The conjoinability relation in Lambek calculus and linear logic

Abstract

In 1958 J. Lambek introduced a calculusL of syntactic types and defined an equivalence relation on types: “x≡ y means that there exists a sequence x=x₁,...,x_n=y (n ≥ 1), such thatx _i →x _i+1 or x_i+ →x _i (1 ≤i ≤ n)”. He pointed out thatx ≡y if and only if there is joinz such thatx →z andy →z.

This paper gives an effective characterization of this equivalence for the Lambeck calculiL andLP, and for the multiplicative fragments of Girard's and Yetter's linear logics. Moreover, for the non-directed Lambek calculusLP and the multiplicative fragment of Girard's linear logic, we present linear time algorithms deciding whether two types are equal, and finding a join for them if they are.