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

TL;DR

This paper explores the generative power of LP-grammars over the Lambek calculus with permutation, establishing their connection to branching vector addition systems with states and demonstrating their ability to generate complex, non-semilinear languages.

Contribution

It proves LP-grammars are equivalent to a modified form of branching vector addition systems with states and introduces a normal form for LP-grammars, expanding understanding of their generative capabilities.

Findings

LP-grammars generate more than permutation closures of context-free languages

LP-grammars are equivalent to a class of branching vector addition systems with states

The class of languages generated by LP-grammars is closed under intersection

Abstract

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