Probabilistic Approach to the Lambda Definability for Fourth Order Types

It has been proved by Loader [3] that Statman-Plotkin conjecture (see [7] and [5]) fails. The Loader proof was done by encoding the word problem in the full type hierarchy based on the domain with 7 elements. The aim of this paper is to show that the lambda definability problem limited to regular fourth order types is decidable in any finite domain. Obviously λ definability is decidable for 1, 2 and 3 order types. As an additional effect of the result described we may observe that for certain types there is no finite context free grammar generating all closed terms. We prove also that probability that randomly chosen fourth order type (or type of the order not grater then 4 ) admits decidable lambda definability problem is zero. 1 Syntax of simple typed lambda calculus We shall consider a simple typed lambda calculus with a single ground type O. The set T of types is defined as follows: O is a type and if τ and μ are types then τ → μ is a type. We will use the following notation: if μ, τ1, τ2, ..., τn are types then by τ1 → τ2 → ... → τn → μ we mean the type τ1 → (τ2 → ... → (τn → μ)...). Therefore, every type τ has the form τ1 → ... → τn → O. By a τ k → μ we mean the type τ... → τ → O with k occurrences of type τ (with τ 0 → μ = μ). If τ has the form τ1 → ... → τn → O, then τi for i ≤ n are called components of type τ and are denoted by τ [i]. For any type τ we define rank(τ) and arg(τ) as follows: arg(O) = 0, rank(O) = 1 and arg(τ1 → ... → τn → O) = n and rank(τ1 → ... → τn → O) = maxi=1...n rank(τ [i]) + 1. We define inductively types τ [i1, ..., ik] by (τ [i1, ..., ik−1])[ik] for 1 ≤ ik ≤ arg(τ [i1, ..., ik−1]). 1 Supported by the State Committee for Scientific Research in Poland (KBN), research grant 7T11C 022 21 2 Email: [email protected] This is a preliminary version. The final version will be published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.nl/locate/entcs