In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

      INFERENCING STRUCTURE      



      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 
      top nodes represent axioms, 
      middle nodes - theorems, 
      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 assume that the factual bottom nodes have high certainty 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, once we 
      believe it 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.

      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

      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