Questions on languages

Let D(⋅, ⋅) be a "distance" operator on phonemes such that phonemes with smaller "distances" sound more similar than phonemes with smaller "distances". (We assume for simplicity that such operators exist and that they can be defined up to isomorphism in a mostly language-independent manner; it is acceptable that they define merely a semimetric rather than a proper metric on any given set of phonemes. If these assumptions are too strong, then for at least the interesting questions below, phonemes may be tagged with their originating language, and D can then be defined strictly in terms of the perception of speakers of the language from which its first argument is drawn.)

Let ℳ be the set of functions from the phonemes of a language a to the phonemes of a language b. Let any M∈ℳ such that ∀x∈a, y∈b: y = M(x) → ¬∃z∈a: D(x,z) < D(x,y) (that is, any M∈ℳ such that M(x) selects a D-closest point in b to x) be called a natural mapping from a to b — a function notionally applied by a speaker of b to obtain familiar phonemes from b from potentially unfamiliar phonemes in a.

We shall write a ⊑ b if there exists an injective natural mapping from a to b; we may then read a ⊑ b as "b phonetically subsumes a" or somesuch. For example, Māori ⊑ Japanese, and Japanese ⊑ French.

Let E be the set of all human languages. What does the graph (E,⊑) look like? (Specifically, what is its graph density?) How strong is the correlation between mutual intelligibility and local symmetry of ⊑? Are there more sources or sinks than one would expect solely from the graph density? Is ⊑ transitive? (Mathematically, it need not be.)

Most relevantly, the question that prompted the definition of ⊑: given languages a, b, and c such that a ⊑ b and a ⋢ c, will a native speaker of a generally notice a qualitative difference between the accent of an accented speaker of b and an accented speaker of c? And is the average intelligibility of speakers of c vs. those of c' related to the degree to which those languages' relevant natural mappings from a fail to be injective?