Soggy Predicates

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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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M_F} , in which an entity Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} has Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x)} copies, where this number of copies is then proportional to the degree to which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} has property Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} . If we assume the degrees of membership of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} are of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k \over N} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x) = m} means Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F(x)} has degree Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle m \over N} .

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} is Soggy, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x} has been observed Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M} times, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p} of these observations have fully manifested property Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} , and the other Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle M-p} observations have manifested Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle F} not at all. Then in the multiset interpretation, we can set

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x) = N * {p \over M} }

More generally we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n(F,x) = N * { \sum_{i=1}^M d(F,x_i) \over M } }

where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i} is the i’th observation of x, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle d(F,x_i)} is the degree to which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle x_i} manifests Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 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