The Computational Compexity of Decision Problem in Additive Extensions of Nonassociative Lambek Calculus

Abstract

We analyze the complexity of decision problems for Boolean Nonassociative Lambek Calculus admitting empty antecedent of sequents ($\mathsf{BFN}{\mathsf{L}}^{\ast }$), and the consequence relation of Distributive Full Nonassociative Lambek Calculus ($\mathsf{DFNL}$). We construct a polynomial reduction from modal logic $\mathsf{K}$ into $\mathsf{BFN}{\mathsf{L}}^{\ast }$. As a consequence, we prove that the decision problem for $\mathsf{BFN}{\mathsf{L}}^{\ast }$ is PSPACE-hard. We also prove that the same result holds for the consequence relation of DFNL, by reducing $\mathsf{BFN}{\mathsf{L}}^{\ast }$ in polynomial time to DFNL enriched with finite set of assumptions. Finally, we prove analogous results for variants of $\mathsf{BFN}{\mathsf{L}}^{\ast }$, including $\mathsf{BFN}{\mathsf{L}}^{\ast }\mathsf{e}$ ($\mathsf{BFN}{\mathsf{L}}^{\ast }$ with exchange), modal extensions of $\mathsf{BFN}{\mathsf{L}}_{\mathsf{i}}^{\ast }$ and $\mathsf{BFN}{\mathsf{L}}_{\mathsf{ei}}^{\ast }$ for $i\in \{\mathsf{K},\mathsf{T},K4,S4,S5\}$.

12 pages