Iteration theories: the equational logic of iterative processes
Iteration theories: the equational logic of iterative processes
Terms and Infinite Trees as Monads Over a Signature
TAPSOFT '89/CAAP '89 Proceedings of the International Joint Conference on Theory and Practice of Software Development, Volume 1: Advanced Seminar on Foundations of Innovative Software Development I and Colloquium on Trees in Algebra and 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
Terminal coalgebras and free iterative theories
Information and Computation
Mathematical Structures in Computer Science
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
From Coalgebraic to Monoidal Traces
Electronic Notes in Theoretical Computer Science (ENTCS)
Equational properties of iterative monads
Information and Computation
Elgot theories: A new perspective on the equational properties of iteration
Mathematical Structures in Computer Science
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The concept of iteration theory of Bloom and Esik summarizes all equational properties that iteration has in usual applications, e.g., in Domain Theory where to every system of recursive equations the least solution is assigned. However, this assignment in Domain Theory is also functorial. Yet, functoriality is not included in the definition of iteration theory. Pity: functorial iteration theories have a particularly simple axiomatization, and most of examples of iteration theories are functorial. The reason for excluding functoriality was the view that this property cannot be called equational. This is true from the perspective of the category Sgn of signatures as the base category: whereas iteration theories are monadic (thus, equationally presentable) over Sgn, functorial iteration theories are not. In the present paper we propose to change the perspective and work, in lieu of Sgn, in the category of sets in context (the presheaf category of finite sets and functions). We prove that Elgot theories, which is our name for functorial iteration theories, are monadic over sets in context. Shortly: from the new perspective functoriality is equational.