Structured recursion for non-uniform data-types

Recursion is a simple but powerful programming technique used extensively in functional languages. For a number of reasons, however, it is often desirable to use structured forms of recursion in programming, encapsulated in so-called recursion operators, in preference to unrestricted recursion. In particular, using recursion operators has proved invaluable for transforming and reasoning about functional programs. The recursion operators map and fold are well-understood for uniform datatypes, and recent work on applications of non-uniform data-types has motivated extending these recursion operators to non-uniform data-types as well. Previous work on defining recursion operators for non-uniform data-types has been hampered by expressibility problems caused by parametric polymorphism. In this thesis, we show how these problems can be solved by taking a new approach, based upon the idea of “algebra families”. We show how these recursion operators can be realized in the functional language Haskell, and present a categorical semantics that supports formal reasoning about programs on non-uniform data-types.