Soggy Predicates

Warning: I suggest to either directly use the subset notation (see below) or introduce a SoggyLink to turn any predicate into a soggy predicate where the domain of that soggy predicate is and only the powerset of the domain the given predicate. Automatically extending the domain of a predicate to its powerset leads to bad foundations.

A Simple Observation Grounded predicate, or Soggy predicate, is an uncertain predicate F so that: For each x, the number F(x) lies in [0,1] and can be interpreted as the average degree to which an arbitrary element of some set O of observations has property F. (Here we assume that the degree to which a specific observation has a property F is itself a number in [0,1])

(Note that the set of observations O need not be the observations actually made by the AI system whose memory contains the predicate — it may be observations made by some other hypothesized entities, etc.)

In this case we have a clear interpretation for

EvaluationLink <s>
   PredicateNode F
   Atom x

So we can say

PredicateNode F <t>

means the average degree to which an arbitrary element y of the AI system’s “default set of observations” satisfies F (i.e. the average over this default observation-set of F(y))…

Or, for an observation-set C, we can say

ContextLink <t1>
   ConceptNode C
   PredicateNode F

where t1 is the average over F(y), where y is counted in the average with a weight proportional to the degree to which y is in C.

So then

PredicateNode F <t>

means, conceptually,

ContextLink <t>
   >default context<
   PredicateNode F

Next, we can then define

EvaluationLink <s>
   PredicateNode F
   Atom x

as being equivalent to

MemberLink <s>
   Atom x
   SatisfyingSet
      PredicateNode F

Basically, this is just defining the membership function of the fuzzy set

   SatisfyingSet
      PredicateNode F

in a particular way.

We can then convert this to

Subset <s>
   Atom x
   SatisfyingSet
      PredicateNode F

because of the way F was originally defined.

This becomes a bit subtle to interpret in the case that the argument of F is a list.

For example,

EvaluationLink <s>
   PredicateNode “eat”
   ListLink
      ConceptNode “cat”
      ConceptNode “mouse”

converts to

SubsetLink <s>
  ListLink
      ConceptNode “cat”
      ConceptNode “mouse”
  SatisfyingSet
      ConceptNode “eat”

which means that s is the average degree to which an observation (in the default overall observation-set) involving “eat”, also involves the pair (eater = cat, eatee=mouse).

Interpretation in terms of Multisets

One way to phrase the above interpretation is to introduce the multiset interpretation of fuzzy membership degrees from

https://pdfs.semanticscholar.org/c18f/ba07ba1dbb99700391df30f6ca8798e5df9f.pdf

and to associate F with a multiset $M_{F}$, in which an entity $x$ has $n(F,x)$ copies, where this number of copies is then proportional to the degree to which $x$ has property $F$. If we assume the degrees of membership of $F$ are of the form $k \over N$, then $n(F,x)=m$ means $F(x)$ has degree $m \over N$.

Suppose $F$ is Soggy, and $x$ has been observed $M$ times, and $p$ of these observations have fully manifested property $F$, and the other $M-p$ observations have manifested $F$ not at all. Then in the multiset interpretation, we can set

$n(F,x)=N*{p \over M}$

More generally we have

$n(F,x)=N*{\sum _{i=1}^{M}d(F,x_{i}) \over M}$

where $x_{i}$ is the i’th observation of x, and $d(F,x_{i})$ is the degree to which $x_{i}$ manifests $F$.

This multiset interpretation has the advantage of

• presenting fuzzy degrees as probabilities (across multisets rather than ordinary sets
• providing a formal justification for the use of fuzzy set (min/max) algebra