Reduce the prime implicant chart by eliminating the essential prime implicant rows and the corresponding columns.[7]
Label the rows of the reduced prime implicant chart , , , , etc.[7]
Form a logical function which is true when all the columns are covered. P consists of a product of sums where each sum term has the form , where each represents a row covering column .[7]
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]
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]
Choose the term or terms composed of the minimum total number of literals, and write out the corresponding sums of prime implicants.[7]
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
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
^This causes exponential blow-up in the number of columns: expanding the product of sums that have terms generally results in a sum with terms.
^"Obituaries - Cedar Rapids - Stanley R. Petrick". The Gazette. 2006-08-05. p. 16. Archived from the original on 2017-04-13. Retrieved 2017-04-12. [...] CEDAR RAPIDS Stanley R. Petrick, 74, formerly of Cedar Rapids, died July 27, 2006, in Presbyterian/St. Luke's Hospital, Denver, Colo., following a 13-year battle with leukemia. A memorial service will be held Aug. 14 at the United Presbyterian Church in Laramie, Wyo., where he lived for many years. [...] Stan Petrick was born in Cedar Rapids on Aug. 6, 1931 to Catherine Hunt Petrick and Fred Petrick. He graduated from Roosevelt High School in 1949 and received a B.S. degree in mathematics from Iowa State University. Stan married Mary Ethel Buxton in 1953. He joined the U.S. Air Force and was assigned as a student officer studying digital computation at MIT, where he earned an M.S. degree. He was then assigned to the Applied Mathematics Branch of the Air Force Cambridge Research Laboratory and while there earned a Ph.D. in linguistics. He spent 20 years in the Theoretical and Computational Linguistics Group of the Mathematical Sciences Department at IBM's T. J. Watson Research Center, conducting research in formal language theory. He had served as an assistant director of the Mathematical Sciences Department, chairman of the Special Interest Group on Symbolic and Algebraic Manipulation of the Association for Computing Machinery and president of the Association for Computational Linguistics. He authored many technical publications. He taught three years at Northeastern University and 13 years at the Pratt Institute. Dr. Petrick joined the University of Wyoming in 1987, where he was instrumental in developing and implementing the Ph.D. program in the department and served as a thesis adviser for many graduate students. He retired in 1995. [...] (NB. Includes a photo of Stanley R. Petrick.)
^Petrick, Stanley R. (1956-04-10). A Direct Determination of the Irredundant Forms of a Boolean Function from the Set of Prime Implicants. Bedford, Cambridge, Massachusetts, USA: Air Force Cambridge Research Center. AFCRC Technical Report TR-56-110.
^Frenzel, James "Jim" F. (2007). "Lecture #10: Petrick's Method". ECE 349 – Background Study in Digital Computer Fundamentals. Moscow, Idaho, USA: Department of Electrical and Computer Engineering, University of Idaho. Archived from the original on 2017-04-12. Retrieved 2019-08-08. [3]