Petrick's method

"Branch-and-bound method" redirects here and is not to be confused with Branch and bound.

In Boolean algebra, Petrick's method^[1] (also known as Petrick function^[2] or branch-and-bound method) is a technique described by Stanley R. Petrick (1931–2006)^[3][4] in 1956^[5][6] for determining all minimum sum-of-products solutions from a prime implicant chart.^[7] Petrick's method is very tedious for large charts, but it is easy to implement on a computer.^[7] The method was improved by Insley B. Pyne and Edward Joseph McCluskey in 1962.^[8][9]

Algorithm

1. Reduce the prime implicant chart by eliminating the essential prime implicant rows and the corresponding columns.^[7]
2. Label the rows of the reduced prime implicant chart ${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$, ${\displaystyle P_{3}}$, ${\displaystyle P_{4}}$, etc.^[7]
3. Form a logical function ${\displaystyle P}$ which is true when all the columns are covered. P consists of a product of sums where each sum term has the form ${\displaystyle (P_{i0}+P_{i1}+}$${\displaystyle \cdots }$${\displaystyle +P_{iN})}$, where each ${\displaystyle P_{ij}}$ represents a row covering column ${\displaystyle i}$.^[7]
4. Apply De Morgan's Laws to expand ${\displaystyle P}$ into a sum of products^{[nb 1]} and minimize by applying the absorption law ${\displaystyle X+XY=X}$.^[7]
5. Each term in the result represents a solution, that is, a set of rows which covers all of the minterms in the table. To determine the minimum solutions, first find those terms which contain a minimum number of prime implicants.^[7]
6. Next, for each of the terms found in step five, count the number of literals in each prime implicant and find the total number of literals.^[7]
7. Choose the term or terms composed of the minimum total number of literals, and write out the corresponding sums of prime implicants.^[7]

Example of Petrick's method

Following is the function we want to reduce:^[10]

${\displaystyle f(A,B,C)=\sum m(0,1,2,5,6,7)=A'B'C'+A'B'C+A'BC'+AB'C+ABC'+ABC}$

The prime implicant chart from the Quine-McCluskey algorithm is as follows:

	0	1	2	5	6	7	⇒	A	B	C
K = m(0,1)							⇒	0	0	—
L = m(0,2)							⇒	0	—	0
M = m(1,5)							⇒	—	0	1
N = m(2,6)							⇒	—	1	0
P = m(5,7)							⇒	1	—	1
Q = m(6,7)							⇒	1	1	—

Based on the ✓ marks in the table above, build a product of sums of the rows. Each column of the table makes a product term which adds together the rows having a ✓ mark in that column:

 (K+L)(K+M)(L+N)(M+P)(N+Q)(P+Q)

Use the distributive law to turn that expression into a sum of products. Also use the following equivalences to simplify the final expression: X + XY = X and XX = X and X + X = X

 = (K+L)(K+M)(L+N)(M+P)(N+Q)(P+Q)
 = (K+LM)(N+LQ)(P+MQ)
 = (KN+KLQ+LMN+LMQ)(P+MQ)
 = KNP + KLPQ + LMNP + LMPQ + KMNQ + KLMQ + LMNQ + LMQ

Now use again the following equivalence to further reduce the equation: X + XY = X

 = KNP + KLPQ + LMNP + LMQ + KMNQ

Choose products with fewest terms, in this example, there are two products with three terms:

 KNP
 LMQ

Referring to the prime implicant table, transform each product by replacing prime implicants with their expression as boolean variables, and substitute a sum for the product. Then choose the result which contains the fewest total literals (boolean variables and their complements). Referring to our example:

KNP    expands to    A'B' + BC' + AC   where K converts to A'B', N converts to BC', etc.
LMQ    expands to    A'C' + B'C + AB

Both products expand to six literals each, so either one can be used. In general, application of Petrick's method is tedious for large charts, but it is easy to implement on a computer.^[7]

Notes

1. This causes exponential blow-up in the number of columns: expanding the product of ${\displaystyle k}$ sums that have ${\displaystyle n_{1},n_{2},...,n_{k}}$ terms generally results in a sum with ${\displaystyle n_{1}n_{2}...n_{k}}$ terms.