Parsing for Semidirectional Lambek Grammar is NP-Complete

We study the computational complexity of the parsing problem of a variant of Lambek Categorial Grammar that we call semidirectional. In semidirectional Lambek calculus SD[ there is an additional nondirectional abstraction rule allowing the formula abstracted over to appear anywhere in the premise sequent's left-hand side, thus permitting non-peripheral extraction. SD[ grammars are able to generate each context-free language and more than that. We show that the parsing problem for semidireetional Lambek Grammar is NP-complete by a reduction of the 3Partition problem. K e y words: computational complexity, Lambek Categorial Grammar 1 I n t r o d u c t i o n Categorial Grammar (CG) and in particular Lambek Categorial Grammar (LCG) have their well-known benefits for the formal treatment of natural language syntax and semantics. The most outstanding of these benefits is probably the fact that the specific way, how the complete grammar is encoded, namely in terms of 'combinatory potentials' of its words, gives us at the same time recipes for the construction of meanings, once the words have been combined with others to form larger linguistic entities. Although both frameworks are equivalent in weak generative capacity both derive exactly the context-free languages , LCG is superior to CG in that it can cope in a natural way with extraction and unbounded dependency phenomena. For instance, no special category assignments need to be stipulated to handle a relative clause containing a trace, because it is analyzed, via hypothetical reasoning, like a traceless clause with the trace being the hypothesis to be discharged when combined with the relative pronoun. Figure 1 illustrates this proof-logical behaviour. Notice that this natural-deduction-style proof in the type logic corresponds very closely to the phrasestructure tree one would like to adopt in an analysis with traces. We thus can derive Bill misses ~ as an s from the hypothesis that there is a "phantom" np in the place of the trace. Discharging the hypothesis, indicated by index 1, results in B i l l misses being analyzed as an s/np from zero hypotheses. Observe, however, that such a bottom-up synthesis of a new unsaturated type is only required, if that type is to be consumed (as the antecedent of an implication) by another type. Otherwise there would be a simpler proof without this abstraction. In our example the relative pronoun has such a complex type triggering an extraction. A drawback of the pure Lambek Calculus !_ is that it only allows for so-called 'peripheral extraction', i.e., in our example the trace should better be initial or final in the relative clause. This inflexibility of Lambek Calculus is one of the reasons why many researchers study richer systems today. For instance, the recent work by Moortgat (Moortgat 94) gives a systematic in-depth study of mixed Lambek systems, which integrate the systems L, NL, NLP, and LP. These ingredient systems are obtained by varying the Lambek calculus along two dimensions: adding the permutation rule (P) and/or dropping the assumption that the type combinator (which forms the sequences the systems talk about) is associative (N for non-associative). Taken for themselves these variants of I_ are of little use in linguistic descriptions. But in Moortgat's mixed system all the different resource management modes of the different systems are left intact in the combination and can be exploited in different parts of the grammar. The relative pronoun which would, for instance, receive category (np\np)/(np --o s) with --o being implication in LP, 1 i.e., it requires 1The Lambek calculus with permutation I_P is also called the "nondirectional Lambek calculus" (Benthem 88). In it the leftward and rightward implication