IntensionalSimilarityLink

A type of Link used to describe intensional similarity between concepts.

PLN Semantics

In PLN the IntensionalSimilarityLink between 2 concepts corresponds to the extensional similarity of their patterns (supersets, accounting for their simplicity). Formally

IntensionalSimilarityLink <TV>
   A
   B

is equivalent to

 ExtensionalSimilarityLink <TV>
    SatisfyingSetLink
       LambdaLink
          $Z
          EvaluationLink
             GroundedPredicateNode "pattern-of"
             ListLink
                $Z
                X
    SatisfyingSetLink
       LambdaLink
          $Z
          EvaluationLink
             GroundedPredicateNode "pattern-of"
             ListLink
                $Z
                Y

where the pattern-of predicate is a GroundedPredicateNode defined so that

$\operatorname {pattern-of} (Z,X)=s(Z)\times (P(Z|X)-P(Z|\neg X))^{+}$

where s(Z) is the prior of Z, reflecting it's simplicity, that is the simpler Z and the stronger it's discriminating power over X, the more it is a pattern of X. For the discriminating power of Z over X we also say that X is attracted to Z, that we can represent with the following link

 AttractionLink
   X
   Z

where the strength of the TV of such attraction link is $(P(Z|X)-P(Z|\neg X))^{+}$. Note that $(x)^{+}$ is the positive part of $x$.

See FormulaPredicateLink for an example of specifying formulas in Atomese.

Properties

Although IntensionalSimilarity is different than ExtensionalSimilarity they have properties in common. For instance

IntensionalSimilarityLink <0, 1>
  A
  NotLink
    A

holds for any A (even fuzzy). That is because given any $Z$, if

$(P(Z|\neg A)<P(Z|A)$

that is Z is a pattern of A, then it entails that

$(P(Z|\neg \neg A)>P(Z|\neg A)$

that is $Z$ is not a pattern of $\neg A$, thus $A$ and $\neg A$ have no pattern in common.

Interestingly though it should be noted that

ExtensionalSimilarityLink <0, 1>
  A
  NotLink
    A

only holds if A is not a fuzzy set, so in that regard extensional similarity is actually more permissive than intensional similarity.