Define SemiMetric and Metric to be traits rather than types

[matthieugomez]

The package Distances groups distances by their mathematical properties (symmetry and triangular inequality), via the type hierarchy Metric <: SemiMetric <: PreMetric.

This type hierarchy is very limiting. In certain contexts, it makes more sense to group distances by their object of application (for instance string distances), or by their general algorithm (for instance, q-gram distances are a certain type of string distances that operate on qgrams). In StringDistances, I would love to be able to define certain functions on these groups. This is not possible, however, due to the lack of multiple inheritance in Julia.

For this reason, I'd prefer SemiMetric/Metric to be traits, rather than types. This would keep the implementation of pairwise/colwise efficient, while making it easier for external packages to define their own abstract types.

[nalimilan]

I'm afraid this would create dispatch issues, notably if we want pairwise to be supported in StatsBase for generic functions (JuliaStats/StatsBase.jl#627), and specialized for distances in Distances. Or we would need to make Metric and friends internal types. But then what would you use traits for since you can't even dispatch on them?

Cc: @dkarrasch

[dkarrasch]

It's hard for me to estimate the impact of this proposal without seeing a draft. In some sense, I think the proposal is consistent with @nalimilan's pairwise implementation in StatsBase.jl. There you have a symmetric keyword, which could be set by a trait symmetric = issymmetric(dist), and in fact this is how I anticipated how one would map the generic pairwises here to those in StatsBase. Currently, one would do that by setting issymmetic(::SemiMetric) = true, but without a type SemiMetric, one would do that one by one, similarly to how we handle parameters in UnionMetrics. Also, if I understand @matthieugomez correctly, then he suggests to replace only Metric and SemiMetric by traits, not PreMetric, so there would be still something that dispatches to Distances.jl. Did I understand that correctly?