Cyclic Shift in the Lambek Calculus

TL;DR

This paper extends the Lambek calculus with a cyclic shift operation, enabling modeling of cyclic language closure, and explores its implications for categorial grammars, recognizing power, and embedding in hypergraph Lambek calculus.

Contribution

It introduces a new cyclic shift operation into the Lambek calculus, proves cut elimination, and analyzes its impact on language recognition and calculus embedding.

Findings

Categorial grammars with cyclic shift can generate non-context-free languages.

The calculus with cyclic shift can be embedded into hypergraph Lambek calculus.

Recognition power differs between the calculus with cyclic shift as a structural rule and the one with it as an operation.

Abstract

We enrich the Lambek calculus with the cyclic shift operation, which is expected to model the closure operator of formal languages with respect to cyclic shifts. We introduce a Gentzen-style calculus and prove cut elimination. Secondly, we turn to categorial grammars based on this calculus and show that they can generate non-context-free languages; besides, we consider a related calculus where the cyclic shift is a structural rule, and compare recognizing power of these two calculi. Thirdly, we attempt to embed the Lambek calculus with the cyclic shift operation in the hypergraph Lambek calculus. This results in considering a ``bracelet'' operation, which can be defined through the cyclic shift, union, and the reversal operation.