Lambek Calculus with Optional Divisions

In this paper, we introduce an extension of the Lambek calculus by optional divisions ( \(\textrm{L}_{opt}\) ). Namely, the right optional division \(A \angle B\) is defined as \(A \wedge (A/B)\) , and the left one is defined as \(A \wedge (B \backslash A)\) . A possible linguistic motivation to consider the new operations is describing verbs with optional arguments, e.g. reads in Tim reads the book. \(\textrm{L}_{opt}\) is a fragment of the multiplicative-additive Lambek calculus, so it would be interesting to compare the two calculi. The main part of the paper is devoted to the following grammar result: finite intersections of context-free languages can be generated by grammars over the calculus \(\textrm{L}_{opt}\) . The proof involves introducing a useful normal form for Lambek grammars, namely, the notion of an interpretable grammar.