Complexity of Nonassociative Lambek Calculus with Classical and Intuitionistic Logic

The Nonassociative Lambek Calculus (NL) represents a logic devoid of the structural rules of exchange, weakening, and contraction, and it does not presume the associativity of its connectives. Its finitary consequence relation is decidable in polynomial time. However, the addition of classical connectives conjunction and disjunction (FNL) makes the consequence relation undecidable. Interestingly, if these connectives are distributive, the consequence relation is decidable in exponential time. This paper provides the proof, that we can merge classical logic with NL (i.e. BFNL) and intuitionistic logic with NL (i.e. HFNL), and still consequence relations are decidable in exponential time.