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Exact ND Propositional Calculus (PC)
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We have seen that for n=2, the 2-dimensional operators work on 2 operands (p,q)
and that we have 2**(2**n)=16 operators:
.......................1.1.1.1.1.1.1
p.q..1.2.3.4.5.6.7.8.9.0.1.2.3.4.5.6
1.1..0.1.1.1.0.0.0.1.1.1.0.0.0.1.0.1
1.0..1.0.1.1.0.1.1.0.0.1.0.0.1.0.0.1
0.1..1.1.0.1.1.0.1.0.1.0.0.1.0.0.0.1
0.0..1.1.1.0.1.1.0.1.0.0.1.0.0.0.0.1
The situation is simple, we know the 16 operators by heart, like the
multiplication table and with a bit of practice can execute and program
all operations of the 2d-PC from memory.
For n>2 PC becomes much more complex. Let's start with n=3 and 3 operands p,q,r:
p.q.r
1.1.1..01111111
1.1.0..10111111
1.0.1..11011111
1.0.0..11101111
0.1.1..11110111
0.1.0..11111011
0.0.1..11111101
0.0.0..11111111.etc.
We have 2**(2**3)=256 operators. Number of operators increases very fast with n.
For n=4 we have 2**(2**4)=65536 and for n=5 2**(2**5)=2**32=4294967296 operators.
And for practical applications 5 is small. We can have 20 symptoms of a disease
or 100 "symptoms" of some breakdown in a jet plane. The respective diagnostic
expert systems would extend over 2**(2**20) and 2**(2**100) operators. A bit to
much to learn by heart, to describe in a textbook, or, for that matter, in the
whole Congress Library. We have to look for some other procedures.
Let's come back to n=3.
Some operators map from n=2 to n=3 as one to one, ex. "and", "or":
p.q.r....and..or
1.1.1....1....1
1.1.0....0....1
1.0.1....0....1
1.0.0....0....1
0.1.1....0....1
0.1.0....0....1
0.0.1....0....1
0.0.0....0....0
For any n they may be evaluated: "and" as product of all operands' certainties
"or" as their maxof value.
However, for n=3 "orr" forks to 3 distinct operators "one-of", "two-of" and
"not-all":
p.q.r....one-of.two-of.not-all
1.1.1....0......0......0
1.1.0....0......1......1
1.0.1....0......1......1
1.0.0....1......0......1
0.1.1....0......1......1
0.1.0....1......0......1
0.0.1....1......0......1
0.0.0....0......0......0
For n=20 "orr" will fork to 20 operators, from one-of to 19-of and not-all.
On this example we see that for higher n's only a few operators can be chosen
from endless lists in function of their utility for a particular problem.
As we have said before, the user has to tailor his logic to his problem by
choosing pertinent operators and designing their evaluation algorithms.
Evaluation algorithms for some operators may become a bit complex even in the
Exact PC. They become really difficult in the Fuzzy.
DIMENSIONS AND CHAINING.
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Inferencing systems using PC are in general network structures. Each node is
a statement. A node may be an aggregate related to several parts and, in turn,
a part related to several aggregates. A syndrom is aggregate with respect to
its symptoms and part with respect to one or more diseases. Aggregate-part
relation is many-to-many: a syndrom may have several symptoms, a symptom may
belong to several syndroms.
By dimension of a node we understand the number of its parts. The lowest level
nodes the "pure parts" having no parts themselves have their certainties set
factually: by a doctor observing symptoms, by security sensors in the circuits
of a plane, etc.
Then in the so called "backward chaining" certainties of aggregates are
recursively evaluated in function of those of their parts. Parts are considered
as operands and operators are determined by the relations part-aggregate.
We can see two rather difficult problems:
1.One cannot talk about dimensionality of the whole inference system, but of
each of its nodes.
2.Each part may be related to aggregate via different operator. Some may be
considered as necessary (and), some as optional (or), some as exclusive (one-of),
etc. Expressions containing several subsets of parts each subset related via
different operators, may be rather difficult to evaluate.
That's about all we will say for the moment about Exact n-d PC. It starts
looking a bit complex, but its complexity and difficulty are childish compared
with those of the Fuzzy, which is the one important for practice.
Therefore, please shoot questions, so that we may clear all details before
going to the real and exciting stuff of the Fuzzy n-d PC.
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