Re: Algebraic sum

ca314159@bestweb.net
Sun, 7 Jun 1998 01:26:31 +0200 (MET DST)

In article <1998060120310700.QAA20182@ladder01.news.aol.com>,
wsiler@aol.com (WSiler) wrote:
>
> The term "algebraic sum" is a buzz word to denote the probabilistic OR, not
> actually the sum A+B, and the formula
>
> A OR B = A + B - A*B
>
> is correct. If this operator is used for ORing quantities, the the
> corresponding AND operator is the probabilistic AND,
>
> A AND B = A * B
>
> so that the probabilistic OR may be expressed as
>
> A OR B = A + B - (A AND B)
>
> and are valid if the operands are statistically indepedent, that is have
zero
> correlation.
>
> These formulas have disadvantages in a fuzzy expert system. As the number of
> operands increases, their OR tends to increase toward a, and their AND
tends
to
> decrease toward zero, limiting the number of operands which can practically
be
> used in fuzzy rules.
>
> William Siler
>
>

http://www.cs.princeton.edu/~ristad/java/overlap/index.html

shows the entropy problem in terms of the three valued relation:

A + B + C - (A AND B) - (A AND C) - (B AND C) + (A AND B AND C)

which suggests a possible means of counting primes using Brun's
approximation and the min-max arrangement for "AND" outlined in the
"Fuzzy Intersection" thread in news:comp.ai.fuzzy

See:

http://www.bestweb.net/~ca314159/BRUN.HTM

What's also interesting is the meaning of the "AND" in terms of
vector algebra as the inner product (A,B) of two vectors A, and B.

(A AND B) in this case is then a measure of orthogonality (correlation)
between these vectors:

(A AND B) =|A| |B| cos(theta) = a1b1 + a2b2

where theta is some angle between the two vectors and a1 and a2 are
perpendicular components of A. Same for B.

>From which Kosko derives the orthogonality relation for the Pythagorean
equation based on (A AND A) = a^2 = a1^2 + a2^2 and we derive the similarly
relations for the complex modulus etc.

The quantum mechanical relation (given in generality):

(A OR B) = A + B + (A AND B)

is used to determine the intensity pattern associated
with Young's two-slit experiment for the case of
like polarizations, but here (A AND B) is allowed to
take on negative values. By suitable translation of the
"interference term" from the physicsally derived -1 to 1
range mapped into the 0 to -1 range the usual relation
can be maintained. (In physics, the invariance of a "relation"
under scaling and translation is of prime importance, not the
actual values which are dependant on some arbitrary zero-point
associated with an arbitrary coordinate system)

"Orthogonality" is also taken in the abstract sense to mean
the degree of correlation, as you state, since two orthogonal
"states" are distinguishable (completely uncorrelated) while
if the vectors representing those states are parallel, we
say that the corresponding states are perfectly correlated.
For some intermediate value of the inner (dot,scalar) product,
the states are said to be partially correlated. So the degree of
orthogonality is equivalent in this sense to a degree of correlation.

The interesting thing is, that in a complex space (unitary) the
inner product is non-communitive, and for wavefunctions, it appears
to be "context sensitive" in its application. For instance, when you
take the inner product of two waves in terms of their amplitudes,
you generally mean the correlation is in relation to the phases.
In terms of a complex inner product (a non-communitive fuzzy "A AND B").
Yet you can also express the correlation relative to the polarization
vectors which is real-valued and due to the underlying symmetry of
polarization the range is from 0 to 90 degrees for the angles and
not from -180 to 180 as in the case of the phases.

It might be interesting to explore the meaning of this to the fuzzy
paradigm, because it is of no little significance to physics, mathematics
and especially to general applied mathematics.

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