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 LOGIC WORKSHOP 
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Inferencing structure is a network consisting of nodes-aggregates pointing 
downwards to nodes-parts. Parts point upwards to their aggregates. The structure 
is recursive: a node may be at the same time aggregate and part, pointing 
respectively to downward and upward nodes. An engine is part of car and aggregate 
of block, cylinder head, pistons, etc. This aggregate-part recursivity stops 
upwards at 'top' nodes, pure aggregates, having parts, but no aggregates and 
downwards at 'bottom' nodes, pure parts, belonging to aggregates, but having 
no parts.
In scientific models the top nodes represent axioms, the middle nodes - theorems 
and the bottom nodes - factual information reducible to observations.
Top-down display, or explosion of the axioms represents deduction, or conceptual 
structure of the model.
Bottom-up display, or implosion of factual nodes represents induction, which 
confirms, or refutes theorems and axioms of the model.
Each node is associated with a variable 'certainty' taking continuously values 
between 0 and 100. Relations between nodes contain fuzzy operators allowing to 
calculate the certainty of aggregates in function of that of parts, while 
executing the inductive, bottom-up implosion.
We may use the bottom-up inferencing to test a model. In this case we are sure 
that the factual bottom nodes are reliable and exact and we induce the certainty 
of theorems and axioms. High certainty confirms the theory. Low certainty of a 
theorem refutes it and the model has to be corrected correspondingly.
 Low certainty of an axiom refutes the axiom and the model which has to be 
scrapped and replaced by a new one. 
Another application of inductive inferencing consists in using the model, 
which we believe to be reliable, to establish essential and complex causes 
behind the apparently arbitrary and meaningless factual data.  High certainty 
of certain aggregates designates them as causes of factual data. Low certainty 
of all aggregates indicates lack of precision of the factual data. We shall 
illustrate this second type of inductive inferencing with a simple example 
of virtual diagnostic.

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EXAMPLE.

Let us imagine a top aggregate 'diagnostics' pointing to three 'diagnosis':

diagnostics 
 diagnosis_1_3 oof 
 diagnosis_2_4 oof
 diagnosis_3_4 oof

'oof' means that the three diagnosis parts are related to the aggregate 
diagnostics via fuzzy operator oof (one of). It means that the three diagnosis 
are mutually exclusive, that only one can be confirmed by getting a high 
certainty.

Each diagnosis becomes in turn aggregate, pointing downwards to parts-syndroms: 

diagnosis_1_3
 syndrom_1 and
 syndrom_3 and
diagnosis_2_4
 syndrom_2 and
 syndrom_4 and
diagnosis_3_4
 syndrom_3 and
 syndrom_4 and

Syndroms are 'necessary' parts of their respective diagnosis, i.e. are related 
to them via fuzzy operators 'and'.

Syndroms in turn point to symptoms via 'and':

syndrom_1
 symptom_a and
 symptom_c and
 symptom_f and
 
syndrom_2
 symptom_b and
 symptom_c and
 symptom_d and
 
syndrom_3
 symptom_b and
 symptom_g and
 symptom_h and
 
syndrom_4
 symptom_a and
 symptom_g and
 symptom_e and
 
The whole model structure is:                                                                                                                                              
1 diagnostics first
 2 diagnosis_1_3 oof mid
  3 syndrom_1 and mid
   4 symptom_a and last
   4 symptom_c and last
   4 symptom_f and last 
  3 syndrom_3 and mid
   4 symptom_b and last
   4 symptom_g and last
   4 symptom_h and last
 2 diagnosis_2_4 oof mid
  3 syndrom_2 and mid
   4 symptom_b and last 
   4 symptom_c and last 
   4 symptom_d and last
  3 syndrom_4 and mid
   4 symptom_a and last 
   4 symptom_g and last
   4 symptom_e and last
 2 diagnosis_3_4 oof mid
  3 syndrom_3 and mid 
   4 symptom_b and last
   4 symptom_g and last
   4 symptom_h and last
  3 syndrom_4 and mid 
   4 symptom_a and last 
   4 symptom_g and last
   4 symptom_e and last

'first' indicates the top, pure aggregate, 'last' - the bottom, pure part, 
'mid' - the middle part-aggregate nodes.
                                   
Let's now attribute certainty to factual bottom nodes, in our case to symptoms. 
It simulates a doctor observing symptoms and attributing his estimate of 
precision and exactitude of these observations.

   4 symptom_a and 98 last
   4 symptom_c and 96 last
   4 symptom_f and 97 last 
   4 symptom_b and 95 last
   4 symptom_g and 98 last
   4 symptom_h and 94 last
   4 symptom_d and 25 last
   4 symptom_e and 18 last

The structure looks now:

1 diagnostics first
 2 diagnosis_1_3 oof mid
  3 syndrom_1 and mid
   4 symptom_a and 98 last
   4 symptom_c and 96 last
   4 symptom_f and 97 last 
  3 syndrom_3 and mid
   4 symptom_b and 95 last
   4 symptom_g and 98 last
   4 symptom_h and 94 last
 2 diagnosis_2_4 oof mid
  3 syndrom_2 and mid
   4 symptom_b and 95 last 
   4 symptom_c and 96 last 
   4 symptom_d and 25 last
  3 syndrom_4 and mid
   4 symptom_a and 98 last 
   4 symptom_g and 98 last
   4 symptom_e and 18 last
 2 diagnosis_3_4 oof mid
  3 syndrom_3 and mid 
   4 symptom_b and 95 last
   4 symptom_g and 98 last
   4 symptom_h and 94 last
  3 syndrom_4 and mid 
   4 symptom_a and 98 last 
   4 symptom_g and 98 last
   4 symptom_e and 18 last

And after executing the bottom-up inductive inferencing:

1 diagnostics 59 first
 2 diagnosis_1_3 oof 80 mid
  3 syndrom_1 and 93 mid
   4 symptom_a and 98 last
   4 symptom_c and 96 last
   4 symptom_f and 97 last 
  3 syndrom_3 and 89 mid
   4 symptom_b and 95 last
   4 symptom_g and 98 last
   4 symptom_h and 94 last
 2 diagnosis_2_4 oof 1 mid
  3 syndrom_2 and 18 mid
   4 symptom_b and 95 last 
   4 symptom_c and 96 last 
   4 symptom_d and 25 last
  3 syndrom_4 and 12 mid
   4 symptom_a and 98 last 
   4 symptom_g and 98 last
   4 symptom_e and 18 last
 2 diagnosis_3_4 oof 6 mid
  3 syndrom_3 and 89 mid 
   4 symptom_b and 95 last
   4 symptom_g and 98 last
   4 symptom_h and 94 last
  3 syndrom_4 and 12 mid 
   4 symptom_a and 98 last 
   4 symptom_g and 98 last
   4 symptom_e and 18 last

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We conclude that diagnosis_1_3 is the most certain with certainty 80 against 
1 and 6 of the others.
The whole diagnostics procedure gets the certainty 59, which is the measure 
of its reliability, quite good for practical applications.
If, nevertheless, we find it not reliable enough, we shall have to repeat 
observations of symptoms with improved precision.

1 diagnostics 59 first
 2 diagnosis_1_3 oof 80 mid
 2 diagnosis_2_4 oof 1 mid
 2 diagnosis_3_4 oof 6 mid

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PARENTS:
 LOGIC WORKSHOP 
 WORKSHOPS 
 ARCHIVE