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In (Van Benthem, 1991) it is proved that all permutation closures ofcontext-free languages can be generated by grammars over the Lambek calculuswith the permutation rule (LP-grammars); however, to our best knowledge, it isnot established whether the converse holds or not. In this paper, we show thatLP-grammars are equivalent to linearly-restricted branching vector additionsystems with states and with additional memory (shortly, lBVASSAM), which aremodified branching vector addition systems with states. Then an example of suchan lBVASSAM is presented, which generates a non-semilinear set of vectors; thisyields that LP-grammars generate more than permutation closures of context-freelanguages. Moreover, equivalence of LP-grammars and lBVASSAM allows us topresent a normal form for LP-grammars and, as a consequence, prove thatLP-grammars are equivalent to LP-grammars without product. Finally, we provethat the class of languages generated by LP-grammars is closed underintersection.

Grammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States