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

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Abstract

For a categoryK of data types, solutions of recursive data-type equationsXT(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.

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Dedicated to Dieter Pumplün on the occasion of his 60th birthday

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Adamek, J., Reiterman, J. Banach's Fixed-Point Theorem as a base for data-type equations. Appl Categor Struct 2, 77–90 (1994). https://doi.org/10.1007/BF00878504

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Mathematics Subject Classifications (1991)

  • 68Q65
  • 18D20

Key words

  • Data type equation
  • complete metric space
  • Banach's theorem
  • categories enriched over metric spaces