A Finite Model Property for Intersection Types

We show that the relational theory of intersection types known as BCD has the finite model property; that is, BCD is complete for its finite models. Our proof uses rewriting techniques which have as an immediate by-product the polynomial time decidability of the preorder ⊆ (although this also follows from the so called beta soundness of BCD). BCD is the relational theory of intersection types presented by Henk Barendregt, Mario Coppo, and Mariangiola Dezani in [2]. Here we consider the theory, without top element, as about a preorder ⊆, a ⊆ a a ⊆ b & ⊆ c ⇒ a ⊆ c a ∧ b ⊆ a a ∧ b ⊆ b c ⊆ a & c ⊆ b ⇒ c ⊆ a ∧ b, and a contravariant-covariant operation →, c ⊆ a & b ⊆ d ⇒ a → b ⊆ c → d satisfying the weak distributive law (c → a) ∧ (c → b) ⊆ c → (a ∧ b). Of course it is well known that if the points of such a preorder are partitioned by the congruence ∼ defined by a ∼ b ⇔ a ⊆ b & b ⊆ a we obtain a semilattice with ∧, that is, a ∧ (b ∧ c) ∼ (a ∧ b) ∧ c a ∧ b ∼ b ∧ a a ∼ a ∧ a where the quotient partial order can be recovered a ⊆ b ⇔ a ∼ a ∧ b.