Reducing recursion to iteration by means of pairs and N-tuples

Abstract

(i) The family $\mathbb{D}$of data systems , considered as heterogeneous term- (anarchic-) algebras with a finite number of supports and constructors, has the property that the family $\mathbb{I}$of iterative functions mapping algebras into algebras does not change if, inside its definition, the primitive recursion scheme replaces the iteration scheme [Böhm 1986]. The same idea is developed here, for other recursion schemes, neither iterative nor primitive recursive, like the simultaneous (mutual) iteration schemes. Illustrations of such schemes are algorithms or functions more or less familiar in computer science, e.g. functions defined by some limited while schemes.

(ii) Since recursion scheme can also be viewed as a rewriting (computation) scheme it makes sense to ask for special representing algebras with the property that the computational complexity of some functions, like the inverses of the constructors, be reasonably comparable with that one of the corresponding computer instructions.

This research has been supported by Grants of the Ministry of Public Instruction, Italy.