Digital-electronics 简明教程

Logical Expression in SOP and POS Form

在重点介绍 logical expression in SSOP (Standard Sum of Products) form 和 SPOS (Standard Product of Sum) form 之前，我们先简要介绍一下“乘积和”和“和的乘积”形式。

Before focusing on logical expression in SSOP (Standard Sum of Products) form and SPOS (Standard Product of Sum) form, let us have a brief introduction the "Sum of Products" and "Product of Sum" forms.

SOP (Sum of Products) Form

SOP or Sum of Products form 是一种表达式逻辑或布尔表达式的形式。在 SOP 中，输入变量的不同乘积项被逻辑 OR 在一起。因此，在 SOP 形式的情况下，我们首先对输入变量进行逻辑 AND 操作，然后使用逻辑 OR 操作将所有这些乘积项求和。

The SOP or Sum of Products form is a form of expressing a logical or Boolean expression. In SOP, different product terms of input variables are logically ORed together. Therefore, in the case of SOP form, we first logically AND the input variables, and then all these product terms are summed together with the help of logical OR operation.

例如 -

For example −

\mathrm{\mathit{f} \lgroup A,B,C \rgroup \: = \: ABC \: + \: \bar{A}BC \: + \: AB \bar{C}}

这是一个包含三个变量的逻辑表达式。在这里，ABC、A’BC 和 ABC' 是三个乘积项，它们被求和以在 SOP 形式中得到表达式。

This is a logical expression in three variables. Here, ABC, A’BC, and ABC' are the three product terms which are summed together to get the expression in SOP form.

POS (Product of Sum) Form

POS or Product of Sum form 是用于表示逻辑表达式的另一种形式。在 POS 形式中，输入变量的不同和项被逻辑 AND 在一起。因此，如果要以 POS 形式表达一个逻辑表达式，我们首先对所有输入变量进行逻辑 OR 操作，然后使用 AND 操作对这些和项进行 AND 操作。

The POS or Product of Sum form is another form used to represent a logical expression. In POS form, different sum terms of input variables are logically ANDed together. Hence, if we want to express a logical expression in POS form, for that we first logically OR all the input variables and then these sum terms are ANDed using AND operation.

例如 -

For example −

\mathrm{\mathit{f} \lgroup A,B,C \rgroup \: = \: \lgroup A \: + \: B \: + \: C \rgroup \lgroup \bar{A} \: + \: B \: + \: C \rgroup \lgroup A \: + \: B \: + \: \bar{C} \rgroup}

在这里，f 是一个包含三个变量的逻辑表达式。从这个示例中可以看出，有三个和项同时进行 AND 操作，从而得到给定表达式的 POS 形式。

Here, f is a logical expression in three variables. From this example, it can be seen that there are three sum terms which are ANDed together to obtain the POS form of the given expression.

现在，我们详细讨论一下标准乘积和 (SSOP) 形式和标准和的乘积 (SPOS) 形式。

Now, let us discuss the Standard Sum of Products (SSOP) form and Standard Product of Sum (SPOS) form in detail.

一个布尔或逻辑表达式可以表示为两种标准形式，即

A Boolean or logical expression can be represented into two standard forms namely,

  1. SSOP Form

  2. SPOS Form

Standard Sum of Products (SSOP) Form

Standard Sum of Products (SSOP) form 是一种表达逻辑表达式的形式，其中逻辑表达式表示为多个乘积项的和，而每个乘积项将包含所有变量的逻辑表达式，以补码或非补码形式。

The Standard Sum of Products (SSOP) form is a form of expressing a logical expression in which the logical expression is represented as the sum of a number of product terms where each product term will contain all the variables of the logical expression either in complemented or un-complemented form.

由于 SSOP 形式的每个乘积项都包含所有变量，因此它也被称为 Expanded Sum of Products 形式。SSOP 形式也称为 Disjunctive Canonical Form (DCF) 、 Canonical Sum of Products Form 或 Normal Sum of Products Form.

Since, each product term of the SSOP form contains all the variables, hence it is also known as Expanded Sum of Products form. The SSOP form is also known Disjunctive Canonical Form (DCF) or Canonical Sum of Products Form or Normal Sum of Products Form.

通过确定对给定逻辑表达式 (称作 f) 具有值 1 的那些组合所对应的所有项的和，我们可以从真值表中简单得到逻辑表达式的标准乘积和形式。

We can simply obtain the standard sum of products form of a logical expression from the truth table by determining the sum of all the terms that correspond to those combinations for which the given logical expression (say f) has the value 1.

我们还可以使用布尔代数从乘积和 (SOP) 形式得到一个表达式的标准乘积和 (SSOP) 形式。

We can also obtain the standard sum of products (SSOP) form of an expression from the sum of products (SOP) form by using Boolean algebra.

例如，

For example,

\mathrm{\mathit{f} \lgroup A,B,C \rgroup \: = \: A \bar{B} \: + \: B \bar{C}}

这是一个包含三个变量的逻辑表达式，但它是以 SOP 形式表示的。我们可以使用布尔代数以如下方式将此表达式转换为 SSOP 形式。

This is a logical expression in three variables, but it is expressed in SOP form. We can convert this expression into SSOP form using Boolean algebra as follows.

\mathrm{\mathit{f}\lgroup A,B,C \rgroup \: = \: A \bar{B} \lgroup C \: + \: \bar{C} \rgroup \: + \: B \bar{C} \lgroup A \: + \: \bar{A} \rgroup}

\mathrm{\mathit{f}\lgroup A,B,C \rgroup \: = \: A \bar{B}C \: + \: A \bar{B} \: \bar{C} \: + \: AB \bar{C} \: + \: \bar{A}BC}

这是给定的逻辑表达式的标准乘积和形式。我们比较容易发现, 在 SSOP 形式中, 每个乘积项都包含逻辑函数的所有变量, 无论是补码形式还是非补码形式。这些乘积项中的每一个都称为 minterm 。一个具有 n 个变量的逻辑函数或表达式可以最多有 2n 个极小项。值为 1 的逻辑表达式的极小项之和称为表达式的标准乘积和形式。

This is the Standard Sum of Products form of the given logical expression. We can notice that in the SSOP form, each product term contains all the variables of the logic function either in complemented or un-complemented form. Each of these product terms is called a minterm. A logical function or expression in ‘n' variables can have maximum 2n minterms. The sum of minterms of a logical expression whose value is 1 is called the standard sum of products form of the expression.

Standard Product of Sum (SPOS) Form

Standard Product of Sums (SPOS) form 是表示逻辑函数的一种形式, 其中逻辑表达式表示为多个和项的乘积, 其中每个和项都将包含逻辑表达式的所有变量, 无论是补码形式还是非补码形式。

The Standard Product of Sums (SPOS) form is a form of expressing a logical function in which the logical expression is represented as the product of a number of sum terms where each sum term will contain all the variables of the logical expression either in complemented or un-complemented form.

SPOS 形式也称为 Conjunctive Canonical Form (CCF) or Expanded Product of Sums Form 或 Normal Product of Sums Form 或 Canonical Product of Sums Form 。

SPOS form is also known as Conjunctive Canonical Form (CCF) or Expanded Product of Sums Form or Normal Product of Sums Form or Canonical Product of Sums Form.

每个术语的 SPOS 形式都是通过考虑输出等于 0 的变量组合得出的。每个术语都是表达式的所有变量的总和。

The SPOS form of each term is derived by considering the combinations of variables for which the output is equal to 0. Each term is a sum of all the variables of the expression.

在 SPOS 形式中, 如果变量在组合中取值为 1, 则该变量以其补码形式出现；如果变量在组合中取值为 0, 则该变量以非补码形式出现。

In the SPOS form, a variable appears in its complemented form if it has a value of 1 in the combination, and it appears in un-complemented form if it has a value of 0 in the combination.

在标准和积形式的情况下, 其中一个包含函数中每个 n 个变量（以补码形式或非补码形式）的术语称为 maxterm 。在 n 个变量的逻辑函数中, 最多可以有 2n 个极大项。值为 0 的逻辑表达式的极大项乘积称为表达式的标准和积形式。

In the case of standard product of sums form, a term which contains each of the n variables of the function in either complemented or un-complemented form is called a maxterm. For a logical function in n variables, there could be at the most 2nmaxterms. The product of maxterms of a logical expression whose value is 0 is called the standard product of sums form of the expression.

类似于 SSOP 形式, 我们可以通过确定满足给定逻辑表达式（假设为 f）值为 0 的变量组合的所有和项的乘积, 从逻辑表达式的真值表中获得标准和积形式。

Similar to the SSOP form, we can obtain the standard product of sums form from the truth table of the logical expression by determining the product of all the sum terms that correspond to those combinations of variables for which the given logical expression (say f) has the value equal to 0.

此外, 逻辑表达式的 SPOS 形式可以通过使用布尔代数获得。

Also, the SPOS form of a logical expression can be obtained by using Boolean algebra.

例如，

For example,

\mathrm{\mathit{f} \lgroup A,B,C \rgroup \: = \: \lgroup \bar{A} \: + \: B \rgroup \: + \: \lgroup A \: + \: \bar{C} \rgroup}

这是一个具有三个变量的逻辑表达式, 但它以 POS 形式表示。我们可以使用布尔代数将此表达式转换成 SPOS 形式, 如下所示。

This is a logical expression in three variables, but it is expressed in POS form. We can convert this expression into SPOS form by using Boolean algebra as follows.

\mathrm{\mathit{f} \lgroup A,B,C\rgroup \: = \: \lgroup \bar{A} \: + \: B \: + \: C \bar{C} \rgroup \: + \: \lgroup A \: + \: \bar{C} \: + \: B \bar{B} \rgroup}

\mathrm{\mathit{f} \lgroup A,B,C \rgroup \: = \: \lgroup \bar{A} \: + \: B \: + \: C \rgroup \lgroup \bar{A} \: + \: B \: + \: \bar{C} \rgroup \lgroup A \: + \: B \: + \: \bar{C} \rgroup \lgroup A \: + \: \bar{B} \: + \: \bar{C} \rgroup}

这是给定逻辑表达式的标准和积 (SPOS) 形式。在这里, 我们注意到, 在 SPOS 形式中, 每个和项都包含逻辑函数的所有变量, 无论是补码形式还是非补码形式。

This is the Standard Product of Sums (SPOS) form of the given logical expression. Here, we can note that in the SPOS form, each sum term contains all the variables of the logic function either in complemented or un-complemented form.