SimilarityLink as ContextLink

[Nil Geisweiller]

Hi,

I've noticed another equivalence

SimilarityLink <TV>
A
B

==

ContextLink <TV>
A OR B
A AND B

1) by def

$$(SimilarityLink A B).TV = \frac{(A \cap B).TV}{(A \cup B).TV}$$

2)

ContextLink <TV>
A OR B
A AND B

==

SubSet <TV>
A OR B
A AND B

let be

f_A the function corresponding to the fuzzy set of A,
f_B the function of B.
f_c the function of (A OR B) with obviously f_c = max(f_A,f_B),
f_d the function of (A AND B) with obviously f_d = min(f_A,f_B),

$$TV = \frac{\sum_x x.TV \times min(f_c(x),f_d(x))} {\sum_x x.TV
\times f_c(x)}$$

$$TV = \frac{\sum_x x.TV \times
min(max(f_A(x),f_B(x)),min(f_A(x),f_B(x)))} {\sum_x x.TV \times
max(f_A(x),f_B(x))}$$

$$TV = \frac{\sum_x x.TV \times min(f_A(x),f_B(x))} {\sum_x x.TV
\times max(f_A(x),f_B(x))}$$

Although it's perhaps not a requirement of the PLN theory (or perhaps
it is?) I think it's fair to assume that

$$(A \cap B).TV = \sum_x x.TV \times min(f_A(x),f_B(x))$$

and

$$(A \cup B).TV = \sum_x x.TV \times max(f_A(x),f_B(x))$$

think of the TV of some concept C as the average intensity at which
pattern C occurs...

So to sum up the similarity between A and B can be seen as the TV of
(A AND B) in the context of (A OR B). Kinda obvious but I didn't see
it before!

Nil

[Nil Geisweiller]

Oh, I made a little mistake, we're obviously talking about
ExtSimilarityLink, not SimilarityLink.

Nil

[Ben Goertzel]

Seems clear, yep... ;)

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--
Ben Goertzel, PhD
CEO, Novamente LLC and Biomind LLC
CTO, Genescient Corp
Chairman, Humanity+
Adjunct Professor of Cognitive Science, Xiamen University, China
Advisor, Singularity University and Singularity Institute
b...@goertzel.org

"My humanity is a constant self-overcoming" -- Friedrich Nietzsche

[Nil Geisweiller]

> Although it's perhaps not a requirement of the PLN theory (or perhaps
> it is?) I think it's fair to assume that
>
> $$(A \cap B).TV = \sum_x x.TV \times min(f_A(x),f_B(x))$$
>
> and
>
> $$(A \cup B).TV = \sum_x x.TV \times max(f_A(x),f_B(x))$$

oh well, of course it's a PLN "requirement", fairly obvious (albeit a
missing normalizing factor) using

A <TV>

==

Subset <TV>
Universe
A

...

Nil