What Are Iteration Theories?

We prove that iteration theories can be introduced as algebras for the monad Rat on the category of signatures assigning to every signature Σ the rational-Σ-tree signature. This supports the result that iteration theories axiomatize precisely the equational properties of least fixed points in domain theory: Rat is the monad of free rational theories and every free rational theory has a continuous completion.

[1]  Naoki Kobayashi,et al.  Resource Usage Analysis for the p-Calculus , 2006, Log. Methods Comput. Sci..

[2]  Zoltán Ésik.  Axiomatizing Iteration Categories , 1999, Acta Cybern..

[3]  G. M. Kelly,et al.  Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads , 1993 .

[4]  Gordon D. Plotkin,et al.  Complete axioms for categorical fixed-point operators , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[5]  Zoltán Ésik,et al.  Fixed-Point Operations on ccc's. Part I , 1996, Theor. Comput. Sci..

[6]  Stefan Milius,et al.  Elgot Algebras † ( Extended Abstract ) , 2006 .

[7]  Zoltán Ésik,et al.  Iteration Theories , 1993, EATCS Monographs on Theoretical Computer Science.

[8]  Joseph A. Goguen,et al.  Rational algebraic theories and fixed-point solutions , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

[9]  Susanna Ginali,et al.  Regular Trees and the Free Iterative Theory , 1979, J. Comput. Syst. Sci..

[10]  Zoltán Ésik,et al.  The Equational Logic of Fixed Points (Tutorial) , 1997, Theor. Comput. Sci..

[11]  Jean Benabou,et al.  Structures algébriques dans les catégories , 1968 .