## Solutions of Generalized Recursive Metric-Space Equations

Citations: | 1 - 1 self |

### BibTeX

@MISC{Birkedal_solutionsof,

author = {Lars Birkedal and Kristian Støvring and Jacob Thamsborg},

title = {Solutions of Generalized Recursive Metric-Space Equations},

year = {}

}

### OpenURL

### Abstract

It is well known that one can use an adaptation of the inverse-limit construction to solve recursive equations in the category of complete ultrametric spaces. We show that this construction generalizes to a large class of categories with metric-space structure on each set of morphisms: the exact nature of the objects is less important. In particular, the construction immediately applies to categories where the objects are ultrametric spaces with ‘extra structure’, and where the morphisms preserve this extra structure. The generalization is inspired by classical domain-theoretic work by Smyth and Plotkin. Our primary motivation for solving generalized recursive metric-space equations comes from recent and ongoing work on Kripke-style models in which the sets of worlds must be recursively defined. For many of the categories we consider, there is a natural subcategory in which each set of morphisms is required to be a compact metric space. Our setting allows for a proof that such a subcategory always inherits solutions of recursive equations from the full category. As another application, we present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. 1

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Citation Context ... present a construction that relates solutions of generalized domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. 1 Introduction Smyth and Plotkin =-=[17]-=- showed that in the classical inverse-limit construction of solutions to recursive domain equations, what matters is not that the objects of the category under consideration are domains, but that the ... |

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Citation Context ...ntractive mixed-variance functor G on D such that a fixed point of G (necessarily unique, by Theorem 3.1) is the same as a fixed point of F that furthermore satisfies a ‘minimal invariance’ condition =-=[12]-=-. Thus, generalized domain equations can be solved in M-categories. The construction generalizes an earlier one [6] which is for the particular category of pointed cpos and strict, continuous function... |

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Citation Context ...s, but that the sets of morphisms between objects are domains. In this work we show that, in the case of ultrametric spaces, the standard construction of solutions to recursive metric-space equations =-=[5, 10]-=- can be similarly generalized to a large class of categories with metric-space structure on each set of morphisms. The generalization in particular allows one to solve recursive equations in categorie... |

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Citation Context ...om a particular instance of an M-category obtained from this construction. The extra metric information in that category (as compared with the underlying O-category) is useful in realizability models =-=[1, 4]-=-. An O-category [17] is a category C where each hom-set C (A,B) is equipped with an ω-complete partial order, usually written ⊑, and where each composition function is continuous with respect to these... |

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Citation Context ...om a particular instance of an M-category obtained from this construction. The extra metric information in that category (as compared with the underlying O-category) is useful in realizability models =-=[1, 4]-=-. An O-category [17] is a category C where each hom-set C (A,B) is equipped with an ω-complete partial order, usually written ⊑, and where each composition function is continuous with respect to these... |

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Citation Context ...ts of V instead of sets, and where the ‘composition functions’ are morphisms in V . Other related work. The idea of considering categories with metric spaces as hom-sets has been used in earlier work =-=[9, 14]-=-. Rutten and Turi [14] show existence and uniqueness of fixed points in a particular category of (not necessarily ultrametric) metric spaces, but with a proof where parts are more general. In other wo... |

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Citation Context ... for maximal generality, but rather for a level of generality that seems right for our applications [8]. 2 Ultrametric spaces We first recall some basic definitions and properties about metric spaces =-=[13, 16]-=-. A metric space (X,d) is 1-bounded if d(x,y) ≤ 1 for all x and y in X. We shall only work with 1-bounded metric spaces. One advantage of doing so is that one can define coproducts and general product... |

29 | Nested Hoare triples and frame rules for higher-order store
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Citation Context ...e the morphisms preserve this additional structure. Our main motivation for solving equations in such categories comes from recent and ongoing work in denotational semantics by the authors and others =-=[7, 15]-=-. There, solutions to such equations are used in order to construct Kripke models over recursively defined worlds: a novel approach that allows one to give semantic models of predicates and relations ... |

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Citation Context ...metric spaces, i.e., recursive equations whose solutions cannot necessarily be described as fixed-points of functors. In contrast, we only consider functorial recursive equations in this work. Wagner =-=[18]-=- gives a comprehensive account of a generalized inverse limit construction that in particular works for categories of metric spaces and categories of domains. Another such construction has recently be... |

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Citation Context ...e the morphisms preserve this additional structure. Our main motivation for solving equations in such categories comes from recent and ongoing work in denotational semantics by the authors and others =-=[7, 15]-=-. There, solutions to such equations are used in order to construct Kripke models over recursively defined worlds: a novel approach that allows one to give semantic models of predicates and relations ... |

19 | The category-theoretic solution of recursive metric-space equations
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(Show Context)
Citation Context ...ver recursively defined worlds: a novel approach that allows one to give semantic models of predicates and relations over languages with dynamically allocated, higher-order store. See Birkedal et al. =-=[8]-=- for examples of such applications. For many of the categories we consider, there is a natural variant, indeed a subcategory, in which each set of morphisms is required to be a compact metric space [2... |

7 | Solutions of functorial and non-functorial metric domain equations
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(Show Context)
Citation Context ...nce of fixed points for a general class of ‘metric-enriched’ categories (as in our Theorem 3.1), nor a general theorem about fixed points in locally compact subcategories (Theorem 4.1.) Alessi et al. =-=[3]-=- consider solutions to non-functorial recursive equations in certain categories of metric spaces, i.e., recursive equations whose solutions cannot necessarily be described as fixed-points of functors.... |

7 | Solving domain equations in a category of compact metric spaces
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Citation Context ...8] for examples of such applications. For many of the categories we consider, there is a natural variant, indeed a subcategory, in which each set of morphisms is required to be a compact metric space =-=[2, 9]-=-. Our setting allows for a general proof that such a subcategory inherits solutions of recursive equations from the full category. Otherwise put, the problem of solving recursive equations in such a ‘... |

6 |
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Citation Context ...d domain equations in the sense of Smyth and Plotkin to solutions of equations in our class of categories. This construction generalizes and improves an earlier one due to Baier and Majster-Cederbaum =-=[6]-=-. The key to achieving the right level of generality in the results lies in inspiration from enriched category theory. We shall not refer to general enriched category theory below, but rather present ... |

3 |
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(Show Context)
Citation Context ...8] for examples of such applications. For many of the categories we consider, there is a natural variant, indeed a subcategory, in which each set of morphisms is required to be a compact metric space =-=[2, 9]-=-. Our setting allows for a general proof that such a subcategory inherits solutions of recursive equations from the full category. Otherwise put, the problem of solving recursive equations in such a ‘... |

1 |
On the influence of domain theory
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(Show Context)
Citation Context ...eneralized inverse limit construction that in particular works for categories of metric spaces and categories of domains. Another such construction has recently been given by Kostanek and Waszkiewicz =-=[11]-=-. Our generalization is in a different direction, namely to categories where the hom-sets are metric spaces. We do not know whether there is a common generalization of our work and Wagner’s work; in t... |