Introduction to a Hypergraph Logic Unifying Different Variants of the Lambek Calculus

Abstract: In this paper hypergraph Lambek calculus ($\mathrm{HL}$) is presented. This formalism aims to generalize the Lambek calculus ($\mathrm{L}$) to hypergraphs as hyperedge replacement grammars extend context-free grammars. In contrast to the Lambek calculus, $\mathrm{HL}$ deals with hypergraph types and sequents; its axioms and rules naturally generalize those of $\mathrm{L}$. Consequently, certain properties (e.g. the cut elimination) can be lifted from $\mathrm{L}$ to $\mathrm{HL}$. It is shown that $\mathrm{L}$ can be naturally embedded in $\mathrm{HL}$; moreover, a number of its variants ($\mathrm{LP}$, $\mathrm{NL}$, $\mathrm{NLP}$, $\mathrm{L}$ with modalities, $\mathrm{L}^\ast(\mathbf{1})$, $\mathrm{L}^{\mathrm{R}}$) can also be embedded in $\mathrm{HL}$ via different graph constructions. We also establish a connection between $\mathrm{HL}$ and Datalog with embedded implications. It is proved that the parsing problem for $\mathrm{HL}$ is NP-complete.