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

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