Boolean Orthogonalizing Combination Methods

Jan Zizka et al. (Eds) : CCSEIT, MoWiN, IT, AIAP, ICBB
pp. 09–22, 2015. © CS & IT-CSCP 2015
BOOLEAN
COMBINATION
Yavuz Can
1,2
Institute for Electronics Engineering (LTE),
Friedrich-Alexander University
ABSTRACT
In this paper a new logical operation method called “
presented. It is used to calculate the difference, but also the complement of a function as well as
the EXOR and EXNOR of two minterms respectively two ternary
respectively two ternary-vector
logical operation method called “orthogonal OR
advantages of both methods are their results, which are already available
form that has an essential advantage for continuing calculations. Since it applies, an orthogonal
disjunctive normal form is equal to orthogonal antivalence normal form, subsequent Boolean
differential calculus will be simplified.
KEYWORDS
Difference-Building, Orthogonality, TVL, Data Memory Request, Computing Time
1. INTRODUCTION
Orthogonality is a special property of Boolean functions because an orthogonal function can be
transformed in another form and so it simplifies the handling for further calculations in
applications of electrical engineering. For example, the orthogonal for
equal to the orthogonal form of an antivalent form,
orthogonalizing methods, which means a new method of OR
the difference of minterms respec
vector-lists (TVL) of disjunctive normal form which result in orthogonal form. On the one hand,
the method “othogonalizing difference
derived from the set theory, but, on the other hand, it provides orthogonal results. It also enables
the replacement of EXOR and EXNOR of two minterms (also TVs) or two functions (also
TVLs). Thus, orthogonal results can be achieved which has essential advanta
calculation in particular in the area of TVL
ing ” offers the logical disjunction of two minterms respectively two TVs with the characteristic
of the resulting in orthogonal form.
Jan Zizka et al. (Eds) : CCSEIT, MoWiN, IT, AIAP, ICBB - 2015
CSCP 2015 DOI : 10.5121/csit.2015.51102
OOLEAN ORTHOGONALIZING
OMBINATION METHODS
Yavuz Can1
and Georg Fischer2
Institute for Electronics Engineering (LTE),
Alexander University, Erlangen-Nuremberg, Germany
yavuz.can@fau.de
georg.fischer@fau.de
ogical operation method called “orthogonalizing difference-building”
the EXOR and EXNOR of two minterms respectively two ternary-vectors or two func
vector-lists is presented. On the basis of this new method a further
gical operation method called “orthogonal OR-ing” is going to be introduced. The
advantages of both methods are their results, which are already available in an orthogonal
differential calculus will be simplified.
, Orthogonality, TVL, Data Memory Request, Computing Time
applications of electrical engineering. For example, the orthogonal form of a disjunctive form is
equal to the orthogonal form of an antivalent form, ‫ܨܰܦ‬௢௥௧௛
ൌ ‫ܨܰܣ‬௢௥௧௛
. This work shows new
orthogonalizing methods, which means a new method of OR-ing as well as a method for building
the difference of minterms respectively ternary-vectors (TV) or functions respectively ternary
lists (TVL) of disjunctive normal form which result in orthogonal form. On the one hand,
the method “othogonalizing difference-building ⊝” is similar to the usual difference
ved from the set theory, but, on the other hand, it provides orthogonal results. It also enables
TVLs). Thus, orthogonal results can be achieved which has essential advantage for further
calculation in particular in the area of TVL-arithmetic. The other method “orthogonalizing OR
offers the logical disjunction of two minterms respectively two TVs with the characteristic
of the resulting in orthogonal form.
DOI : 10.5121/csit.2015.51102
Nuremberg, Germany
building” is
vectors or two functions
lists is presented. On the basis of this new method a further
is going to be introduced. The
in an orthogonal
m of a disjunctive form is
. This work shows new
ing as well as a method for building
vectors (TV) or functions respectively ternary-
lists (TVL) of disjunctive normal form which result in orthogonal form. On the one hand,
” is similar to the usual difference-building
ved from the set theory, but, on the other hand, it provides orthogonal results. It also enables
ge for further
arithmetic. The other method “orthogonalizing OR-
offers the logical disjunction of two minterms respectively two TVs with the characteristic

10 Computer Science & Information Technology (CS & IT)
2. THEORETICAL FOUNDATIONS
2.1. Isomorphism: Set Theory as Basis for Switching Algebra
In Figure1a), the set difference from the set theory is shown. If the set difference is called‫,ܵܯ‬
the remaining set of these two sets is the set of all elements belonging to set ‫,ܯ‬ but not to set ܵ
(‫ܯ‬without ܵ). In Figure1b), the union of set ܵଵ and ܵଶ, that isܵଵ ∪ ܵଶ, is illustrated [2][3].
Figure 1: a) Set Difference ‫;ܵܯ‬ b) Set Union ܵଵ ∪ ܵଶ
Due to the isomorphism the set difference and the set union out of the set theory are expressed in
the propositional logic as follows:
‫ܥ‬ଵ ൌ ‫ܵܯ‬ ൌ ‫ܯ‬ ∩ ܵ̅ ↣ ‫ܯ‬ ∧ ܵ̅ (1)
‫ܥ‬ଶ ൌ ܵଵ ∪ ܵଶ ↣ ܵଵ ∨ ܵଶ (2)
Thus, an Equation (3) to build the difference of two minterms in the switching algebra is defined
out of (1). The minuend minterm is indentified with݉ெ ≔ ⋀ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴ , the subtrahend minterm
is indentified with ݉ௌ ≔ ⋀ ‫ݔ‬௡ି௦
௡ିଵ
௦ୀ଴ and with ݊ ∈ ℕfollows:
݉ெ݉ௌ ൌ ൭ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ൭ሥ ‫ݔ‬௡ି௦
௡ିଵ
௦ୀ଴
൱ ≔ ൭ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ∧ ൭ሥ ‫ݔ‬௡ି௦
௡ିଵ
௦ୀ଴
൱
തതതതതതതതതതതതതതത
ൌ
ൌ ൭ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ∧ ൭ሧ ‫̅ݔ‬௡ି௠
௡ିଵ
௦ୀ଴
൱ ൌ ሺ‫ݔ‬௡ ⋅ ‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଵሻெ ∧ ሺ‫ݔ‬௡ ⋅ ‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଵሻௌ (3)
An Equation (4) building the disjunction of two minterms where the first summand minterm is
identified with ݉ௌభ
≔ ⋀ ‫ݔ‬௡ି௦భ
௡ିଵ
௦భୀ଴ and the second summand minterm is identified with ݉ௌమ
≔
⋀ ‫ݔ‬௡ି௦మ
௡ିଵ
௦మୀ଴ is defined out of (2). And with ݊ ∈ ℕ follows:
݉ௌభ
∨ ݉ௌమ
ൌ ቌሥ ‫ݔ‬௡ି௠
௡ିଵ
௦భୀ଴
ቍ ∨ ቌሥ ‫ݔ‬௡ି௦
௡ିଵ
௦మୀ଴
ቍ ൌ ሺ‫ݔ‬௡ ⋅ ‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଵሻௌభ
∨ ሺ‫ݔ‬௡ ⋅ ‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଵሻௌమ
(4)
Equation (3) and (4) only indicate the value of the variable at the corresponding position in terms
of its index. That means ‫̅ݔ‬௜is the complement value of variable at thei-th place, whereas‫ݔ‬௜ hows
the value of the variable at the i-th place. The general validity is proved by the method of
mathematical induction.

Computer Science & Information Technology (CS & IT) 11
2.2. Ternary-Vector-List (TVL)
Ternary-Vector-Lists are representations of Boolean functions which can take one of three
possible values for each point. A non-negated variable is characterized by ‘1’, a negated by ‘0’
and not included by ‘-‘. Boolean equations represented by TVL can be treated computationally
easier [5][10]. A TVL consists of m-rows (number of conjunctions or disjunctions contained in
the function) and n-columns (number of independent variables) and is characterized with݊ ∈
ℕand ‫ݐ‬ ∈ {0, 1, −}. Any form of a function, that means disjunctive form (DF), conjunctiveform
(KF), equivalent form (EF) and antivalence form (AF), can be represented as a TVL. As no
operators (∨, ∧, ⨀, ⨁ ) in the TVL presentation are given, the designations of the matrix by D(f),
K(f), E(f) and A(f) show the type of the TVL [5][10][12].
‫ݔ‬௡ … ‫ݔ‬ଶ‫ݔ‬ଵ
ܸܶ‫ܮ‬ ൌ ൦
ܸܶଵ
ܸܶଶ
∶
ܸܶ௠
൪ ≔
1. ‫ݓ݋ݎ‬
2. ‫ݓ݋ݎ‬
∶
݉. ‫ݓ݋ݎ‬
൦
‫ݐ‬ଵ௡
‫ݐ‬ଶ௡
∶
‫ݐ‬௠௡
…
…
∶
…
‫ݐ‬ଵଶ
‫ݐ‬ଶଶ
∶
‫ݐ‬௠ଶ
‫ݐ‬ଵଵ
‫ݐ‬ଵଶ
∶
‫ݐ‬௠ଵ
൪ ൌ [‫ݐ‬௠௡]
(5)
The rule for AND-ing (∧) of two TVs (ܸܶ௜,௝) which represent an average TV (ܸܶ௞), is defined by
Table 1:
Table 1. Rule for AND-ing of two TVs
∧ 0 1 -
0 0 × 0
1 × 1 1
- 0 1 -
if × ≥ 1
else
×: empty set
ܸܶ௞ ൌ×
ܸܶ௞ ൌ ܸܶ௜ ∧ ܸܶ௝
The rule for OR-ing (∨) of two TVs (ܸܶ௜,௝), which represent the union of both TVs, is defined as
follows:
ܸܶ௜ ∨ ܸܶ௝ ൌ ൤
ܸܶ௜
ܸܶ௝
൨
(6)
The difference-building of two TVs with ܸܶெ ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ெ and ܸܶௌ ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ௌ is
distinguished out of Equation (3):
ܸܶெ ܸܶௌ ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ெ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ௌ ≔ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ெ ∧ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ௌ
തതതതതതതതതതതതതതതതതതതത ൌ
ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ெ ∧ ൦
‫ݐ‬̅௡ −
− ‫ݐ‬௡ିଵതതതതതത
⋮
−
⋮
−
… −
… −
⋮
…
⋮
‫ݐ‬̅ଵ
൪
ௌ
(7)
2.3. Orthogonality
A function or TVL is orthogonal if its minterms respectively its TVs are disjoint to one another in
pairs at least in one column. Consequently, these minterms (݉௜,௝) or these TVs (ܸܶ௜,௝) then have
no common covering after their logical conjunction. An orthogonal function has no redundant
minterm. Thus, the following Equations in (8) can be formulated for the proof of orthogonality
[17]:

3. ORTHOGONALIZING D
3.1. Method
The method of orthogonalizing difference
corresponds to the removal of the intersection which is formed between the minuend
subtrahend ݉ௌ, from the minuend
several disjoint minterms, which cover all of the remaining 1s an
other. For this purpose, an example is shown in a K
group of 2 (subtrahend: ‫ݔ‬ଷ‫ݔ‬ଶ‫ݔ‬ଵ) is subtracted from group of 8 (minuend:
of several blocks (1st
Block, 2nd
Block, 3
‫ݔ‬ସณ
୑୧୬୳ୣ୬ୢ
⊝ ‫ݔ‬ଷᇣ
ୗ୳ୠ୲୰ୟ୦ୣ୬ୢ
• The first literal of the subtrahend, here
minterm of the minuend, here
• Then the second literal, here
minuend and to the first literal
‫ݔ‬ସ‫ݔ‬ଷ‫ݔ‬ଶതതത.
• Following the next literal, hier
the minuend and to the first literal
third block of the difference is
• This process is continued until all literals of the subtrahend are singly complemented and
linked by AND-ing to the minuend in a separate minterm.
The Equation (9) is applied to calculate the orthogonalizing difference
In this case, the formula does not show the value of the individual literals, but it shows whether
the associated literal exists complement or not. The indices declare only the order of literals
which have to be calculated. In thi
formed from [18] is applied to describe the orthogonalizing difference
mathematically easier way:
Computer Science & Information Technology (CS & IT)
DIFFERENCE-BUILDING ⊝
The method of orthogonalizing difference-building ⊝ illustrated in the Karnaugh map (Fig
corresponds to the removal of the intersection which is formed between the minuend
from the minuend ݉ெ, which means ݉ெ ሺ݉ெ ∧ ݉ௌሻ. The result consists of
several disjoint minterms, which cover all of the remaining 1s and are pairwise orthogonal to each
other. For this purpose, an example is shown in a K-map with 4 variables (Fig. 2
) is subtracted from group of 8 (minuend: ‫ݔ‬ସ). The resu
Block, 3rd
Block) which are pairwise orthogonal to each other.
Figure 2: Example of ⊝ in a K-map
‫ݔ‬ଷ‫ݔ‬ଶ‫ݔ‬ଵᇣᇧᇤᇧᇥ
ୗ୳ୠ୲୰ୟ୦ୣ୬ୢ
ൌ ‫ݔ‬ସ‫ݔ‬ଷതതതถ
ଵ౩౪୆୪୭ୡ୩
∨ ‫ݔ‬ସ‫ݔ‬ଷ‫ݔ‬ଶതതതᇣᇧᇤᇧᇥ
ଶ౤ౚ୆୪୭ୡ୩
∨ ‫ݔ‬ସ‫ݔ‬ଷ‫ݔ‬ଶ‫ݔ‬ଵതതതᇣᇧᇧᇤᇧᇧᇥ
ଷ౨ౚ୆୪୭ୡ୩ᇣᇧᇧᇧᇧᇧᇧᇧᇧᇧᇤᇧᇧᇧᇧᇧᇧᇧᇧᇧᇥ
ୈ୧୤୤ୣ୰ୣ୬ୡୣ
The first literal of the subtrahend, here ‫ݔ‬ଷ, is taken complement and AND
minterm of the minuend, here ‫ݔ‬ସ. Consequently, the first block of the difference is
Then the second literal, here‫ݔ‬ଶ, is taken complement and AND-ing to the minterm of the
minuend and to the first literal ‫ݔ‬ଷ of the subtrahend. Therefore, the second block is
Following the next literal, hier ‫ݔ‬ଵ, is taken complement and AND-ing to the minterm of
the minuend and to the first literal ‫ݔ‬ଷ and second literal ‫ݔ‬ଶ of the subtrahend. Thus, the
of the difference is ‫ݔ‬ସ‫ݔ‬ଷ‫ݔ‬ଶ‫ݔ‬ଵതതത.
This process is continued until all literals of the subtrahend are singly complemented and
ing to the minuend in a separate minterm.
is applied to calculate the orthogonalizing difference-building of two minterms.
which have to be calculated. In this case, the formula ሺ⋁ ‫ݔ‬పഥ
௡ೕ
௜ୀଵ
ൌ ‫ݔ‬ଵതതത ∨ ‫ݔ‬ଵ‫ݔ‬ଶതതത ∨. .∨
formed from [18] is applied to describe the orthogonalizing difference-building in a
illustrated in the Karnaugh map (Fig. 2)
corresponds to the removal of the intersection which is formed between the minuend ݉ெand the
The result consists of
d are pairwise orthogonal to each
. 2),in which a
). The result consists
Block) which are pairwise orthogonal to each other.
, is taken complement and AND-ing to the
. Consequently, the first block of the difference is ‫ݔ‬ସ‫ݔ‬ଷതതത.
ing to the minterm of the
of the subtrahend. Therefore, the second block is
ing to the minterm of
of the subtrahend. Thus, the
This process is continued until all literals of the subtrahend are singly complemented and
building of two minterms.
ത ∨ ‫ݔ‬ଵ‫ݔ‬ଶ ⋅. .⋅ ‫ݔ‬௡തതതሻ
building in a

݉ெ ⊝ ݉ௌ ൌ ൭ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ⊝ ൭ሥ ‫ݔ‬௡ି௦
௡ିଵ
௦ୀ଴
൱ ≔ ൭ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ∧ ቌ ሧ ‫̅ݔ‬௡ି௦
௡ೕିଵ
௦ୀ଴
ቍ ൌ
ൌ ሺ‫ݔ‬௡ ⋅ ‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଵሻெ ∧ ሺ‫̅ݔ‬௡ ∨ ‫ݔ‬௡‫̅ݔ‬௡ିଵ ∨. .∨ ‫ݔ‬௡ ⋅ ‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଵሻௌ (9)
The result may differ depending on the starting literal. There are many equivalent options. They
only differ in the form of coverage. The minterms of the difference are pairwise disjoint to each
other, so that the result is already availabe in an orthogonal form. The the result corresponds to
the minuend if the subtrahend is already orthogonal to the minuend (݉ெ ⊥ ݉ௌ):
ܑ܎: ݉ௌ ⊈ ݉ெ‫:ܖ܍ܐܜ‬ ݉ெ ⊝ ݉ௌ ൌ ݉ெ (10)
The number of the minterms (blocks) in the result called ݊ corresponds to the number of the
variables presented in the subtrahend and are not presented in the minuend at the same time. The
number of the possible results can be defined by ݊! for ݊ > 0. The orthogonalizing difference-
building of two TVs (ܸܶெ, ܸܶௌ), is defined by a corresponding Equation (11):
ܸܶெ ⊝ ܸܶௌ ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ெ ⊝ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ௌ ≔
ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ெ ∧ ൦
‫ݐ‬̅௡ −
‫ݐ‬௡ ‫ݐ‬௡ିଵതതതതതത
⋮
‫ݐ‬௡
⋮
‫ݐ‬݊
… −
… −
⋮
…
⋮
‫ݐ‬̅ଵ
൪
ௌ
(11)
By the use of this new method, two calculation procedures - building the difference and the
subsequent orthogonalization - can be performed in one step. That means that the orthogonalizing
difference-building ⊝represents the composition of the difference-building and the subsequent
orthogonalization Orth [6].
3.2. Analysis
3.2.1. Mathematical
The general validity is proved by mathematical induction:
Basis: n = 1
൭ሥ ‫ݔ‬ଵି௠
ଵିଵ
௠ୀ଴
൱ ∧ ൭ሧ ‫̅ݔ‬ଵି௦
ଵିଵ
௦ୀ଴
൱ ൌ ሺ‫ݔ‬ଵሻெ ∧ ሺ‫̅ݔ‬ଵሻௌ
ሺ‫ݔ‬ଵሻெ ∧ ሺ‫̅ݔ‬ଵሻௌ ൌ ሺ‫ݔ‬ଵሻெ ∧ ሺ‫̅ݔ‬ଵሻௌ
Statement is true:
Inductive step: n = n + 1
൭ሥ ‫ݔ‬ሺ௡ାଵሻି௠
௡
௠ୀ଴
൱ ∧ ቌሧ ‫̅ݔ‬ሺ௡ାଵሻି௦
௡ೕ
௦ୀ଴
ቍ ൌ ൫‫ݔ‬ሺ௡ାଵሻ‫ݔ‬௡‫ݔ‬௡ିଵ. . ‫ݔ‬ଵ൯
ெ
∧ ቀ‫̅ݔ‬ሺ௡ାଵሻ ∨ ‫ݔ‬ሺ௡ାଵሻሺ‫̅ݔ‬௡ ∨ ‫ݔ‬௡‫̅ݔ‬௡ିଵ ∨. .∨ ‫ݔ‬௡ ⋅. .⋅ ‫ݔ‬ଵሻቁ
ௌ
൭‫ݔ‬ሺ௡ାଵሻ ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ∧ ቌ‫̅ݔ‬ሺ௡ାଵሻ ∨ ‫ݔ‬ሺ௡ାଵሻ ሧ ‫̅ݔ‬௡ି௦
௡ೕିଵ
௦ୀ଴
ቍ ൌ ൭‫ݔ‬ሺ௡ାଵሻ ሥ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴
൱ ∧ ቌ‫̅ݔ‬ሺ௡ାଵሻ ∨ ‫ݔ‬ሺ௡ାଵሻ ሧ ‫̅ݔ‬௡ି௦
௡ೕିଵ
௦ୀ଴
ቍ
The new Equation (3) is equated with the usual Equation (9) to show the equivalence. Since the
term of the minuend (⋀ ‫ݔ‬௡ି௠
௡ିଵ
௠ୀ଴ ) is equal on both sides, it can be neglected:

൭ሧ
௡ି
௦ୀ
‫̅ݔ‬௡ ∨ ‫̅ݔ‬௡ିଵ
Due to the axiom of absorption with
confirmed. They only differ in their form of coverage. The right side is the orthogonal form of the
left side. This proves that the orthogonalizing difference
the difference-building derived out of the set theory.
3.2.2. Computing Time
A study between usual and new method in computing time depe
(dimension݀݅݉[݅]) of the minuend and subtrahend indicates almost identical co
The new method (orth_Diff) has a slightly longer computing time with increasing dimension than
the method of difference-building (
shows a maximum difference of just 1.2
complexity. The supplementation instruction per loop iteration results in a multiplicative
constant which explains the minimal difference. Against it, the results are orthogonal. The
comparison in Figure 3 illustrates that the method
increasing dimension ݀݅݉[݅] as the composition of method
orthogonalizing Orth [19].
Figure 3: Comparison of computing time of
In Diagram (Fig. 4) the explanation for the higher computing time of the composition is provided.
With increasing dimension the computing
Diff decreases with respect to the composition. In dimension
same percentage value of the whole computing time. Thereafter, the percentage value of
low and at the same time the percentage value of
൭ሧ ‫̅ݔ‬௡ି௦
ିଵ
ୀ଴
൱ ൌ ቌ ሧ ‫̅ݔ‬௡ି௦
௡ೕିଵ
௦ୀ଴
ቍ
ଵ ∨. .∨ ‫̅ݔ‬ଵ ൌ ‫̅ݔ‬௡ ∨ ‫ݔ‬௡‫̅ݔ‬௡ିଵ ∨. .∨ ‫ݔ‬௡‫ݔ‬௡ିଵ ⋅. .⋅ ‫ݔ‬ଶ‫̅ݔ‬ଵ
Due to the axiom of absorption with ‫ݔ‬పഥ ∨ ‫ݔ‬௜‫ݔ‬ఫഥ ൌ ‫ݔ‬పഥ ∨ ‫ݔ‬ఫഥ, the equivalence between two methods is
proves that the orthogonalizing difference-building supplies equivalent results as
building derived out of the set theory.
A study between usual and new method in computing time depending on the length
of the minuend and subtrahend indicates almost identical computing times.
has a slightly longer computing time with increasing dimension than
building (Diff). The comparison of the average values of both methods
shows a maximum difference of just 1.2µs, which can be explained by the consideration
The supplementation instruction per loop iteration results in a multiplicative
the minimal difference. Against it, the results are orthogonal. The
illustrates that the method orth_Diff has faster computing
] as the composition of method Diff and the subsequent method of
Figure 3: Comparison of computing time of orth_Diff⊝ and composition of Diff and
) the explanation for the higher computing time of the composition is provided.
With increasing dimension the computing time of the method Orth increases, whereas the method
decreases with respect to the composition. In dimension ݀݅݉[14] both functions have the
same percentage value of the whole computing time. Thereafter, the percentage value of
the same time the percentage value of Orthincreases.
, the equivalence between two methods is
building supplies equivalent results as
nding on the length
mputing times.
has a slightly longer computing time with increasing dimension than
The comparison of the average values of both methods
, which can be explained by the consideration
The supplementation instruction per loop iteration results in a multiplicative
the minimal difference. Against it, the results are orthogonal. The
has faster computing time with
uent method of
and Orth
) the explanation for the higher computing time of the composition is provided.
increases, whereas the method
both functions have the
same percentage value of the whole computing time. Thereafter, the percentage value of Diffruns

Figure 4: Percentage value of whole computing of
At the dimension݀݅݉[50]the share of computing time of
percentage value of Diffis 6.7 %. Finally,
the new method orth_Diff brings a significant advantage due to the orthogonal results [19].
3.2.3. Complexity
The complexity of both methods is analyzed by the evaluation of their implemented functions.
The dimension of the inserted minterms respectively ternary
size ݆. Figure 5 shows the pseudocode of
according to the rules of the
accordingly comparisons and two inte
complexity is ܱሺ݆ଶ
ሻ [19]. The complexity of the function
not required and may be replaced by three linear operations with the complexity of
Figure 5
However, the additive and multiplicative constants have no influence on the
omitted for this reason. Although the total complexity of the usual method is minor but it
increases in total by an additional code for orthogonalization.
Percentage value of whole computing of Diff and Orth
the share of computing time of Orth is already 93.7
%. Finally, the measurement of computing time demonstrates that
brings a significant advantage due to the orthogonal results [19].
The dimension of the inserted minterms respectively ternary-vectors is determined as the input
shows the pseudocode of orth_Diff. For the function orth_Diff
according to the rules of the ܱ-calculus many complexities of ܱሺ1ሻ for instructions and
accordingly comparisons and two interleaved nested complexities ofܱሺ1ሻ. As the result, the total
[19]. The complexity of the function Diff is ܱሺ݆ሻ because the inner loops are
not required and may be replaced by three linear operations with the complexity of ܱ
5: Pseudocode of orth_Diff with complexity
However, the additive and multiplicative constants have no influence on the complexity and are
increases in total by an additional code for orthogonalization.
15
is already 93.7 % and the
the measurement of computing time demonstrates that
brings a significant advantage due to the orthogonal results [19].
s determined as the input
orth_Diff is obtained
for instructions and
esult, the total
because the inner loops are
ܱሺ1ሻ (Fig. 6).
complexity and are

Figure 6: Pseudocode of Diff with complexity
3.2.4. Data Memory Request
First, the memory request of the data structures TV and TVL is determined for this comparison
[19]. A TV consists of an array of ݉ ⋅ byte elements. Each byte represents an element of the
ternary variables. This means that the memory request of the data structures depends on the
dimension of the ternary variables. In C# the data type Byte consists of 8Bits. For a 64 bit system
a minimum memory request ்ܾܵ௏ሺ݉ሻfor the data structure of TV with an addressing pointer is
calculated by:
்ܾܵ௏ሺ݉ሻ ൌ ݉ ⋅ 8 ‫ݏݐ݅ܤ‬ᇣᇧᇧᇤᇧᇧᇥ
௠⋅஻௬௧௘ିா௟௘௠௘௡௧௦
+ 32 ‫ݏݐ݅ܤ‬ᇣᇧᇤᇧᇥ
௉௢௜௡௧௘௥ ௔௧ ்௏
(12)
Since a TVL consists of a list of TVs. The memory request of a TVL depends on the number of
ternary variables and also the number of the consisting TVs called ݇. The easiest way to
implement a list is the use of a linked list. Each entry in the list has a pointer at its follower. Thus,
a minimal memory request ்ܾܵ௏௅ሺ݇, ݉ሻis calculated by:
்ܾܵ௏௅ሺ݇, ݉ሻ ൌ ݇ ⋅ ்ܾܵ௏ሺ݉ሻ + 32 ‫ݏݐ݅ܤ‬ᇣᇧᇤᇧᇥ
௉௢௜௡௧௘௥ ௔௧ ்௏௅
(13)
For this reason, the minimal total memory requestܾܵ௧௢௧ for the operation of two TVs, the
difference-building and orthogonalizing difference-building in this case is:
ܾܵ௧௢௧ ൌ 2 ⋅ ்ܾܵ௏ሺ݉ሻᇣᇧᇧᇧᇤᇧᇧᇧᇥ
்௏ಾ&்௏ೄ
+ ்ܾܵ௏௅ሺ݇, ݉ሻᇣᇧᇧᇧᇤᇧᇧᇧᇥ
ோ௘௦௨௟௧ି்௏௅
(14)
The theoretical memory request has to be calculated equally for both function. Because of that,
they have the same minimal memory request. Addional memory request for a function for
orthogonalization is not needed, because an orthogonal TVL of the orthogonalizing difference is
already provided. That shows that the new method, which is the composition of two functions has
the advantage of reducing the memory request in addition to faster computing time. The minimal
theoretical memory request depending on the dimension ݀݅݉[݅]and the number of n which
primarily affects the memory usage is illustrated in Figure 7. Thereby, ݀݅݉[݅] is varied at a
constant n and afterwards in the reverse case and analyzed on memory request. For
constant݀݅݉[݅]and changing ݊the memory usage is higher than in the reverse case. It
applies∆ܾܵ௧௢௧ሺ݊ሻ ൌ ∆ܾܵ௧௢௧ሺ݀݅݉[݅]ሻ.

Figure 7: Minimal theoretical memory request
3.3. Applications of Orthogonalizing Difference
3.3.1. Orthogonalizing Difference
By Equation (14) (presented in TVL
it is possible to calculate an orthogonal difference of two functions respectively two TVLs. One
of them is the minuend function
subtrahend function ݂ௌ. The minuend is subducted the common set of subtrahend and minuend;
the result is represented in an orthogonal form. The associated minterms (TVs) have the
corresponding index-notation, which means
subtrahend:
‫ܦ‬൫݂ெ
௢௥௧௛
൯ ⊝ ‫ܦ‬ሺ݂ௌሻ ൌ ൦
ܸܶଵெ
ܸܶଶெ
∶
ܸܶ௠ெ
ൌ ൦
ሺܸܶଵ
ሺܸܶଶ
ሺܸܶ௠ெ
In this case it is important to stress that any outcome of each individual
considered. That means, if the combination of
complete the appropriate row to 0
building has already been proved in general validity, there is no need proof for generel validity in
this case, because all the single links are generally valid. Therefore, out of logical conclusion,
Equation (14) is generally valid. If two functions are disjoint to each other (
difference cannot be formed and it follows:
ܑ܎
Figure 7: Minimal theoretical memory request
Applications of Orthogonalizing Difference-Building
Difference-Building of two Functions
(presented in TVL-arithmetic) based on the orthogonalizing difference
ion ݂ெ
௢௥௧௛
which has to be orthogonal and the other one is the
. The minuend is subducted the common set of subtrahend and minuend;
notation, which means ‫ܯ‬ stands for the minuend and ܵ stands for the
ெ
ெ
௠ெ
൪ ⊝ ൦
ܸܶଵௌ
ܸܶଶௌ
∶
ܸܶ௠ௌ
൪ : ൌ
ሺ ଵெ ⊝ ܸܶଵௌሻ ∧ ሺܸܶଵெ ⊝ ܸܶଶௌሻ ∧. .∧ ሺܸܶଵெ ⊝ ܸܶ௠ௌ
ሺ ଶெ ⊝ ܸܶଵௌሻ ∧ ሺܸܶଶெ ⊝ ܸܶଶௌሻ ∧. .∧ ሺܸܶଶெ ⊝ ܸܶ௠ௌ
∶
ሺ ௠ெ ⊝ ܸܶଵௌሻ ∧ ሺܸܶ௠ெ ⊝ ܸܶଶௌሻ ∧. .∧ ሺܸܶ௠ெ ⊝ ܸܶ௠ௌ
In this case it is important to stress that any outcome of each individual ⊝-linkings has to be
considered. That means, if the combination of ሺܸܶଵெ ⊝ ܸܶଵௌሻ ൌ 0 arises for example, this will
omplete the appropriate row to 0 because of: ‫ݔ‬௜ ∧ 0 ൌ 0. Since the orthogonalizing difference
ingle links are generally valid. Therefore, out of logical conclusion,
is generally valid. If two functions are disjoint to each other (݂ெ
difference cannot be formed and it follows:
ܑ܎: ݂ௌ ⊈ ݂ெ‫:ܖ܍ܐܜ‬ ݂ெ ⊝ ݂ௌ ൌ ݂ெ
17
arithmetic) based on the orthogonalizing difference-building
which has to be orthogonal and the other one is the
. The minuend is subducted the common set of subtrahend and minuend;
stands for the
ሻ
ሻ
௠ௌሻ
൪
(14)
linkings has to be
arises for example, this will
Since the orthogonalizing difference-
ingle links are generally valid. Therefore, out of logical conclusion,
⊥ ݂ௌ) then a
(15)

3.3.2. Orthogonal Complement of a Function
A further application is the building of an orthogonal complement of a function of the disjunctive
normal form. In the set theory the set difference of universal set ‫ܩ‬and a set ‫ܣ‬corresponds to set
‫ܣ‬̅which is the complement of the given set ‫ܣ‬:
‫ܩ‬ ‫ܣ‬ ൌ ‫ܣ‬̅ → ‫ܩ‬ ∩ ‫ܣ‬̅ ൌ ‫ܣ‬̅ (16)
Transferred to the switching (Boolean) algebra the complement of a function ݂̅ሺ‫ݔ‬ሻ can be
determined from the difference of a unit function ݂൫‫ݔ‬൯ ൌ 1 and the function ݂ሺ‫ݔ‬ሻ:
݂ሺ1ሻ ݂ሺ‫ݔ‬ሻ ൌ ݂̅ሺ‫ݔ‬ሻ → ݂ሺ1ሻ ∧ ݂̅ሺ‫ݔ‬ሻ ൌ ݂̅ሺ‫ݔ‬ሻ (17)
As both methods are equivalent, Equation (17) can be formulated with ⊝ to find out an
orthogonal complement:
݂ሺ1ሻ ⊝ ݂ሺ‫ݔ‬ሻ ൌ ݂̅൫‫ݔ‬൯
௢௥௧௛
or ‫ܦ‬ሺ1ሻ ⊝ ‫ܦ‬ሺ݂ሻ ൌ ‫ܦ‬ሺ݂̅௢௥௧௛
ሻ (18)
3.3.3. Orthogonal EXOR of two Functions
The EXOR-operation of two minterms ݉௜,௝and also of two functions ݂௜,௝can be calculated as:
݂௜ ⊕ ݂௝ ൌ ݂௜݂̅௝ ∨ ݂̅௜݂௝ ൌ
ൌ ሺ݂௜ ∨ ݂௝ሻ ∧ ሺ݂̅௜ ∨ ݂̅௝ሻ ൌ
ൌ ሺ݂௜ ∨ ݂௝ሻ ∧ ሺ݂ప ∧ ݂ఫሻതതതതതതതതതത ൌ
ൌ ሺ݂௜ ∨ ݂௝ሻሺ݂௜ ∧ ݂௝ሻ ൌ (19)
It may be formulated with ⊝ due to the equivalence to get orthogonal result:
݂௜ ⊕ ݂௝ ൌ ሺ݂௜ ∨ ݂௝ሻ ⊝ ሺ݂௜ ∧ ݂௝ሻ (20)
By AND-ing and OR-ing of the functions ݂௜,௝an inherent relation between these two functions is
constructed.Thus, the minuend function does not need to be orthogonal. The advantage is the
orthogonal result again. By using ⊝ the difficulties which arise with the building of the
complement of a function are circumvented. The form of the function changes by building the
complement and the transformation back to its original form requires a more sophisticated
calculation.
3.3.4. Orthogonal EXNOR of two Functions
The EXNOR-operation of two minterms ݉௜,௝or two function݂௜,௝is basically the complement of the
EXOR-operation of the same minterms or functions. The EXNOR can be expressed by the
complement of EXOR, which also can be formulated by using ⊝. Accordingly, the result is
orthogonal:
݂௜ ⊙ ݂௝ ൌ ݂ప ⊕ ݂ఫ
തതതതതതതതത ൌ ݂ሺ1ሻ ⊝ ൫݂௜ ⊕ ݂௝൯ ൌ ݂ሺ1ሻ ⊝ [ሺ݂௜ ∨ ݂௝ሻ ⊝ ሺ݂௜ ∧ ݂௝ሻ] (21)

v
3.3.5. Orthogonalizing OR-ing
Another method, the orthogonalizing OR-ing, which is based on the orthogonalizing difference-
building, is formed in the following. The orthogonalizing OR-ing is a variant of building the
disjunction of two summand-minterms (ܵଵ, ܵଶ) whereby the result is orthogonal. Orthogonalizing
OR-ing is going to be explained by an example in a K-map with 4 variables (Fig. 8). Two
summand-minterms (a group of 8 and a group of 4) are orthogonalizing OR-ed and a result
consisting of several blocks appears; the several blocks are pairwise orthogonal to each other.
Figure 8: Orthogonalizing OR-ing in K-map
The idea out of the K-map is noted as propositional logic which a Boolean form is going to be
derived. By the use of orthogonalizing OR-ing, the intersection set of first and second summand-
minterm (ܵଵ, ܵଶ) is removed from the first summand-minterm ܵଵand the second summand-
minterm ܵଶ is linked by a disjunction to that subtraction:
The Boolean form of orthogonalizing OR-ing of two minterms ݉ௌభ
ൌ ⋀ ‫ݔ‬௡ିௌభ
௡ିଵ
ௌభୀ଴ and ݉ௌభ
ൌ
⋀ ‫ݔ‬௡ିௌమ
௡ିଵ
ௌమୀ଴ is defined by Equation (24), which does not give the value of separate literals, but it
shows whether the value of the variable is complement or not. The indices only indicate the order
of the variables to be calculated:

v
By swapping the two summands in their position, the result changes. Both solutions, however, are
equivalent because the same set is covered. They only differ in the form of coverage which also
be seen in Figure 8, in which both possible solutions are presented. But in order to represent the
orthogonal result with a minimum of minterms to work in the TVL-representation with low
memory request, the summand-minterm with more literals has to be accepted as the first
summand-minterm,because the commutativity applies for and also for ⊝. The following
Equation (25) is used for the orthogonalizing OR-ing of two ternary vectors
ܸܶௌభ
ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ௌభ
and ܸܶௌమ
ൌ [‫ݐ‬௡, ‫ݐ‬௡ିଵ, . . , ‫ݐ‬ଵ]ௌమ
:
The number of the minterms in the result called ݊ corresponds to the number of the variables
presented in the second summand ݉ௌమ
and are not presented in the first summand ݉ௌభ
at the same
time; plus1 for the second summand ݉ௌమ
as the last linked minterm. The number of the possible
results can be charged by ݊!for ݊ > 0. Depending on the starting literal the result may differ.
There are many equivalent options which only differ only in the form of coverage. If both
minterms are disjoint (orthogonal) to each other, the result corresponds to the disjunction of both
minterms:
By the use of orthogonalizing OR-ing , two calculation procedures - OR-ing and subsequent
orthogonalizing - can be performed in one step. That means that the orthogonalizing OR-ing
is the composition of OR-ing (∨) and the subsequent orthogonalization Orth.
It is not necessary to prove this method for general validity because it includes the already general
method of orthogonalization difference-building. The Equations (23) and (2) are equalized to
indicate the equivalence of orthogonalization OR-ing and usual OR-ing:
ܵଵܵଶ
തതത ∨ ܵଶ ൌ ܵଵ ∨ ܵଶ (27)
Due to the axiom of the absorption the equivalence is verified. The right side is the orthogonal
form of the left side which means they only differ in the form of coverage. So, the results of both
sides are equal.

4. CONCLUSION
This work shows that the method of the orthogonalizing difference-building is generally valid and
is also equivalent to the usual method of difference-building. In contrast to the composition,
orthogonalizing difference-building has faster computing time with increasing dimension. In
addition, the method does not require additional memory request for an additional function for
orthogonalization because this method already provides orthogonal results. The orthogonalizing
difference-building is used to calculate the orthogonal difference of two minterms respectively
two TVs, two functions or two TVLs. It is also employed to determine the complement of a
function as well as the EXOR and EXNOR of two functions to achieve an orthogonal result.
Another method, the orthogonalizing OR-ing of two minterms or TVs, is developed out of the
orthogonalizing difference-building. The application of ternary-vector-list is amplified by these
new methods to implement simple and quick elementary functions. Due to the inner
orthogonalization further processing steps in the TVL arithmetic are considerably simplified
because the orthogonal form of disjunctive normal form has the advantage to consider it as
antivalence normal form. Thus, additional calculation such as the differential calculus are
remarkably facilitated.
REFERENCES
[1] Zander, H. J.: Logischer Entwurf binärer Systeme. 3. bearb. Auflage. Berlin, Germany: Verl.Technik,
1989, ISBN 3-341-00526-9.
[2] Bronstein, I.N.; Semendjajew, K.A.; Musiol, G.; Mühlig, H.:Taschenbuch der Mathematik. 7.
vollständig überarbeitete und ergänzte Auflage. Frankfurt am Main, Germany: wissenschaftlicher
Verl. Harri Deutsch GmbH, 2008, ISBN 978-3-8171-2007-9.
[3] Popula, L.: Mathematik für Ingenieure und Naturwissenschaften, Band 1. 13. durchgelesene Auflage.
Wiesbaden, Germany: Viewer + Teubner Verlag | Springer Fachmedien. Wiesbaden GmbH, 2011,
ISBN 978-3-8348-1749-5.
[4] Matthes, W.: Datenzugriffsprinzipien in objektorientierten Rechnerarchitekturen. Preprint.
Technische Universität Karl-Marx-Stadt (Chemnitz), 1989.
[5] Matthes, W.: Spezielle Hardware zur Verarbeitung von Ternärvektorlisten. Dissertation. Technische
Universität Karl-Marx-Stadt (Chemnitz), 1987.
[6] Posthoff, Ch.; Steinbach, B.: Binäre Gleichungen - Algorithmen und Programme.wissenschaftliche
Schriftreiche. Technische Universität Karl-Marx-Stadt (Chemnitz), 1979.
[7] Posthoff, Ch.; Steinbach, B.: Binäre dynamische Systeme - Algorithmen und Programme.
wissenschaftliche Schriftenreihe. Technischen Hochschule Karl-Marx-Stadt (Chemnitz), 1979.
[8] Posthoff, Ch.; Steinbach, B.: Binäre dynamische Systeme. Berlin: Oldenbourg R. Verlag GmbH,
1981, ISBN 348625071X.
[9] Bochmann, D.; Zakrevskij, A.D.; Posthoff, Ch.: Boolesche Gleichungen. Theorie - Anwendungen -
Algorithmen.Berlin: VEB Verlag Technik, 1984, ISBN 3211958150.
[10] Kühnrich, M.: Ternärvektorlisten und deren Anwendung auf binäre Schaltnetzwerke. Dissertation.
Technische Hochschule Karl-Marx-Stadt (Chemnitz), 1979.
[11] Posthoff, C.; Bochmann, D.; Haubold, K.: Diskrete Mathematik. 1. Auflage. Leipzig, DDR: BSB
Teubner, 1986, ISSN 0465-3769.
[12] Kempe, G.: Tupel von TVL als Datenstruktur für Boolesche Funktionen. Dissertation (A). Technische
Universität Bergakademie Freiberg, 2003.
[13] Whitesitt, J.E.: Boolesche Algebra und Ihre Anwendungen. Band 3. Braunschweig, Germany: Friedr.
Vieweg + Sohn GmbH, 1969.
[14] Pomberger, G.; Dobler, H.: Algorithmen und Datenstrukturen - eine systematische Einführung in die
Programmierung. 1. Auflage. München, Germany: Addison-Wesley Verlag, 2008, ISBN-10:
3827372682.
[15] Saake, G.; Sattler, K.-U.: Algorithmen und Datenstrukturen - Eine Einführung mit Java. 3. Auflage.
Heidelberg, Germany: dpunkt.verlag GmbH, 2006, ISBN-10: 3898643859.
[16] Wagenknecht, Ch.: Algorithmen und Komplexität. 1. Auflage. Leipzig, Germany: Carl Hanser Verlag
GmbH & Co. KG, 2003, ISBN-10: 3446223142.

[17] Bochmann, D.: Binäre Systeme. Ein BOOLEAN Buch. Hagen, Germany: LiLoLe-Verlag GmbH,
2006, ISBN 3-934447-10-4.
[18] Crama, Y.; Hammer, P.L.: Boolean Functions. Theory, Algorithms, and Applications.New York,
USA: Cambridge University Press, 2011, ISBN 978-0-521-84751-3.
[19] Kassim,H.; Can, Y.; Sattler, M.S.:Untersuchung eines neuen Algorithmus zur Berechnung
orthogonalisierter Differenz. Bachelor-Thesis, Lehrstuhl für Zuverlässige Schaltungen und Systeme,
Friedrich-Alexander-Universität Erlangen Nürnberg, Germany, 2014.
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(ICCESEN2014), 24-29 October, 2014, Antalya-Turkey.
AUTHORS
Yavuz Canwas born in Erlangen, Germany, in 1979. He received his Diploma (Dipl.-Ing.)
degree in me-chatronics from Friedrich-Alexander-University Erlan-gen-Nürnberg,
Germany, in 2010. He is currently a Research Assistant of Prof. Georg Fischer working
toward his Ph.D. degree in the Institute for Electronics Engineering at Friedrich-Alexander
University in Erlangen. His research interest include orthogonality of Boolean functions and
Ternary-Vector-List.
Georg Fischer was born in Lower Rhine region, Germany, in 1965. He received the
Diploma degree in electrical engineering with focus on communications, micro-wave and
electro-dynamics from RWTH Aachen University, Aachen, Germany, in 1992, and the Dr.-
Ing. degree in electrical engineering from the University of Paderborn, Paderborn,
Germany, in 1997. From 1993 to 1996, he was a Research Assistant with the University of
Paderborn, where he was involved with adaptive antenna array systems for mobile satellite
communications. From 1996 to 2008, he performed research with Bell Laboratories,
Lucent (later Alcatel-Lucent), where he focused on the RF and digital architecture of mobile
communication basestations for global system for mobile communications (GSM), Universal Mobile
Telecommunications System (UMTS), and features for network coverage and capacity enhancements. In
2000, he became a Bell Labs Distinguished Member of Technical Staff (DMTS), and in 2001, a Bell Labs
Consulting Member of Technical Staff (CMTS). He was also a Chairman with the European Telecom-
munications Standards Institute (ETSI) during the physical layer standardization of the GSM-EDGE
system. From 2001 to 2007, he was a Part-Time Lecturer with the University of Erlangen–Nuremberg,
Erlangen, Germany, during which time he lectured on basestation RF and digital technology. Since April
2008, he has been a Professor of electronics engineering with the University of Erlangen–Nuremberg. He
holds over 50 patents concerning microwave and communications technology. His research interests are in
transceiver design, analog/digital partitioning, digital signal processing, con-verters, enhanced amplifier
architectures, duplex filters, metamaterial structures, GaN transistor technology and circuit design, and RF
microelectromechanical systems (MEMS) with specific emphasis on frequency agile, tunable, and
reconfigurable RF systems for software-define radio (SDR) and cognitive radio (CR) applications. His new
research interests concentrate on medical electronics like using microwaves for detection of vital
parameters. Georg Fischer is a Senior Member of the IEEE Microwave Theory and Techniques Society
(MTT-S)/Antennas and Propagation Society (AP-S)/Communications Society (COMSOC)/ Vehicular
Technology Society (VTC) and Engineering in Medicine and Biology Society (EMBS). He is a member of
VDE-ITG and the European Microwave Association (EUMA). He was the co-chair of the European
Conference on Wireless Technology (ECWT) of European Microwave Week Conference (EUMW 2007).
For EUMW 2013, Nuremberg, Germany and GeMiC 2015, Nuremberg, Germany he has served as the
General Technical Program Committee (TPC) chairman.

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