80 research outputs found
Convergence and quantale-enriched categories
Generalising Nachbin's theory of "topology and order", in this paper we
continue the study of quantale-enriched categories equipped with a compact
Hausdorff topology. We compare these -categorical compact
Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that
the presence of a compact Hausdorff topology guarantees Cauchy completeness and
(suitably defined) codirected completeness of the underlying quantale enriched
category
The Convergence of Filters on Quantales and Its Hausdorffness
In this paper, we introduce the definition of conergence of filters on quantale. Some characterizations of finit completeness and compactness of quantales are studied. At last, the Hausdorff property in quantale using the converence structure is presented.key words: Quantale; Point; ideal; Congerencen of filter; Hausdorff Propert
An Algebraic Framework for Compositional Program Analysis
The purpose of a program analysis is to compute an abstract meaning for a
program which approximates its dynamic behaviour. A compositional program
analysis accomplishes this task with a divide-and-conquer strategy: the meaning
of a program is computed by dividing it into sub-programs, computing their
meaning, and then combining the results. Compositional program analyses are
desirable because they can yield scalable (and easily parallelizable) program
analyses.
This paper presents algebraic framework for designing, implementing, and
proving the correctness of compositional program analyses. A program analysis
in our framework defined by an algebraic structure equipped with sequencing,
choice, and iteration operations. From the analysis design perspective, a
particularly interesting consequence of this is that the meaning of a loop is
computed by applying the iteration operator to the loop body. This style of
compositional loop analysis can yield interesting ways of computing loop
invariants that cannot be defined iteratively. We identify a class of
algorithms, the so-called path-expression algorithms [Tarjan1981,Scholz2007],
which can be used to efficiently implement analyses in our framework. Lastly,
we develop a theory for proving the correctness of an analysis by establishing
an approximation relationship between an algebra defining a concrete semantics
and an algebra defining an analysis.Comment: 15 page
Approximation in quantale-enriched categories
Our work is a fundamental study of the notion of approximation in
V-categories and in (U,V)-categories, for a quantale V and the ultrafilter
monad U. We introduce auxiliary, approximating and Scott-continuous
distributors, the way-below distributor, and continuity of V- and
(U,V)-categories. We fully characterize continuous V-categories (resp.
(U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in
the same ways as continuous domains are characterized among all dcpos. By
varying the choice of the quantale V and the notion of ideals, and by further
allowing the ultrafilter monad to act on the quantale, we obtain a flexible
theory of continuity that applies to partial orders and to metric and
topological spaces. We demonstrate on examples that our theory unifies some
major approaches to quantitative domain theory.Comment: 17 page
Enriched lower separation axioms and the principle of enriched continuous extension
[EN] This paper presents a version of the lower separation axioms and the principle of enriched continuous extension for quantale-enriched topological spaces. As a remarkable result, among other things, we point out that in the case of commutative Girard quantales the principle of continuous extension holds for projective modules in Sup.The authors acknowledge support from the Basque Government (grant IT1483-22). The first named author also acknowledges support from a postdoctoral fellowship of the Basque Government (grant POS-2022-1-0015)
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