Disjointed sum of products by a novel technique of orthogonalizing ORing

: This work presents a novel combining method called 'orthogonalizing ORing ∨ (cid:102) ' which enables the building of the union of two conjunctions whereby the result consists of disjointed conjunctions. The advantage of this novel technique is that the results are already presented in an orthogonal form which has a significant advantage for further calculations as the Boolean Differential Calculus. By orthogonalizing ORing two calculation steps - building the disjunction and the subsequent orthogonalization of two conjunctions - are performed in one step. Postulates, axioms and rules for this linking technique are also defined which have to be considered getting correct results. Additionally, a novel equation, based on orthogonalizing ORing, is set up for orthogonalization of every Boolean function of disjunctive form. Thus, disjointed Sum of Products can be easily calculated in a mathematical way by this equation.

There are four standard forms f S (x) of switching function [1,6,19] which are either connected by conjunctions c i (x) = n j= x j = x · . . . · x n− · xn or by disjunctions d i (x) = n j= x j = x ∨ . . . ∨ x n− ∨ xn [3]. Conjunctions are connected by disjunctions in the disjunctive form DF (1) or by antivalence-operations in the antivalence form AF (3) and disjunctions are connected by conjunctions in the conjunctive form CF (2) or by equivalence-operations in the equivalence form EF (4). The connection of canonical conjunctions or disjunctions is named as normal form: disjunctive normal form DNF, antivalence normal form ANF, conjunctive normal form CNF and equivalence normal form ENF. Therefore, the orthogonal form of a DF can also be named as disjointed Sum of Products (dSOP) which is the set of products terms (conjunctions) whose disjunction equals f (x) in non-orthogonal form, such that no two product terms cover the same 1. Consequently, dSOP consist of non intersecting cubes. Deriving an orthogonal form of DF is a classical problem of Boolean theory. In this work, this problem is supported by a contribution of a novel solution.

Characteristic of orthogonality
The characteristic of orthogonality is a special attribute of a switching function [1,5,6,[15][16][17][18]. The orthogonal form of a switching function is characterized by conjunctions or disjunctions which are disjointed to one another in pairs. This means, that for each pair of conjunctions, one of them contains a direct Boolean variable (x i ) and the other contains the negation (x i ) of the same Boolean variable. Consequently, the intersection of each pair of these conjunctions (c i,j (x)) results in 0, as shown in Eq. (5). In contrast, the disjunction of each pair of orthogonal disjunctions (d i,j (x)) results in 1, as shown in Eq. (6).
De nition 2.1 (Orthogonality of Conjunctions or Disjunctions). Two conjunctions c i (x) and c j (x) are orthogonal to each other if the following applies: Two disjunctions d i (x) and d j (x) are orthogonal to each other if the following applies: The orthogonal form of a f S (x) enables its transformation in another form, which will have equivalent function values. This means, that the native form and the transformed form have the same function values if the same input values are used in each case. Therefore, orthogonalization simpli es the handling for further calculation steps, especially in the application of electrical engineering, e.g. for further calculation step as the Boolean Di erential Calculus (BDC) [2,3] by which all possible test patterns for a combinational circuit can be determined. Test patterns are used to detect feasible faults in combinational circuits. Additionally, it facilitates the calculation of BDC particularly in Ternary-Vector-List (TVL) arithmetic [11,12,[16][17][18]. TVL is a kind of matrix which simpli es the computational representation of Boolean functions and their computational handling of tasks in a facilitated way.  [1,3,10,14,15,21]: Proof of Theorem 2.2. By using (5) for orthogonal conjunctions c i,j (x) the relation in (7) applies. The respective proof is brought by the following relation. The disjunction of two conjunctions c i,j (x) on the right side is equivalent to the antivalence operation of the same two conjunctions on the right side. This is the procedure of reshaping of a disjunctive form in the antivalence form. For the case that both conjunctions are orthogonal the last term on the right side results in 0. An antivalence-operation with 0 is to be neglected, as x i ⊕ = x i applies. Consequently, this leads to the relation in (7).
Proof of Theorem 2.3. A CF can be represented as an EF by using the following condition in (10). By using (6) for orthogonal disjunctions d i,j (x) in this case the relation in (9) applies. The conjunction of two disjunctions d i,j (x), that means a CF, on the left side is equivalent to the right side which illustrates the equivalence operation of the same two disjunctions. If these both disjunctions d i,j (x) are orthogonal then the union of them results in 1, as shown by the last term on the right side. An equivalence-operation with 1 is to be neglected, as x i = x i applies. Consequently, the relation in (9) applies. (10)

Elementary operations of two conjunctions
In this section the elementary operations (intersection, union, di erence-building) of conjunctions, which are deduced out of the set theory due to the isomorphism, are de ned for the switching algebra. These formulas for di erent operations of conjunctions specify the order in which the variables of the given conjunction have to be linked. That means, if a variable is displayed negated, the corresponding literal of the given conjunction must be negated at this point. The number of variables in their respective conjunction is de ned by n or n ′ .
Theorem 3.2 (Union of two conjunctions). The union of any two conjunctions c i,j (x) with n, n ′ ∈ N is given by:

Theorem 3.3 (Di erence-building of two conjunctions). The di erence-building of a conjunction cm(x) as minuend and another conjunction cs(x) as subtrahend with n, n ′ ∈ N is calculated by following equation, which
is deduced out of the set theory [4,13]. That means, for the di erence of two sets M − S it applies M ∩S which is transferred to the switching algebra due to the isomorphism. Consequently, the di erence-building of two conjunctions is the intersection of the minuend and the complement of the subtrahend. By building the di erence several conjunctions arise which are not disjointed (orthogonal) to each other.

Orthogonalizing di erence-building
The technique of orthogonalizing di erence-building is used to calculate the orthogonal di erence of two conjunctions (c i,j (x)) whereby the result is orthogonal. This method is generally valid and equivalent to the usual method of di erence-building [6][7][8][9]. Orthogonalizing di erence-building is the composition of two calculation steps -the di erence-building and the subsequent orthogonalization.

De nition 4.1 (Orthogonalizing di erence-building). Orthogonalizing di erence-building corresponds to the removal of the intersection which is formed between the minuend conjunction cm(x) and the subtrahend conjunction cs(x) from the minuend cm(x), which means cm(x) − (cm(x) ∧ cs(x)); the result is orthogonal.
Orthogonalizing di erence-building of two conjunctions with n, n ′ ∈ N is de ned as: In this case, the formula ( [10] is used to describe the orthogonalizing di erence-building in a mathematically easier way, where x j , x j , . . . , xn j are literals of the subtrahend conjunction cs(x). This method is explained by the following example and description. Additionally, this example is also illustrated in a K-map ( Figure 1).
It is a result of several conjunctions ( st cube, nd cube, rd cube) which are orthogonal to each other and cover all of the remaining 1s.
-The rst literal of the subtrahend, here x , is complemented and builds the intersection with the minuend, here x . Consequently, the rst conjunction of the di erence isx x . -Then the second literal, here x , is complemented and forms the intersection with the minuend, and the rst literal x of the subtrahend is built. Therefore, the second conjunction is x x x . -The next literal, here x , is complemented and forms the intersection with the minuend, and the rst literal x and second literal x of the subtrahend is built. Thus, the third term of the di erence is x x x x . -This process is continued until all literals of the subtrahend are singly complemented and linked by building the intersection with the minuend in a separate conjunction.

Orthogonalizing ORing
By a further novel technique 'orthogonalizing ORing ∨ f ', which is based on the orthogonalizing di erencebuilding , the union of two conjunctions (c i,j (x)) can be calculated of two conjunctions (c i,j (x)) whereby the result is orthogonal.
De nition 5.1 (Orthogonalizing ORing). The intersection of a conjunction cs (x), called as the rst summand, and a second conjunction cs (x), called as the second summand, is removed by the method from the rst summand cs (x), and the second summand cs (x) is linked to that subtraction by a disjunction; the result is orthogonal. This procedure is labeled as orthogonalizing ORing and is de ned with n, n ′ ∈ N as: This method of orthogonalizing ORing is explained by the Example 5.2 which is also illustrated in a K-map (Fig. 2).
It is a result of several conjunctions or cubes which are disjointed to each other in pair (Fig. 2). The following points explain this unprecedented technique: -The rst literal of the second summand cs (x), here x , is complemented and ANDed to the rst summand cs (x). Consequently, the rst conjunction of the orthogonal result arises,x x .
-Then the second literal of the second summand cs (x) is complemented, here x , and ANDed with the rst literal of the second summand cs (x) to the rst summand cs (x). Therefore, the second conjunction of the orthogonal result is developed, x x x . -This process is continued until all literals of the second summand cs (x) are singly complemented and linked by ANDing to the rst summand cs (x) in a separate term. -At last the second summand cs (x) is added to the heretofore calculated conjunctions. By swapping the position of the summands, the result changes as shown in the following: However, both solutions are equivalent because the same 1s are covered. They only di er in the form of their coverage. But in order to represent an orthogonal result with a fewer number of conjunctions, the conjunction with more literals has to be accepted as the rst summand. That is possible because orthogonalizing ORing has the property of commutativity. By the mathematical induction the general validity of Eq. (15) is given in Proof 1. The number of conjunctions in the result, called nx, corresponds to the number of literals presented in the second summand cs (x) and not presented in the rst summand cs (x) at the same time; in addition, a 1 is added to nx, which stands for the second summand cs (x) as the last linked term. It applies: Furthermore, the number of possible results, which primarily depends on nx, can be charged by: Depending on the starting literal the result may di er. There are many equivalent options which only di er in the form of their coverage. This novel technique contains the composition of two calculation procedures -the union '∨' and the subsequent orthogonalization. The result out of orthogonalizing ORing is orthogonal in contrast to the result out of the usual method of union ∨. Both results are di erent in their representations but cover the same 1s. Hereinafter, the proof of this equivalence between orthogonalizing ORing and union is exempli ed. On the left side it is denoted the orthogonalizing ORing of two sets S , S and on the right side the union of the same sets. Due to the axiom of absorption, the equivalence between orthogonalizing ORing and ORing is veri ed. The right side is the orthogonal form of the left side, which are equivalent but di erent in their form of coverage. Proof.

. Orthogonalizing ORing between a DF and a conjunction
The orthogonalizing ORing of an orthogonal DF f (x) orth and a conjunction cs (x) can be reached by Eq. (19).
With ls ∈ N + as the number of conjunctions in the given function the following applies: The orthogonal form is generated: Next step is the application of Eq. (19). Each conjunction of f (x) orth is combined by orthogonalizing ORing with c (x). However, the adding of the second summand at last is ful lled after each combining step.
In the K-map on the left side the cubes of f (x) and c (x) are represented, and on the K-map on the right side the result after the procedure of orthogonalizing ORing is illustrated (Fig. 3). The following postulates are necessary for getting correct results after each operation of orthogonalizing ORing.
-If two conjunctions are already orthogonal to each other (cs (x) ⊥ cs (x)) the result corresponds to the disjunction of both conjunctions: -If the rst conjunction is the subset of the second conjunction (cs (x) ⊂ cs (x)) it follows: -and in the reverse case (cs (x) ⊂ cs (x)) it follows: .

. Axioms for variables
The following rules apply for the linking of variables and constants.
-The orthogonalizing ORing of a variable and constant 0 results in the variable itself (23) and the orthogonalizing ORing of a variable and constant 1 results in 1 (24). x x i ∨ f = . (24) -Furthermore, the orthogonalizing ORing of a variable with the same variable leads to the variable itself (25) and the orthogonalizing ORing of a variable with its negated form leads to the union of both (26). Consequently, this results in 1 at last

. . Axioms for conjunctions
Following axioms for conjunctions are deduced out of the axioms for variables.
-The neutral element of orthogonalizing ORing is 0: -The result of orthogonalizing ORing between 0 and a conjunction c i (x) is this conjunction c i (x) itself: -The orthogonalizing ORing of a conjunction c i (x) with the unit-term 1 leads to 1: -The result of orthogonalizing ORing between 1 and a conjunction c i (x) is 1: -The result of the orthogonalizing ORing between two the same conjunctions c i (x) results to this conjunction itself: -The orthogonalizing ORing of a conjunction c i (x) with its complement c i (x) results in an unit-term 1:

. . Commutativity
Commutativity is the property of operation which allows the changing of the terms in their position in such that the value of the expression will not change. As the novel method of orthogonalizing ORing is commutative, the sequence of its execution can be changed.
The value of both sides are equivalent and orthogonal. They can only di er in the form of coverage. The following Example 5. 4 gives an overview about the commutative property.
The left side di ers only in the form of coverage in contrast to the right side, which is shown in the corresponding K-maps (Fig. 4). Both sides consist of disjointed cubes. Associativity is the property of an operation which allows the rearranging of the parentheses in such that the value of the expression will not change. As orthogonalizing ORing is associative, the position of the parentheses can be changed.
The value of both sides are equivalent and orthogonal. Only the form of their coverage can be di erent. Following Example 5.5 illustrates this characteristic of associativity.

Example 5.5.
( Both sides are homogeneous and orthogonal as shown in the K-maps in Figure 5. They only di er in their form of coverage. The distributive property of an operation allows the exclusion of the same term. That means, that a term can be factored out. Hereby, the orthogonality of both sides has to be insisted. In this case, the distributive law for ANDing out applies for left and right side.
The validity of the distributive property is given by the following proof: Both sides are equivalent and orthogonal. They can only di er in the form of their coverage. This charasteristic of distributivity is demonstrated by the following Example 5.6 whereby both sides result to the same term.

Disjointed sum of products
Based on this technique of orthogonalizing ORing a novel Equation (36) is formed which enables the orthogonalization of every disjunctive form DF. That means, this formula enables the transformation of a SOP in a homogeneous dSOP. With m ∈ N as the number of conjunctions c i (x) that are included in the given DF, which has to be orthogonalized, it follows that: The explanation for Equation (36) is provided as follows: -Orthogonalizing ORing is realized between the rst and the second conjunction, c (x)∨ f c (x), by Eq. (15). After that, orthogonalizing ORing is calculated between the result of c (x) ∨ f c (x) and the third conjunction, c (x)∨ f c (x) ∨ f c (x), by Eq. (19). -This procedure is continued until the last conjunction cm(x).
The general validity of (36) is proven by the following mathematical induction: Proof.
-∀(m) ∈ N, (m) ≥ m applies A(m): The orthogonal result may di er depending on the order of the conjunctions because of the property of commutativity. Thus, the conjunctions can be changed if necessary. However, all solutions are equivalent and orthogonal. In Example 6.1 it is given an overview of the use of Eq. (36) Example 6.1. Function f (x) = x ∨ x ∨ x x has to be orthogonalized by Eq. (36).
Function f (x) orth is the orthogonal form of function f (x), illustrated in the K-maps (Fig. 6).
By rearrangement of the order of the consisting conjunctions of a DF we obtain fewer number of conjunctions in the derived orthogonal form. This procedure of sorting is carried out from large to small. That means, it takes place from conjunctions of higher number of variables to conjunctions of fewer number of variables.
The following Example clari es this advantage of sorting.

Example 6.2. Function f (x) =
x ∨ x ∨ x x has to be orthogonalized after resorting.
The orthogonalized form of the sorted DF contains fewer number of conjunctions (see Fig. 6). The analysis of a measurement, as shown in Fig. 7, gives an overview of the comparison of the orthogonalization process depending on sorting. . This comparison illustrates a reduction of conjunctions in the orthogonal form when the DF was sorted before. This reduction is approximately 19% on average (see Table 2). Additionally, this feature allows the reducing of operation for subsequent calculation steps. Hereby, the relation in (37) is con rmed by the comparison in Fig. 7. The number of conjunctions of an orthogonalized DF P s c(x) (f (x) orth sort ) which was sorted before (sortORTH[∨]) is smaller than the number of conjunctions of the orthogonalized DF P c(x) (f (x) orth ) which was not sorted before (ORTH[∨]):

Conclusion
This work showed a novel technique for building a union of disjointed conjunctions which is called as orthogonalizing ORing. Its results are orthogonal. Orthogonalizing ORing is used to calculate the orthogonal form of building the union of two conjunctions. This linking technique replaces two calculation stepsbuilding a union and the subsequent orthogonalization -by one step. Orthogonalizing ORing is valid in general, which was proven by the mathematical induction, and is also equivalent to the usual method of union ∨. Additionally, postulates related to commutativity, distributivity and associativity and axioms for this method are also de ned. Furthermore, every Boolean function of disjunctive form or every Sum of Products, respectively, can easily be orthogonalized mathematically by a novel equation which is based on this linking technique of orthogonalizing ORing. By this orthogonalization a disjointed Sum of Products can be reached in a simpler way. The general validity was also proven by the mathematical induction. An additional step of sorting before the step of orthogonalization achieves a reduction of approximately 19% of the number of conjunctions in the orthogonal result. This feature was illustrated by a measurement whereby the orthogonalization took place before and after sorting.