# On the expressive strength of the finitely typed lambda — terms

## Abstract

In the setting of the language of finitely typed λ-terms with if-then else and fixpoints we investigate the question: In which respect do higher types bear on the expressive strength of programming languages?

We restrict attention to the set of closed λ-terms of first-order type, the set of programs. (Terms of first-order type have type i →...→ i → i, i for individuals, they have subterms of arbitrary types.) The set of programs can be naturally classified into an infinite syntactic hierarchy: A program is in the n'th level of this hierarchy, i. e. a level-n-program, if n is an upper bound on the functional level of its subterms.

Using a novel diagonalization technique over a class of finite interpretations, such that the set of cardinalities of the interpretations of this class has no finite upper bound, we show: Level-(n+1)-programs define more functions (in the sense of the theory of program schemes) than level-n-programs. Using reductions to already established hierarchies KfTiUr 87 shows: Level-(n+2)-programs define more functions than level-n-programs.

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