SDR: SDRInheritance for matching, and its truth value

In https://github.com/patham9/ANSNA/wiki/Cycle:-The-operating-cycle it was mentioned that an event can match to multiple concepts with a truth value "penalty" according to the match. Here are the details behind that:

Let S and P be a SDR. We want that S can be said to be a special case of P, or can stand for P, denoted by S --> P, if most of the bits in P also occur in S, but not necessarily vice versa. So S=SDRSet(black,cat) should be a special case of P=SDRSet(cat). It has most the features of cat, but also has the blackness feature, meaning a black cat can effectively stand for, or be treated as a cat too.

Following this intuition, a truth value of S --> P can be established as follows: Let's define each 1-bit in the SDR to be a NAL-sentence with frequency 1 and confidence c_default. Each of these bits, at position i, in S, basically encode: <S --> (/,_,1_at,{i})>., or written differently: S_i = 1

By the definition of truth value, we know that a case of positive evidence for S --> P, is, that something is in the intension of S and in the intension of P. This can be a certain term related to bit 5 being a 1-bit, like (/,_,1_at,{5}) is. We also know, that a case of negative evidence would be an element in the intension of P, that is not in the intension of S. Given that, and knowing that each bit-related statement has the same confidence and frequency 1 or 0, we can define the set of positive evidence as: E_p:={i | P_i=S_i=1} and the set of negative evidence as: E_n:={i | S_i=1 and P_i=0}. With E_total:=union(E_p,E_s), the truth value of S --> P can now be defined as a tuple T_match=(f_match,c_match) with f_match:=|E_p|/|E_total| and c_match:=|E_total|/(|E_total|+1).

If the event E has truth value T_E, to apply the penalty of "treating it as concept C", the truth value becomes SDR_Inheritance(S,P):=deduction(T_match,T_E), which will then be used in the inference rule within the concept for deriving further tasks.

That is motived by that if event E is a special case of the pattern it is encoded by, SDRE, and SDRE is a special case of SDRC, as the match determined, so we have E --> SDRE with truth value T_E and SDRE --> SDRC with truth value T_match. Using the deduction rule as specified in \cite{NAL}, we end up with E --> SDRC, allowing to treat the event as if it would have the SDR SDRC.