# Logical Expression in SSOP and SPOS Form

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

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.

For example −

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

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

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.

For example −

$$\mathrm{\mathit{f}\lgroup A,B,C\rgroup=\lgroup A+B+C\rgroup\lgroup\overline{A}+B+C\rgroup\lgroup A+B+\overline{C}\rgroup}$$

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.

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

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

• SSOP (Standard Sum of Products) Form

• SPOS (Standard Product of Sum) Form

## Standard Sum of Products (SSOP) Form

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.

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.

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.

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.

For example,

$$\mathrm{\mathit{f}\lgroup A,B,C\rgroup=A\overline{B}+B\overline{C}}$$

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.

$$\mathrm{\mathit{f}\lgroup A,B,C\rgroup=A\overline{B}\lgroup C+\overline{C}\rgroup+B\overline{C}\lgroup A+\overline{A}\rgroup}$$

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

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

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 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.

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.

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.

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.

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.

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

For example,

$$\mathrm{\mathit{f}\lgroup A,B,C\rgroup=\lgroup \overline{A}+B\rgroup+\lgroup A+\overline{C}\rgroup}$$

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.

$$\mathrm{\mathit{f}\lgroup A,B,C\rgroup=\lgroup \overline{A}+B+C\overline{C}\rgroup+\lgroup A+\overline{C}+B\overline{B}\rgroup}$$

$$\mathrm{\mathit{f}\lgroup A,B,C\rgroup=\lgroup \overline{A}+B+C\rgroup\lgroup \overline{A}+B+\overline{C}\rgroup\lgroup A+B+\overline{C}\rgroup\lgroup A+\overline{B}+\overline{C}\rgroup}$$

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.

### Conclusion

This is all about the Standard Sum of Products (SSOP) form and Standard Product of Sums (SPOS) form in Boolean algebra. From the above discussion, we may conclude that the SSOP and SPOS forms are the forms to express a logical expression in the expanded form, where each term (product term or sum term) will contain all the variables of the logical expression.

Updated on: 03-Oct-2023

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