Question

B.16 [30] <§§B.2, B.3> Give an algorithm for constructing the sum-of- products representation for an arbitrary logic equation consisting of AND, OR, and NOT. The algorithm should be recursive and should not construct the truth table in the process.

Short answer

The required algorithm:

A=B’: This is for some equation of “B”. After that, it is needed to find the “sum-of-products” of the “B”, which is denoted by the “$\tilde{\text{B}}$”. Hence the outcome will be $\text{A=}\widetilde{\text{B'}}$.

A=B+C: It is needed to find the “sum-of-products” of the “B” and “C”, which is denoted by the “ $\tilde{\text{B}}$”. Hence the outcome will be$\text{A=}\tilde{\text{B}}\text{+}\tilde{\text{C}}$ .

A=B.C: After that, it is needed use $\mathrm{A}=\mathrm{A}\text{'}\text{'}=(\mathrm{A}\text{'}+\mathrm{B}\text{'})\text{'}$ it is needed to find the “sum-of-products” of the “B’ ” and “C’ ”. Hence the outcome will be obvious.

Finally, it can be used as an example X’’=X for simplifying the required outcome.

In the above algorithm, “A” is initialized as some equation of logic.

Step-by-step solution

Define the concept.

“A” is initialized as some equation of logic.

• A=B’: This is for some equation of “B”. After that, it is needed to find the “sum-of-products” of the “B”, which is denoted by the “ $\tilde{\text{B}}$”. Hence the outcome will be.

• A=B+C: It is needed to find the “sum-of-products” of the “B” and “C”, which is denoted by the “ $\tilde{\text{B}}$”. Hence the outcome will be $\text{A=}\tilde{\text{B}}\text{+}\tilde{\text{C}}$.

• A=B.C: After that, it is needed use $\mathrm{A}=\mathrm{A}\text{'}\text{'}=(\mathrm{A}\text{'}+\mathrm{B}\text{'})\text{'}$ It is needed to find the “sum-of-products” of the “B’ ” and “C’ ”. Hence the outcome will be obvious.

Determine the calculation.

Let “A” is referred to as some equation of logic.

Hence the three possibilities are there.

A=B’: This is for some equation of “B”. After that, it is needed to find the “sum-of-products” of the “B”, which is denoted by the “$\tilde{\text{B}}$”. Hence the outcome will be $\text{A=}\widetilde{\text{B'}}$.

A=B+C: It is needed to find the “sum-of-products” of the “B” and “C”, which is denoted by the “$\tilde{\text{B}}$”. Hence the outcome will be$\text{A=}\tilde{\text{B}}\text{+}\tilde{\text{C}}$ .

A=B.C: After that, it is needed use $\mathrm{A}=\mathrm{A}\text{'}\text{'}=(\mathrm{A}\text{'}+\mathrm{B}\text{'})\text{'}$ it is needed to find the “sum-of-products” of the “B’ ” and “C’ ”. Hence the outcome will be obvious.

Finally, it can be used as an example X’’=X for simplifying the required outcome.