The typed λ-calculus is not elementary recursive

Historically, the principal interest in the typed λ-calculus is in connection with Godel\´s functional ("Dialectica") interpretation\´of intuitionistic arithmetic. However, since the early sixties interest has shifted to a wide variety of applications in diverse branches of logic, algebra, and computer science. For example, in proof-theory, in constructive logic, in the theory of functionals, in cartesian closed categories, in automatic theorem proving, and in the semantics of natural languages. In almost all such applications there is a point at which one must ask, for closed terms t₁ and t₂, whether t₁ β-converts to t₂. We shall show that in general this question cannot be answered by a Turing machine in elementary time. We shall also investigate the computational complexity of related questions concerning the typed. λ-calculus (for example, the question of whether a given type contains a closed term).