Recursively defined metric spaces without contraction

  • Authors:

  • Franck van Breugel;Claudio Hermida;Michael Makkai;James Worrell

  • Affiliations:

  • York University, Department of Computer Science, 4700 Keele Street, Toronto, M3J 1P3, Canada;Instituto Superior Técnico, SQIG-IT and CLC, Av. Rovisco Pais, 1049-001, Lisbon, Portugal;McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, H3A 2K6, Canada;Oxford University, Computing Laboratory, Parks Road, Oxford, OX1 3QD, United Kingdom

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2007

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Abstract

In this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1-bounded complete metric spaces and nonexpansive functions. In contrast to previous approaches, we do not assume that the endofunctors are locally contractive, and our results do not depend on Banach's fixed-point theorem. Our approach is particularly suitable for constructing models of systems that feature quantitative data. For instance, using the Kantorovich metric on probability measures we construct a quantitative model for probabilistic transition systems. The metric in our model can reasonably be seen as measuring the behavioural distance between states of the system; it depends exclusively on the transition probabilities and not on an arbitrary discount factor.