Unifying structured recursion schemes

Abstract

Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many dis-parate generalisations have been proposed that enjoy similar cal-culational benefits. There have also been attempts to unify the various generalisations: two prominent such unifications are the 're-cursion schemes from comonads' of Uustalu, Vene and Pardo, and our own 'adjoint folds'. Until now, these two unified schemes have appeared incompatible. We show that this appearance is illusory: in fact, adjoint folds subsume recursion schemes from comonads. The proof of this claim involves standard constructions in category theory that are nevertheless not well known in functional programming: Eilenberg-Moore categories and bialgebras.

Classical (co)recursion: Mechanics

“…Consequently, corecursion tends to be relegated to a notion of coalgebras (Rutten, 2019), because only the language of category theory speaks clearly enough about their duality. This can be seen in the study of corecursion schemes, where coalgebraic "anamorphisms" (Meijer et al, 1991) and "apomorphisms" (Vos, 1995;Vene & Uustalu, 1998) are the dual counterparts to algebraic "catamorphisms" (Meertens, 1987;Meijer et al, 1991;Hinze et al, 2013) and "paramorphisms" (Meertens, 1992). Yet the logical and computational status of corecursion is not so clear.…”