Corecursion Research Articles

Formal Languages, Formally and Coinductively

Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs of classic results in language theory. In this article, we study this representation in the Isabelle proof assistant. We define regular operations on infinite tries and prove the axioms of Kleene algebra for those operations. Thereby, we exercise corecursion and coinduction and confirm the coinductive view being profitable in formalizations, as it improves over the set-of-words view with respect to proof automation.

Primitive (Co)Recursion and Course-of-Value (Co)Iteration, Categorically

In the mainstream categorical approach to typed (total) functional programming, datatypes are modelled as initial algebras and codatatypes as terminal coalgebras. The basic func- tion definition schemes of iteration and coiteration are modelled by constructions known as cata- morphisms and anamorphisms. Primitive recursion has been captured by a construction called paramorphisms. We draw attention to the dual construction of apomorphisms, and show on ex- amples that primitive corecursion is a useful function definition scheme. We also put forward and study two novel constructions, viz., histomorphisms and futumorphisms, that capture the powerful schemes of course-of-value iteration and its dual, respectively, and argue that even these are helpful.