Lambda Deenability Is Decidable for Second Order Types and for Regular Third Order Types

It has been proved by Loader 1] that Statman-Plotkin conjecture (see 4] and 2]) 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 deenability problem limited for second order types and regular third order types is decidable in any nite domain. Obviously deenability is decidable for 0 and 1 order types. As an additional eeect of the result described we may observe that for certain types there is no nite grammar generating all closed terms. 1. Syntax of simple typed calculus We shall consider a simple typed lambda calculus with a single ground type O. The set TY PES is deened as follows: O is a type and if and are types then ! is a type. We will use the following notation: if ; 1; 2; :::;n are types then are denoted by i]. For any type we deene rank() and arg() as follows: arg(O) = rank(O) = 0 and arg(1 ! ::: ! n ! O) = n and rank(1 ! ::: ! n ! O) = maxi=1:::n rank(i]) + 1. We deene inductively types i1; :::; ik] by (i1; :::;ik?1]))ik] for 1 ik arg(i1; :::;ik?1]). Deenition 1.1. Type is called regular if rank() 3 and every component of has arg 1. This implies that only components allowed for regular types are O, For any type there is given a denumerable set of variables V (). Any type variable is a type term. If T is a term having type ! and S is a type term , then TS is a term which has type. If T is a type term and x is a type variable , then x:T is a term having type !. The axioms of equality between terms have the form of conversions and the convertible terms will be written as T = S. Term T is in a long normal form if T = x1:::xn:yT1:::Tk, where y is an xi for some i n or y is a free variable, Tj for j k are in a long normal form and yT1...Tk is a type O term. Long normal forms exist and are unique for conversions (compare 3]). A closed term is a term without free variables. By Cl() we mean a set of all closed type …