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 {\em semidirectional}. In semidirectional Lambek calculus $\text{SDL}$ there is an additional non-directional abstraction rule allowing the formula abstracted over to appear anywhere in the premise sequent’s left-hand side, thus permitting non-peripheral extraction. $\text{SDL}$ grammars are able to generate each context-free language and more than that. We show that the parsing problem for semidirectional Lambek Grammar is NP-complete by a reduction of the 3-Partition problem.