Boolean Algebra And Its Applications

Transcription

1 oolean lgebra nd Its pplications Introduction Let Ω be a set consisting of two elements denoted by the symbols 0 and 1, i.e. Ω = {0, 1}. Suppose that three operations has been defined: the logical sum + : Ω Ω Ω, the logical multiplication : Ω Ω Ω and the complementation Ω Ω conditions are satisfied: _ : in a way that the following The set Ω with the operations of addition, multiplication and complementation mentioned above is called boolean algebra. One of the possible implementations of boolean algebra are electronic circuits called gates. They operate on one, two or more input signals and produce uniquely defined output signals. mount the simplest and most frequently used gates are the gate, the ND gate and the Inverter. They are designed to perform the addition, multiplication or complementation on input signals, respectively: Z = + Z = Z = G I Theorem For arbitrary elements,, Z of boolean algebra Ω the following equalities are valid: (1) 0+ = ; (2) 1 + = 1; (3) + = ; (4) + = 1; (5) 0 = 0; (6) 1 = ; (7) = ; (8) = 0; (9) ( ) = ; (10) + = + ; (11) + ( + Z) = ( + ) + Z ; 1

2 (12) = ; (13) ( Z) = ( ) Z ; (14) ( + Z) = ( ) + ( Z); (15) + ( Z) = ( + ) ( + Z); (16) ( ) = ; + (de Morgan s law) (17) ( ) = + ; (de Morgan s law). The above properties may be proved by perfect induction called also 0-1 method applying so-called truth tables. s an example we shall verify the equality (11): Z +Z +(+Z) + (+)+Z In view of the equalities (11) and (13) the addition and multiplication in boolean algebra can be generalized on the case of any finite number of arguments. It is done by means of the mathematical induction. Namely, if n > 2, we put = ; 1 1 n n... = ( 1 n 1) (... ). 1 In the same way the and ND gates are generalized on cases of any finite number of input signals. Traditionally, to shorten notation, we apply the same rules concerning the order of operations and usage of parentheses or brackets as in ordinary algebra. lso it is allowed to write instead of, etc. Theorem For arbitrary elements,, Z of boolean algebra Ω the following equalities are valid: (18) + Z = ; (19) ( + Z) = ; (20) + = + ; (21) + Z+ Z = + Z. Proof. We could apply the 0-1 method but there is a shorter way. We shall use the properties (1) - (17) stated in the previous theorem. The equality (18): + Z = 1+ Z = ( 1+ Z) = 1=. The equality (19): (18) ( + Z) = ( ) + ( Z) = + Z. = The equality (20): = ( + ) = ( + ) ( + ) = = + 1 (18) ( + ) + ( 0+ ) = +. The equality (21): + Z+ Z = + Z+ Z = + Z + = + Z 1= + Z n 1 ( ) ( ). n n 2

3 Remark. It can be proved that any expression of boolean algebra can be transformed to any of two possible so-called canonical forms: the sum of products and the product of sums. We assume here that a single term can be considered as a one argument sum or as a one argument product. Therefore the expressions + and are in the form of sum of products as well as in the form of product of sums. xamples Let us simplify two boolean algebra expressions. a) W = ( + )( + )( + Z) = ( )( + Z) = ( + + )( + Z) = + Z+ + Z+ + Z = Z+ ( Z+ Z) = Z+ Z ( + ) = Z+ Z = Z. The other, simpler method of reduction: W = ( + )( + ) ( + Z) = ( + )( + Z) = ( + Z) = + Z = Z ( ). b) V = ( Z+ Z+ Z ) = ( ( Z+ Z) + ( Z+ Z )) = ( Z ( + ) + ( Z+ Z )) = ( Z 1+ 1) = ( + Z) or V = + Z. The first result is in the canonical form of product of sums, the second in the canonical form sum of products. We can put a question: which one solution is simpler? The answer becomes obvious if we look at the realizations of both by means of gates: Z G It is evident that the left solution is better because its implementation requires only two gate whereas the right one three gates. n important consequence of that is that if we apply the first solution to an electronic circuit, it will cost less. The next problem important from informatics point of view is how to design boolean algebra expressions that satisfied required input-output conditions. We shall discuss it by means of few examples. xample Let us find an expression dependent on input variables, that returns an output variable Z according to conditions described by the table: Input Z If we are looking for a solution in the form of sum of products, we append an additional column consisting of products as follows: Z G G 3

4 Input Z Products The required conditions satisfies the sum of that products from the last column for which the output of Z is 1: Z = + +. We can simplify the above expression: Z = ( + ) + ( + ) = ( + ) + ( + ) = + = +. To get a solution in the form of product of sums, we append an additional column consisting of sums as follows: Input Z Sums The solution we are looking for is equal to the product of that sums from the last column for which the output of Z is 0: Z = +. We notice that we have obtained the same result though methods we applied were different. The representation of the derived expression by the gate network can be presented in one of the following forms: I lub xample Let us find an expression dependent on input variables,, Z that returns an output variable according to conditions described by the table: Input Z We are to find solutions in both canonical forms so we append two columns: 4

5 Input Z Products Sums Z * + + Z Z + + Z * Z * + + Z Z + + Z * Z * + + Z Z + + Z * Z * + + Z Z + + Z * The solution in the form of sum of products: = ( Z + Z ) + ( Z + Z ) = Z ( + ) + Z ( + ) = Z + Z = Z ( + ) = Z. The solution in the form of product of sums: = ( + + Z )( + + Z ) ( + + Z )( + + Z ) = ( + Z ) + ( ) ( + Z ) + ( ) ( )( ) ( )( )= ( + Z )( + Z ) = Z + ( ) = Z. lso in this example the final form of result is the same. Its implementation requires one inverter only: Z I = Z xample Let us find an expression dependent on input variables,, Z that returns an output variable according to conditions described by the table: We derive solutions in both canonical forms: Input Z

6 Input Z Iloczyny Sumy Z + + Z * Z + + Z * Z * + + Z Z * + + Z Z + + Z * Z + + Z * Z * + + Z Z + + Z * The solution in the form of sum of products: = ( Z + Z) + ( Z + Z ) = ( Z + Z) + Z ( + ) = + Z. The solution in the form of product of sums: = ( + + Z)( + + Z ) ( + + Z)( + + Z ) ( + + Z )( + + Z ) ( )( )( )= (( + ) + ( Z Z ))(( + ) + Z Z)( ( + Z ) + ) = (( + )( + ))( + Z ) = ( + ) ( + Z ) = ( + Z ). We notice that the final solutions are not the same and the second is simpler. It will be evident if we look at implementations of them by means of gates: Z G G Z G part from three gates that have been introduced before two other gates, i.e. NND gate and N gate are often used in the computer industry. They are defined as follows: N Z = NND Z = + Z = ( + ) = Z = ( ) = Remark. The importance of the NND and N gates is due to the fact that any boolean expression can be realized by means of the NND gates only or by the N gates only. This is a consequence of the following schemes: 6

7 N gates only N = N N ( ) = + N N N ( ) ( ) = NND gates only NND + = + NND NND ( + ) = NND NND NND ( ) + ( ) = + In practice there is no need to apply the above schemes. s was mentioned earlier, all boolean algebra expressions can be transformed to one of two canonical forms: the sum of products or the product of sums. In the first case if we replace all and ND gates by NND gates, instead of the solution in the form ND-to- we shall obtain the logically equivalent solution in the form NNDto-NND. In the second case the replacement of and ND gates by N gates gives the logically equivalent solution in the form N-to-N. Notice that the logical equivalence does not mean that the related circuits are physically the same. xamples a) G C D G D + D+ F G F G G NND + C D NND C + D NND ( + ) + ( C + D) + ( + F + G) = + D+ F G F G NND + F + G 7

8 b) G + D D C G D NND ( ) + ( ) = + NND ( + ) + ( C + D+ ) = + D D C NND ( D) + = C + D C + + c) + C D C+ D G ( ) ( C D) ( F G ) F G + F+ G N C D N D N ( ) ( D) ( F G) = ( + ) ( C+ D) ( + F+ G) d) F G N F G + G ( + ) ( C+ D+ ) D C C + D+ 8

9 N ( ) ( ) = N ( ) ( C D ) = ( + ) ( C+ D + ) D C N ( D) = D Integrated Circuits It is very common in the computer industry that implementations of boolean algebra expressions are realized by integrated circuits in the form of containers consisting of a certain number of gates. The communication between gates is realized by means of external connections of pins related to input and output lines of that gates. Usually printed-circuits boards equipped with metallic strips are in use. xample More advanced method of realization of gate networks applies internal connections. The process of finalization of an integrated circuit is realized by the procedure of so-called programming. 9

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