neither is thus essential. If a prime implicant is essential then, as would be expected, it is necessary to include it in the minimized boolean equation. In some cases, the essential prime implicants do not cover all minterms, in which case additional procedures for chart reduction can be employed. The simplest "additional procedure" is trial and error, but a more systematic way is Petrick's method. In the current example, the essential prime implicants do not handle all of the minterms, so, in this case, the essential implicants can be combined with one of the two non-essential ones to yield one equation: f = BC'D' + AB' + AC or f = BC'D' + AD' + AC Both of those final equations are functionally equivalent to the original, verbose equation: f = A'BC'D' + AB'C'D' + AB'C'D + AB'CD' + AB'CD + ABC'D' + ABCD' + ABCD. See also Blake canonical form Buchberger's algorithm – analogous algorithm for algebraic geometry Petrick's method Qualitative comparative analysis (QCA) References Further reading (viii+635 pages) (NB. This book was reprinted by Chin Jih in 1969.) (47 pages) (4 pages) (7 pages) (22 pages) (16 pages) (NB. QCA, an open source, R based implementation used in the social sciences.) External links Quine-McCluskey algorithm implementation with a search of all solutions, by Frédéric Carpon. Karċma 3, A set of logic synthesis tools including Karnaugh maps, Quine-McCluskey minimization, BDDs, probabilities, teaching module and more. Logic Circuits Synthesis Labs (LogiCS) - UFRGS, Brazil. BFunc, QMC based Boolean logic simplifiers supporting up to 64 inputs / 64 outputs (independently) or 32 outputs (simultaneously), by António Costa Python Implementation by Robert Dick, with an optimized version. Python Implementation for symbolically reducing Boolean expressions. Quinessence, an open source implementation written in Free Pascal by Marco Caminati. minBool an implementation by Andrey Popov. QMC applet, an applet for a step by step analyze of the QMC- algorithm by Christian Roth C++ implementation SourceForge.net C++ program implementing the algorithm. Perl Module by Darren M. Kulp. Tutorial Tutorial on Quine-McCluskey and Petrick's method. Petrick C++ implementation (including Petrick) based on the tutorial above. C program Public Domain console based C program on SourceForge.net. For a fully worked out example visit: http://www.cs.ualberta.ca/~amaral/courses/329/webslides/Topic5-QuineMcCluskey/sld024.htm The Boolean Bot: A JavaScript implementation for the web: http://booleanbot.com/ open source gui QMC minimizer Computer Simulation Codes for the Quine-McCluskey Method, by Sourangsu Banerji. Boolean algebra Willard Van Orman Quine