Abstract
It is known that a quasimetric space can be represented by means of a metric space; the points of the former space become closed subsets of the latter one, and the role of the quasimetric is assumed by the Hausdorff quasidistance. In this paper, we show that, in a slightly more special context, a sharpened version of this representation theorem holds. Namely, we assume a quasimetric to fulfil separability in the original sense due to Wilson. Then any quasimetric space can be represented by means of a metric space such that distinct points are assigned disjoint closed subsets.This result is tailored to the solution of an open problem from the area of approximate reasoning. Following the lines of E. Ruspini’s work, the Logic of Approximate Entailment (mathsf {LAE}) is based on a graded version of the classical entailment relation. We present a proof calculus for mathsf {LAE} and show its completeness with regard to finite theories.
Highlights
A quasimetric is defined to a metric; the assumption of symmetry, is dropped
We focus here on the so-called Logic of Approximate Entailment, LAE for short, which was introduced by Rodríguez in his PhD thesis (2002)
We apply this result in order to show that the calculus LAE, which is known to be complete for the logic LAEq based on quasimetric spaces, is complete for LAE, which is based on metric spaces
Summary
A quasimetric is defined to a metric; the assumption of symmetry, is dropped. This generalisation of the concept of a distance naturally occurs in many real-world situations. An often cited example is the time that a walker needs to get from one place to another one within a mountainous area. Quasimetric spaces are closely related to weighted directed graphs. The latter play a significant role in Communicated by A.
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