# Optimal classification/Rypka/equations

## Equations

### Truth table size-related equations

These equations determine the truth table size from the highest value of logic and the number of characteristics in the set and in the group for the bounded class[1] of elements and denote the range of the truth table values for the set and the group.[2]

#### Group size

${\displaystyle Q=V^{C}}$, where:
• Q is the group size[3]
• V is the highest value of logic in the group,
• C is the highest number of characteristics in the group.

#### Set size

${\displaystyle R=V^{K}}$, where:
• R is the set size[4]
• V is the highest value of logic in the group,
• K is the highest number of characteristics in the set.

### Theoretical and Empirical Separatory Equations

#### Element-related equations

##### Maximum number of pairs of elements to separate

Maximum number of pairs of elements to separate refers to triangularization of the matrix to permit comparison of each element with every other element to determine the number of pairs that are separable or disjoint. Pairs are separable or disjoint whenever the logic values of the elements that make up a pair are different. In theory, therefore the maximum possible number of pairs that can be separated is determined by the following equation:[5]

${\displaystyle p_{max}={\frac {\left[{G(G-1)}\right]}{2}}}$, where:[6]
• pmax is the maximum number of pairs to separate, and
• G is the number of elements in the bounded class.
##### Order of elements

The elements are arranged in descending order according to their truth table value, i.e., the value calculated as the sum of each characteristic's logic state value times the highest value of logic raised to the power of the order of the characteristic.[7] The element truth table value allows elements to be sorted and identified as unique or equivalent.

${\displaystyle e_{i}=\sum _{j=0}^{C}\left[v_{i,j}V^{(C-j)}\right]}$, where:
• ei is the element truth table value in the group,
• V is the highest value of logic in the group,
• v is the value of logic of each characteristic in the group,
• j is the jth characteristic index, where:
j = 0..C and where:
• C is the number of characteristics in the group,
• i is the ith element index, where:
i = 0..G and where:
• G is the number of elements in the bounded class.

#### Characteristic-related equations

##### Theoretical separation
###### The general identification equation
${\displaystyle S_{j}={\frac {1-{V^{-j}}}{1-V^{-C}}}}$, where:[8]
• Sj is the theoretical separatory value per jth characteristic,
• C is the highest number of characteristics in the group,
• V is the highest value of logic in the group and
• j is the jth characteristic index in the target set, where:
j = 0..K and where:
• K is the number of characteristics in the target set.
###### Minimal number of characteristics to result in theoretical separation
${\displaystyle t_{min}={\frac {\log G}{\log V}}}$, where:[9]
• tmin is the minimal number of characteristics to result in theoretical separation,
• G is the number of elements in the in the bounded class and
• V is the highest value of logic in the group.
##### Empirical separation
###### Target set truth table values

${\displaystyle t_{i}=\sum _{j=0}^{K}v_{i,j}V^{(K-j)}}$, where:[10]

• vi,j is the element's attribute value,
• i is the ith element's index value, where,
i = 0...G' where G is the number of elements in the bounded class, and,
• j is the jth characteristic's index value, where,
j = 0...K and where,
• K is the number of characteristics in the target set,
• V is highest value of logic in the group,
• V(K-j) is the positional value of the jth characteristic.

${\displaystyle n_{t_{i}}=n_{t_{i}}+1}$, where,

nti contains the multiset count for each truth table value[11].
###### Initial separation

${\displaystyle S_{j}={\frac {\left[(G^{2})-\sum _{l=0}^{R}n_{l}^{2}\right]}{2}}}$, where:[12]

• Sj is the initial empirical separatory value for each characteristic, where,
j = 0...C and is the index of the jth characteristic in the group and C is the number of characteristics in the group, and,
l = 0...R and is the truth table value of the jth characteristic, where R is the truth table size, where:
R = V0, and,
• V is the highest value of logic in the group and,
• 0 is the target set exponent for a single characteristic, and,
• G is the number of elements in the bounded class.
###### Subsequent separation

${\displaystyle S={\frac {\left[(G^{2})-\sum _{l=0}^{R}n_{l}^{2}\right]}{2}}}$, where:

• Sj is the initial empirical separatory value for each characteristic, where,
l = 0...R and is the target set truth table index value, where R is the target set truth table size value, where:
R = VK, and,
• V is the highest value of logic in the group and,
• K is the number of characteristics in the target set, and,
• G is the number of elements in the bounded class.

## Notes

1. See page 158, Table II of the primary reference
2. See page 172 of the primary reference.
3. group - Columns C1 to C5 (0 to C, C=5)
4. Target set - Columns C1 to C3 (0 to K, K=3)
5. See page 176 Table XI of the primary reference.
6. Triangular Number
7. See Number Systems - essentially using the characteristic values to compute a network or memory address, followed by sorting.
8. See page 153, page 167, Fig. 2. & page 175 of the primary reference
9. See page 157, Primary Schemes footnote of the primary reference
10. As the characteristic with the greatest separatory value is moved to the next most significant position, K is incremented to expand the target set size from two characteristics to the number of characteristics in the group or the the number of characteristics which result in 100% separation. For the initial target set with one characteristic the separatory value is computed individually for each characteristic in the group to find the initial characteristic with the highest separatory value.
11. coefficient of association, ${\displaystyle coa={\frac {n_{t_{i}}}{n_{R}}}}$, see page 172 of the primary reference
12. Must be applied initially to each characteristic. (The equations have been implemented in w:Mathcad and using the Visual Basic programming language. See Application Example - Flag Recognition