From Osher Doctorow
David Grynkiewicz and Oriol Serra, respectively NSF postdoctoral
Fellow USA and Catalan Research Council Grant recipient Spain), in
arXiv: 0710.3127 math.CO, math.NT, prove theorems such as the
following where /A/ is the cardinality of set A:
1) /A + B/ > = (2 - 1/s)(/A/ + /B/) - 2s + 1
provided that / /A/ - /B/ / < = s for s > = 2 integer and if /A/ + /B/
= 4s^2 -6s + 3, A and B being finite sets in R^2 which are non-
empty, where h1(A, B) > = s for h1(A, B) defined as:
2) h1(A, B) = minimum number of t such that there exist 2t parallel
but not necessarily distinct lines l1, ..., lt, l1', ..., lt' such
that A is a subset of U(li) and B is a subset of U(li ' ) where U is
the union for i = 1 through t.
The set A + B, called Minkowski sum or just sumset, is the set of
elements a + b such that a is an element of A and b is an element of
B, and the theorem is part of Inverse Additive Theory.
Although the above authors don't discuss this, it is rather easy to
convert the above theorem and other related ones that they prove to
PI. Simply assume that all cardinalities are normalized into the
interval [0, 1]. Then since P(A-->B) = 1 + y - x for y < = x and x, y
probabilities and the same for P ' (A-->B), the two expressions
differing in that in the first x = P(A) and y = P(AB) while in the
second x = P(A) and y = P(B) < = P(A), we could write (1) as:
3) /A + B/ > = P((2s --> (2 - 1/s)(/A/ + /B/))
but we should divide (2 - 1/s) by a normalization constant C which can
be taken as some very large integer, for example 10^1000. We will
understand C to be included in (3) here, or alternatively we can
assume s to be mapped into (-infinity, -2) since then 2s^2 - 2s + 1 >
0 always (if it equalled 0, the solution would be a non-real complex
number).
Notice that according to the discussion preceding (3), the upper bound
-s (s being mapped into (-infinity, -2) in (3) is exerting a Causal
influence on (2 - 1/s)(/A/ + /B/), but the condition that makes -s an
upper bound involves /A/ - /B/. So the difference in cardinalities
between these two sets A and B (for large /s/ for example) exerts
Probable Causation on /A/ + /B/, the Probable Causation however being
bounded by /A + B/ (normalized of course).
I should comment that the difference in the cardinalities is more
Causal (in the sense of being the Cause) than its sum (in the sense of
being the Effect), while the cardinality of the sumset turns out to be
an upper bound on the Probable Causation/Influence (PI) itself. This
agrees with the general result that differences as in 1 - x + y are
more fundamental in PI than sums as for example 1 + x + y. The
interpretation of the cardinality of the sumset in terms of PI as an
upper bound is unexpected, however, and may indicate a deeper
connection between normalized cardinality and PI as well as a deeper
connection of the sumset and PI.
Osher Doctorow
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