Non Homomorphic Reductions of Data Structures

In this paper we study diierent kinds of reductions of data types. By reduction we mean applying the higher order function fold to a data structure. An appropriate fold function can be deened for any recursive data type. These reductions have been presented as homomorphisms by several authors 2, 6]. Although many useful functions on data structures can be programmed as instances of fold, there are some that cannot. This is due to the fact that they are not mathematical homomorphisms. We show some examples of these functions. Then, we introduce two generalizations of fold (one for lists and the other for binary trees) in terms of which many non homomorphic mappings can be deened. Some examples are presented. A second problem addressed in the paper is the relationship between the deenitions of some particular reductions in diierent data types. We show that the deenition of a particular reduction, e.g. to insert an element in a data structure, in terms of fold (either the generalized version or the usual one), looks the same for diierent data structures. In many cases, one can be obtained by transforming the other. We deene a hierarchy of data types according to the amount of \structural" information they possess. It is shown how some reductions of a data structure A with more structural information than another one B can be obtained by composing the homomorphism from A to B that \forgets" structural information, with the homomorphism reducing B.