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4.4 Boolean Arithmetical Equations

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4.4 Boolean Arithmetical Equations •3rd numerical part: It is the ordinal number, represented by the ordinality, ω = “ nnn;@XX−121... XX ” of the Boolean Arithmetical Field (BAFi). Then, this Numerical Transform (NT) of the given Boolean Function, has the following notation: "NTfXX<A16 ,− ,..., XX ,= ααααn ... .;@ nXX ; − ... XX " nn121 21− 2102 nn121 where we have, from TABLE 2: α = 16 21n − f 11, ,... 11 , ... α2 = f1600, ,... 10 , α1 = f1600, ,... 01 , α = 0 f1600, ,... 00 , NOTE: As we have seen in Section 3.5 of Chapter 3, this abscissa of the NT can be obtained by the following modes: 1st MODE: through a Truth Table, as seen above; 2nd MODE: through the application of the distributive properties of the NTs into the literal expressions of the given Boolean Function we get the Boolean Arithmetical Variables (BAVs) of the Boolean Arithmetical Field (BAFi) previously chosen. 3rd MODE: through the “Doubling Operators” which applies the (4), (6) and (7) Properties of the Numerical Transforms of Section 3.3 of Chapter 3. 4.4 BOOLEAN ARITHMETICAL EQUATIONS A formal Boolean Equation System is a logical interrelation between two Boolean Functions of any given variables. This logical interrelation will be a “logical implication” that means in linguistics a relation between premises and its conclusions, as an active voice (sufficient condition) or passive voice (necessary condition), or both (necessary and sufficient conditions). To simplify the visualisation of these logical interrelations between Boolean Functions, the Boolean Algebra symbols were adopted here. One should bear in mind that the properties of the isomorphism guarantee that these interrelations are maintained in Boolean Arithmetic. 104 These logical interrelations in Boolean Functions are represented by the following five fundamentals symbols: "Symbol 1": X' (read "not X") stands for the opposite of X, i.e., if X is true then X' is false and vice versa. "Symbol 2": X & Y (read "X and Y") should be considered true if and only if both X and Y are true. This symbol shall also be called the "logical conjunction of X and Y", that is, both X and Y should be understood as occurring at the same time ("conjunction time"). "Symbol 3": X v Y (read "X or Y, or both") should be considered true if and only if at least one of the two sentences, X or Y, is true. This symbol is also called "inclusive or" ("in-or"), and it means that both X and Y can be true simultaneously, i.e., X or Y should be understood as occurring in different times, or even simultaneously, if one of them is a particular case of the other ("disjunction time"). This symbol has the meaning of the inclusive "or" (or "ïn-or"), in the sense of the Latin "vel" ("X or Y, or both") and in forensic language, is sometimes expressed by the barbarism "and/or". "Symbol 4": X → Y (read "if X then Y"; or "X only if Y"; or "X is a sufficient condition for Y"; or "a necessary condition for X is Y"; or "in order that X it is necessary that Y") should be considered false if and only if X is true and Y is false. This symbol is also called "logical implication" and in Boolean Arithmetical Equations studies, it is to be taken to indicate a relation of premise and conclusion or of cause and effect. Tautologically this symbol has a passive voice, that is, a reverse expression, "Y ← X" (read "Y if X"; or "Y is a necessary condition for X"; or "a sufficient condition for Y is X"; or "in order that Y it is sufficient that X"; or "Y is implicated by X"). The symbol "Y ← X" stands for the sentence, which is false, if and only if X is true and Y is false. Therefore, in both cases, "X → Y" and "Y ← X", the truth of Y can be inferred from that of X. "Symbol 5": X = Y (read "X logically equal to "Y") stands for the sentence which is true if and only if X and Y are both true or both false. Therefore, X and Y have the same truth value. This expression, "X = Y", also means that also means that both have either true or false values and should occur at the same time. It means, in reality, that the following two expressions occur simultaneously: X → Y and X ← Y, or, 105 (X → Y) & (X ← Y) ≡ (X = Y), and vice-versa: (X = Y) ≡ (X → Y) & (X ← Y) In Logical Linguistics these last two expressions justify the classical notation, "...if and only if..." ("...if..."), which means that the reciprocal of a true assertion is also true. Observations: 1st : There is another symbol, called "exclusive-or" ("ex-or") that means "X or Y, but not both" ("X,or,Y"), in the sense of Latin "aut", that can be expressed by a combination of two of the fundamental symbols, i.e., the negation ("symbol 1") with the logical equality ("symbol 5"), i. e., (X = Y)', read: "either X,or,Y". This combination of symbols stands for the sentence which is true, if and only if, X and Y never have the same true value. In that case we can also read "X,or,Y" ("...,or,..."), that is, "X" or "Y" should be understood as occurring in different times, but never simultaneously. It should be also understood that neither of them is a particular case of the other. 2nd Since logical usage varies regarding the terms "sum" and "product" we are in accordance with Hilbert et al. [01], so these expression are avoided. Then, we shall call "X & Y" the conjunction of "X" and "Y" (i.e. "sum" or "logical addition"), and "XvY" the disjunction of "X" and "Y" (i.e. "product" or "logical product"). HHH Any Boolean Equation of "n" variables "{Xn, ^−1,..., 2, 1)" , refers to all possible logical interrelations between two Boolean Functions of these same variables, that is, the separation of these functions into two members mainly by one of the following symbols: "...→...", or, "...←..." ("Symbol 4"), or, "...=..." ("symbol 5"). Therefore, we have the following three types of Boolean Equations: → (1) fjn(X,... ,X2,X1) gjn(X,... ,X2,X1), or ← (2) fjn(X,... ,X2,X1) gjn(X,... ,X2,X1), or = (3) fjn(X,... ,X2,X1) gjn(X,... ,X2,X1) 106 → In case (1), when ""fgjj, the function of the first member ""f j (minoring function) implies the function of the second member ""g j (majoring function). Then, if we take hindrance as the unit of logical function, the following expression is obtained: WV= ()4 fj & ZZ where "&" means the logical addition operation (the "and" operation). Case (1) means that all values of ""f j are within the set of values of ""g j and therefore the result of their logical addition, ""f j , belongs to both functions simultaneously (see expression (4)). In order to find a new classical expression logically equivalent to expression (4), ' the multiplication of both members ""f j and ""g j by "(f j )" , which is the dual of ""f j , must be carried out. Thus: ' ' ' (5) (fj ) v(fjj ) & (f )v(gjj) = (f )v(fj ) ' But as (f j)v(f j) = 0 , then we have: ' (6) (fj )v(gj ) = 0 Therefore, the following Boolean Equation logically equivalent to expression (1) is obtained: ' (7) [fjn (X ,...,X 21 ,X )] v gjn (X ,...,X 2 ) = 0 ← In case (2), when "fj gj ", the first member function " f j " (minoring function) is implied in the second member function " g j " (majoring function). Then, if hindrance is taken as the unit of logical function, the following expression is obtained: (8) fj & gj = gj Case (2) means that all values of “ g j ” are within the set of values of “ f j ” and therefore the result of their logical addition, “ g j ”, belongs to both functions simultaneously (see (8)). In order to find a new classical expression logically equivalent to expression ' (8), the multiplication of both members " f j " and " g j " by ( gj ), which is the dual of " g j ", must be carried out. Thus: ' ' ' (9) (f j )v(g j ) & (g j)v(g j) = (g j )v(g j) 107 ' But, as (gjj )v(g ) = 0, then we have: ' (10) (fjj )v(g ) = 0, Therefore, the following Boolean Equation logically equivalent to expression (2) is obtained: ' (11) fjn (X ,...,X 21 ,X ) v [g jn(X ,...,X 21 ,X )] = 0 In case (3), that is, when "fj = gj", the first member function " f j " is logically W equal to the second member function " Z ", and conversely. In this case, we have both cases described above, (1) and (2), at the same conjunction time, that is: fj = gj, is logically equal to the following: → ← fj gj, and conversely, fj gj, at the same time; Therefore, according to (6) and (10), we have: →∨=' (13) fjjggjj { (f )()0 (12) fjj = g and ←∨=' (14) fjjggjj { (f )()0 In order to obtain the classical expression (12) the logical addition of both expressions (13) and (14) must be carried out. Thus: ' ' (15) (fj' ) v (gjjj) & (f ) v (g ) = 0 or, ' ' (16) [fjn (X ,...,X 1 )] v gjn(X ,..,X 1 ) & f jn (X ,...,X 1 ) v [g jn(X ,...,X 1 )] = 0 As a conclusion, any Boolean Equation of types (1), (2) or (3) could be reduced to their corresponding logically equivalent Classical Boolean Equation. NOTE: As it will be seen in Rudeanu [02], he came across Boolean Equations in a work of his teacher, Prof. Gr. C. Moish. In the beginning of the Preface, Prof. Moisch, tells us some opportune historical words about Boolean Functions and that subject: “Boolean functions are those functions with arguments and values in a Boolean algebra which are defined by Boolean expressions, i.e., by expressions built up by superpositions starting from indeterminates and 108 using the Boolean connectives “U”, “.”, “ ’ ”.
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