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Functions over free algebras definable in the simply typed lambda calculus. (English) Zbl 0792.03006
Summary: We show that a function over a free algebra is definable in the simply typed $$\lambda$$-calculus (modulo the Böhm-Berarducci embedding) iff it is generated by predicative monotonic recurrence. Monotonic recurrence here is the generalization of iteration-with-parameters from $$\mathbb{N}$$ to arbitrary free algebras, and our predicativity condition uses the notion of tiers introduced by the author [Feasible mathematics, Proc. Math. Sci. Inst. Workshop, Ithaca/NY 1989, Prog. Comput. Sci. Appl. Log. 9, 281-291 (1990; Zbl 0772.03020)]. In fact, we show that the same functions are generated by tiered monotonic recurrence whether 2 tiers or all finite tiers are used.

##### MSC:
 03B40 Combinatory logic and lambda calculus
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##### References:
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