布尔表达式和正则表达式_布尔表达式约简的对偶原理和规则

布尔表达式和正则表达式

对偶原理 (Duality Principle)

According to this principle, if we have postulates or theorems of Boolean Algebra for one type of operation then that operation can be converted into another type of operation (i.e., AND can be converted to OR and vice-versa) just by interchanging '0 with 1', '1 with 0', '(+) sign with (.) sign' and '(.) sign with (+) sign'. This principle ensures if a theorem is proved using postulates of Boolean algebra, then the dual of this theorem automatically holds and we need not prove it again separately.

根据此原理，如果我们为一种类型的运算提供了布尔代数的假设或定理，则只需将“ 0”与“-”互换即可将其转换为另一种类型的运算(即， AND可以转换为OR，反之亦然 ) 。 1' ， '1与0' ， '(+)与(。)符号'和'(。)与(+)符号' 。这个原理确保了，如果使用布尔代数的假设来证明一个定理，那么该定理的对偶就会自动成立，我们不需要再次证明它。

Some Boolean expressions and their corresponding duals are given below,

下面给出了一些布尔表达式及其对应的对偶，

Given Expression	Dual	Given Expression	Dual
0 = 1	1 = 0	A. (A+B) = A	A + A.B = A
0.1 = 0	1 + 0 = 1	AB = A + B	A+B = A.B
A.0 = 0	A + 1 = 1	(A+C) (A +B) = AB + AC	AC + AB = (A+B). (A+C)
A.B = B. A	A + B = B + A	A+B = AB + AB +AB	AB = (A+B).(A+B).(A+B)
A.A = 0	A + A = 1	AB + A + AB = 0	((A+B)).A.(A+B) = 1
A. (B.C) = (A.B). C	A+(B+C) = (A+B) + C

给定表达	双	给定表达	双
0 = 1	1 = 0	A.(A + B)= A	A + AB = A
0.1 = 0	1 + 0 = 1	AB = A + B	A + B = A。乙
A.0 = 0	A +1 = 1	(A + C)( A + B)= AB + A C	AC + A B =(A + B)。 (A + C )
AB =B。	A + B = B + A	A + B = AB + A B + A B	AB =(A + B)。( A + B)。(A + B )
A. A = 0	A + A = 1	AB + A + AB = 0	(( A + B ))。 A。 (A + B)= 1
答：(BC)=(AB)。 C	A +(B + C)=(A + B)+ C

减少布尔表达式 (Reducing Boolean Expressions)

Every Boolean expression must be reduced to its simplest form before realizing it because each logic operation in the expression is carried out using hardware. Thus, realizing the simplest expression requires less circuitry hence reduces the cost of the system. Also, it is highly reliable and less complex in nature. For reducing the Boolean expression, we use the axioms and laws of Boolean algebra (see them in our previous article). Some instructions for reducing the given Boolean expression are listed below,

在实现每个布尔表达式之前，必须将其简化为最简单的形式，因为表达式中的每个逻辑运算都是使用硬件执行的。因此，实现最简单的表达需要较少的电路，从而降低了系统成本。而且，它是高度可靠的并且本质上不太复杂。为了简化布尔表达式，我们使用布尔代数的公理和定律(请参阅上一篇文章中的内容)。下面列出了一些用于简化给定布尔表达式的说明，

Remove all the parenthesis by multiplying all the terms if present.

通过将所有项相乘(如果存在)来删除所有括号。
Group all similar terms which are more than one, then remove all other terms by just keeping one.

将所有不止一个的相似术语归为一组，然后仅保留一个就删除所有其他术语。

Example: ABC + AB +ABC + AB = ABC +ABC + AB +AB = ABC +AB

例如：ABC + AB + ABC + AB = ABC + ABC + AB + AB = ABC + AB
A variable and its negation together in the same term always results in a 0 value, it can be dropped.

在同一术语中，变量及其取反的结果始终为0，可以将其删除。

Example: A. BCC + BC = A. 0 + BC = BC

例如：A. BC C + BC = A. 0 + BC = BC
Look for the pair of terms that are identical except for one variable which may be missing in one of the terms. The larger term can be dropped.

寻找一对相同的术语，只是其中一个术语可能缺少一个变量。较大的项可以删除。

Example: ABCD + ABC = ABC(D + 1) = (ABC).1 = ABC

例如：AB CD + AB C = AB C ( D + 1)=(AB C ).1 = AB C
Look for the pair of terms that have the same variables, with one or more variables complimented. If a variable in one term of such a variable is complimented while in the second term it is not, then such terms can be combined into a single term with that variable dropped.

查找具有相同变量的一对术语，并补充一个或多个变量。如果在此变量的一个术语中对一个变量进行了补充，而在第二个术语中则没有，则可以将这些术语合并为一个单独的变量，并删除该变量。

Example

例

ABCD + ABCD = ABC(D + D) = (ABC).1 = ABC
OR
AB(C+D) + AB(C+D) = AB [(C+D) + (C+D)] = AB. 1 =AB

AB CD + AB C D = AB C ( D + D)=(AB C ).1 = AB C
要么
AB(C + D)+ AB (C + D) = AB [(C + D)+ (C + D) ] = AB。 1 = AB

翻译自: https://www.includehelp.com/basics/duality-principle-and-rules-for-reduction-of-boolean-expressions.aspx

布尔表达式和正则表达式