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PLAN OF CA PREREQUISITES
ca1 introduction to propositional calculus
ca2 introduction to predicate logic
caa 2D exact propositional calculus
cab ND exact propositional calculus
cac implication
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Site Plan
EXACT ND PROPOSITIONAL CALCULUS (PC)
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. For practical applications
5 is small. We may 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
Inference systems using PC are in general network structures.
Each node is an Assertion. A node may be considered as an
aggregate related top-down to several parts and as a part
related bottom-up to several aggregates. A syndrom is an
aggregate of its symptoms and a part of a disease. Relation
Aggregate-part is "many-to-many": a syndrom may have several
symptoms, a symptom may belong to several syndroms. Dimension
of a node is the number of its parts.
