Banach's Fixed-Point Theorem as a base for data-type equations

For a categoryK of data types, solutions of recursive data-type equationsX ℞T(X), whereT is an endofunctor ofK, can be constructed by iteratingT on the unique arrowT1 → 1. This is well-known forK enriched over complete posets and forT locally continuous (an application of the Kleene Fixed-Point Theorem). We prove this forK enriched over complete metric spaces and forT contracting (an application of the Banach Fixed-Point Theorem). Moreover, we prove that each such recursive equation has a unique solution. Our results generalize the approach of P. America and J. Rutten.

[1]  J. W. de Bakker,et al.  Processes and the Denotational Semantics of Concurrency , 1982, Inf. Control..

[2]  Michael A. Arbib,et al.  Algebraic Approaches to Program Semantics , 1986, Texts and Monographs in Computer Science.

[3]  Gordon D. Plotkin,et al.  The category-theoretic solution of recursive domain equations , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[4]  Maurice Nivat,et al.  Metric Interpretations of Infinite Trees and Semantics of non Deterministic Recursive Programs , 1980, Theor. Comput. Sci..

[5]  Pierre America,et al.  Solving Reflexive Domain Equations in a Category of Complete Metric Spaces , 1989, J. Comput. Syst. Sci..

[6]  Jiří Adámek,et al.  Free algebras and automata realizations in the language of categories , 1974 .

[7]  Mitchell Wand.  Fixed-Point Constructions in Order-Enriched Categories , 1979, Theor. Comput. Sci..

[8]  Jirí Adámek.  Recursive Data Types in Algebraically omega-Complete Categories , 1995, Inf. Comput..

[9]  Daniel J. Lehmann,et al.  Data types , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).