Independence of the equational axioms for iteration theories
Journal of Computer and System Sciences
Iteration theories: the equational logic of iterative processes
Iteration theories: the equational logic of iterative processes
Theoretical Computer Science
Models of Sharing Graphs: A Categorical Semantics of Let and Letrec
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Uncertain Programming
Recursion and corecursion have the same equational logic
Theoretical Computer Science - Category theory and computer science
Abstract Syntax and Variable Binding
LICS '99 Proceedings of the 14th Annual IEEE Symposium on Logic in Computer Science
Complete Axioms for Categorical Fixed-Point Operators
LICS '00 Proceedings of the 15th Annual IEEE Symposium on Logic in Computer Science
Infinite trees and completely iterative theories: a coalgebraic view
Theoretical Computer Science
Terminal coalgebras and free iterative theories
Information and Computation
Mathematical Structures in Computer Science
Elgot Theories: A New Perspective of Iteration Theories (Extended Abstract)
Electronic Notes in Theoretical Computer Science (ENTCS)
An Algebra for Kripke Polynomial Coalgebras
LICS '09 Proceedings of the 2009 24th Annual IEEE Symposium on Logic In Computer Science
Completely iterative algebras and completely iterative monads
Information and Computation
Semantics of higher-order recursion schemes
CALCO'09 Proceedings of the 3rd international conference on Algebra and coalgebra in computer science
Iterative reflections of monads
Mathematical Structures in Computer Science
Equational properties of iterative monads
Information and Computation
A Sound and Complete Calculus for Finite Stream Circuits
LICS '10 Proceedings of the 2010 25th Annual IEEE Symposium on Logic in Computer Science
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
From corecursive algebras to corecursive monads
CALCO'11 Proceedings of the 4th international conference on Algebra and coalgebra in computer science
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Bloom and ??sik's concept of iteration theory summarises all equational properties that iteration has in common applications, for example, in domain theory, where to every system of recursive equations, the least solution is assigned. This paper shows that in the coalgebraic approach to iteration, the more appropriate concept is that of a functorial iteration theory (called Elgot theory). These theories have a particularly simple axiomatisation, and all well-known examples of iteration theories are functorial. Elgot theories are proved to be monadic over the category of sets in context (or, more generally, the category of finitary endofunctors of a locally finitely presentable category). This demonstrates that functoriality is an equational property from the perspective of sets in context. In contrast, Bloom and ??sik worked in the base category of signatures rather than sets in context, and there iteration theories are monadic but Elgot theories are not. This explains why functoriality was not included in the definition of iteration theories.