Proving theorems of the second order Lambek calculus in polynomial time

Abstract

In the Lambek calculus of order 2 we allow only sequents in which the depth of nesting of implications is limited to 2. We prove that the decision problem of provability in the calculus can be solved in time polynomial in the length of the sequent. A normal form for proofs of second order sequents is defined. It is shown that for every proof there is a normal form proof with the same axioms. With this normal form we can give an algorithm that decides provability of sequents in polynomial time.