Theoretical Computer Science
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Theoretical Computer Science
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Theoretical Computer Science
From Iterative Algebras to Iterative Theories
Electronic Notes in Theoretical Computer Science (ENTCS)
Electronic Notes in Theoretical Computer Science (ENTCS)
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Electronic Notes in Theoretical Computer Science (ENTCS)
The Recursion Scheme from the Cofree Recursive Comonad
Electronic Notes in Theoretical Computer Science (ENTCS)
Elgot theories: A new perspective on the equational properties of iteration
Mathematical Structures in Computer Science
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Mathematical Structures in Computer Science
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Electronic Notes in Theoretical Computer Science (ENTCS)
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Completely iterative theories of Calvin Elgot formalize (potentially infinite) computations as solutions of recursive equations. One of the main results of Elgot and his coauthors is that infinite trees form a free completely iterative theory. Their algebraic proof of this result is extremely complicated. We present completely iterative algebras as a new approach to the description of free completely iterative theories. Examples of completely iterative algebras include algebras on complete metric spaces. It is shown that a functor admits an initial completely iterative algebra iff it has a final coalgebra. The monad given by free completely iterative algebras is proved to be the free completely iterative monad on the given endofunctor. This simplifies substantially all previous descriptions of these monads. Moreover, the new approach is much more general than the classical one of Elgot et al. A necessary and sufficient condition for the existence of a free completely iterative monad is proved.