The SubsetLink denotes extensional inheritance between two Atoms. An example of a SubsetLink would be:

Subset dog animal


## PLN Semantics

In PLN the truth value of a SubsetLink relationship corresponds to a conditional probability over concepts, formally:

SubsetLink <TV>
A
B


corresponds to the conditional probability ${\displaystyle P(B|A)}$ with truth value estimate:

${\displaystyle TV.strength={\frac {\sum _{x}f_{\wedge }(({\text{Member}}\ x\ B).s,({\text{Member}}\ x\ A).s)}{\sum _{x}({\text{Member}}\ x\ A).s}}}$
${\displaystyle TV.count=|A|}$

where (Member x y).s is the degree to which x is a member of concept y, which can also be represented in the AtomSpace via a MemberLink. In case the truth values of such MemberLinks have confidences smaller than 1, then, due to being sums of random variables, convolution products must be used. In practice that isn't currently done due to its computational cost, even though it's likely that for some sub-cases, such as when the confidence is identical across all member links, an accurate estimate should be cost effective.

It should also be noted that such formula only works under the assumption that all members are equally weighted, that is the mass function over their singletons is uniform. Formally

${\displaystyle \exists m\forall x\forall A\ ({\text{Subset}}\ A\ ({\text{Set}}\ x)).s=m}$

If such assumption does not hold, or no knowledge of members is available, then a more general formula can be applied. Let ${\displaystyle {\mathcal {P}}}$ be a partition of A, then the truth value of such subset is calculated as follows:

${\displaystyle TV.strength=\sum _{X\in {\mathcal {P}}}({\text{Subset}}\ A\ ({\text{And}}\ B\ X)).s}$

A formula for its confidence remains to be determined, although the size of the partition could be used as an approximation of its count.