Corecursive Algebras, Corecursive Monads and Bloom Monads

An algebra is called corecursive if from every coalgebra a unique coalgebra-to-algebra homomorphism exists into it. We prove that free corecursive algebras are obtained as coproducts of the terminal coalgebra (considered as an algebra) and free algebras. The monad of free corecursive algebras is proved to be the free corecursive monad, where the concept of corecursive monad is a generalization of Elgot's iterative monads, analogous to corecursive algebras generalizing completely iterative algebras. We also characterize the Eilenberg-Moore algebras for the free corecursive monad and call them Bloom algebras

Structured general corecursion and coinductive graphs [extended abstract]

Bove and Capretta's popular method for justifying function definitions by general recursive equations is based on the observation that any structured general recursion equation defines an inductive subset of the intended domain (the "domain of definedness") for which the equation has a unique solution. To accept the definition, it is hence enough to prove that this subset contains the whole intended domain. This approach works very well for "terminating" definitions. But it fails to account for "productive" definitions, such as typical definitions of stream-valued functions. We argue that such definitions can be treated in a similar spirit, proceeding from a different unique solvability criterion. Any structured recursive equation defines a coinductive relation between the intended domain and intended codomain (the "coinductive graph"). This relation in turn determines a subset of the intended domain and a quotient of the intended codomain with the property that the equation is uniquely solved for the subset and quotient. The equation is therefore guaranteed to have a unique solution for the intended domain and intended codomain whenever the subset is the full set and the quotient is by equality.Comment: In Proceedings FICS 2012, arXiv:1202.317

Inductive and Coinductive Components of Corecursive Functions in Coq

In Constructive Type Theory, recursive and corecursive definitions are subject to syntactic restrictions which guarantee termination for recursive functions and productivity for corecursive functions. However, many terminating and productive functions do not pass the syntactic tests. Bove proposed in her thesis an elegant reformulation of the method of accessibility predicates that widens the range of terminative recursive functions formalisable in Constructive Type Theory. In this paper, we pursue the same goal for productive corecursive functions. Notably, our method of formalisation of coinductive definitions of productive functions in Coq requires not only the use of ad-hoc predicates, but also a systematic algorithm that separates the inductive and coinductive parts of functions.Comment: Dans Coalgebraic Methods in Computer Science (2008

Language Constructs for Non-Well-Founded Computation

Recursive functions defined on a coalgebraic datatype C may not converge if there are cycles in the input, that is, if the input object is not well-founded. Even so, there is often a useful solution; for example, the free variables of an infinitary λ-term, or the expected running time of a finite-state probabilistic protocol. Theoretical models of recursion schemes have been well studied

The Sierpinski Carpet as a Final Coalgebra

We advance the program of connections between final coalgebras as sources of circularity in mathematics and fractal sets of real numbers. In particular, we are interested in the Sierpinski carpet, taking it as a fractal subset of the unit square. We construct a category of square sets and an endofunctor on it which corresponds to the operation of gluing copies of a square set along segments. We show that the initial algebra and final coalgebra exist for our functor, and that the final coalgebra is bi-Lipschitz equivalent to the Sierpinski carpet. Along the way, we make connections to topics such as the iterative construction of initial algebras as colimits, corecursive algebras, and the classic treatment of fractal sets due to Hutchinson.Comment: In Proceedings ACT 2021, arXiv:2211.0110