic base for various tasks, especially in natural language processing. We have shown that the positive version of RCGs, as simple LMGs or integer indexing LFPs, exactly covers the class PTIME of languages recognizable in deterministic polynomial time. Since the composition operations of RCGs are not restricted to be linear and non-erasing, its languages (RCLs) are not semi-linear. Therefore, RCGs are not MCS and are more powerful than LCFR systems, while staying computationally tractable: its sentences can be parsed in polynomial time. However, this formalism shares with LCFR systems the fact that its derivations are CF (i.e. the choice of the operation performed at each step only depends on the object to be derived from). As in the CF case, its derived trees can be packed into polynomial sized parse forests. For a CFG, the components of a parse forest are nodes labeled by couples (A, p) where A is a nonterminal symbol and p is a range, while for an RCG, the labels have the form (A, p-') where # is a vector (list) of ranges. Besides its power and efficiency, this formalism possesses many other attractive proper-