Efficient Minimisation of Boolean Functions

2008 ◽  
Vol 45 (4) ◽  
pp. 321-326 ◽  
Author(s):  
V. C. Prasad
Keyword(s):  
Boolean Functions ◽  
Minimal Set ◽  
Undergraduate Level ◽  
Popular Method ◽  
Prime Implicants

Quine Mc Cluskey's (QM) method is a popular method for minimisation of Boolean functions. This method is widely taught at undergraduate level. In this paper simple modifications are suggested to make it more efficient. They allow us to avoid repetitions in the QM method. Further, a minimal set of prime implicants is easily obtained.

scholarly journals Quality of Minimal Sets of Prime Implicants of Boolean Functions

2017 ◽  
Vol 63 (2) ◽  
pp. 165-169
Author(s):  
V. C. Prasad
Keyword(s):  
Power Consumption ◽  
Boolean Function ◽  
Boolean Functions ◽  
New Technique ◽  
Minimal Set ◽  
Reduce Power Consumption ◽  
Minimal Sets ◽  
Prime Implicants

Abstract Two new problems are posed and solved concerning minimal sets of prime implicants of Boolean functions. It is well known that the prime implicant set of a Boolean function should be minimal and have as few literals as possible. But it is not well known that min term repetitions should also be as few as possible to reduce power consumption. Determination of minimal sets of prime implicants is a well known problem. But nothing is known on the least number of (i) prime implicants (ii) literals and (iii) min term repetitions , any minimal set of prime implicants will have. These measures are useful to assess the quality of a minimal set. They are then extended to determine least number of prime implicants / implicates required to design a static hazard free circuit. The new technique tends to give smallest set of prime implicants for various objectives.

scholarly journals Utilization of the Karnaugh Map in Exploring Cause-effect Relations Modeled by Partially-defined Boolean Functions

2021 ◽  
pp. 117-137
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Systematic Investigation ◽  
Exact Matching ◽  
Minimal Set ◽  
Minimal Sets ◽  
Prime Implicants ◽  
Switching Functions ◽  
Programming Techniques ◽  
Karnaugh Map

This paper utilizes a modern regular and modular eight-variable Karnaugh map in a systematic investigation of cause-effect relationships modeled by partially-defined Boolean functions (PDBF) (known also as incompletely specified switching functions). First, we present a Karnaugh-map test that can decide whether a certain variable must be included in a set of supporting variables of the function, and, otherwise, might enforce the exclusion of this variable from such a set. This exclusion is attained via certain don’t-care assignments that ensure the equivalence of the Boolean quotient w.r.t. the variable, and that w.r.t. its complement, i.e., the exact matching of the half map representing the internal region of the variable, and the remaining half map representing the external region of the variable, in which case any of these two half maps replaces the original full map as a representation of the function. Such a variable exclusion might be continued w.r.t. other variables till a minimal set of supporting variables is reached. The paper addresses a dominantly-unspecified PDBF to obtain all its minimal sets of supporting variables without resort to integer programming techniques. For each of the minimal sets obtained, standard map methods for extracting prime implicants allow the construction of all irredundant disjunctive forms (IDFs). According to this scheme of first identifying a minimal set of supporting variables, we avoid the task of drawing prime-implicant loops on the initial eight-variable map, and postpone this task till the map is dramatically reduced in size. The procedure outlined herein has important ramifications for the newly-established discipline of Qualitative Comparative Analysis (QCA). These ramifications are not expected to be welcomed by the QCA community, since they clearly indicate that the too-often strong results claimed by QCA adherents need to be checked and scrutinized.

Boolean functions with long prime implicants

2013 ◽  
Vol 113 (19-21) ◽  
pp. 698-703 ◽  
Author(s):  
Ondřej Čepek ◽  
Petr Kučera ◽  
Stanislav Kuřík
Keyword(s):  
Boolean Functions ◽  
Prime Implicants

scholarly journals Using Variable-Entered Karnaugh Maps in Determining Dependent and Independent Sets of Boolean Functions

2012 ◽  
Vol 1 (2) ◽  
pp. 45-67
Author(s):  
Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati Ali Muhammad Ali Rushdi and Hussain Mobarak Albarakati
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Independent Sets ◽  
Solution Procedure ◽  
Black Box ◽  
Important Class ◽  
Time Saving ◽  
Prime Implicants ◽  
Simultaneous Processing ◽  
Karnaugh Map

An important class for Boolean reasoning problems involves interdependence among the members of a set T of Boolean functions. Two notable problems among this class are (a) to establish whether a given subset of T is dependent, and (b) to produce economical representations for the complementary families of all dependent subsets and independent subsets of T. This paper solves these two problems via a powerful manual pictorial tool, namely, the variableentered Karnaugh map (VEKM). The VEKM is utilized in executing a Label-and-Eliminate procedure for producing certain prime implicants or consequents used in tackling the two aforementioned problems. The VEKM procedure is a time-saving short cut indeed, since it efficiently handles the three basic tasks demanded by the solution procedure, which are: (a) To combine several Boolean relations into a single one, (b) to compute conjunctive eliminants of a Boolean function, and (c) to derive the complete sum (CS) of a Boolean function. The VEKM procedure significantly reduces the complexities of these tasks by introducing useful shortcuts and allowing simultaneous processing. The VEKM procedure is described in detail, and then demonstrated via two illustrative examples, which previously had only black-box computer solutions as they were thought to be not amenable to manual solution. The first example deals with switching or bivalent functions while the second handles 'big' Boolean functions. Both examples indicate that the VEKM procedure proposed herein enjoys the merits of insightfulness, simplicity and efficiency

scholarly journals Utilization of Eight-Variable Karnaugh Maps in the Exploration of Problems of Qualitative Comparative Analysis

2021 ◽  
pp. 57-84
Author(s):  
Ali Muhammad Ali Rushdi ◽  
Raid Salih Badawi
Keyword(s):  
Comparative Analysis ◽  
Boolean Function ◽  
Boolean Functions ◽  
Qualitative Comparative Analysis ◽  
Programming Approach ◽  
Minimal Set ◽  
First Case ◽  
Desirable Feature ◽  
Karnaugh Map ◽  
Variable Problem

Qualitative Comparative Analysis (QCA) is an emergent methodology of diverse applications in many disciplines. However, its premises and techniques are continuously subject to discussion, debate, and (even) dispute. We use a regular and modular Karnaugh map to explore a prominent recently-posed eight-variable QCA problem. This problem involves a partially-defined Boolean function (PDBF), that is dominantly unspecified. Without using the algorithmic integer-programming approach, we devise a simple heuristic map procedure to discover minimal sets of supporting variables. The eight-variable problem studied herein is shown to have at least two distinct such sets, with cardinalities of 4 and 3, respectively. For these two sets, the pertinent function is still a partially-defined Boolean function (PDBF), equivalent to 210 = 1024 completely-specified Boolean functions (CSBFs) in the first case, and to four CSBFs only in the second case. We obtained formulas for the four functions of the second case, and a formula for a sample fifth function in the first case. Although only this fifth function is unate, each of the five functions studied does not have any non-essential prime implicant, and hence each of them enjoys the desirable feature of having a single IDF that is both a unique minimal sum and the complete sum. According  to our scheme of first identifying a minimal set of supporting variables, we avoided the task of drawing prime-implicant loops on the initial eight-variable map, and  postponed this task till the map became dramatically reduced in size. Our map techniques and results are hopefully of significant utility in future QCA applications.

scholarly journals On the generation of prime implicants

1979 ◽  
Vol 2 (1) ◽  
pp. 167-186
Author(s):  
Bernd Reusch ◽  
Lothar Detering
Keyword(s):  
Boolean Functions ◽  
Normal Forms ◽  
General Theorem ◽  
General Representation ◽  
Prime Implicants

It is shown that various well-known normal forms for Boolean functions can be derived from a very general representation of subfunctions. Another general theorem on the relation between the prime implicants of a function and the prime implicants of its subfunctions is used to prove correct various methods of generating prime implicants.

scholarly journals A Method of Two-Level Simplification of Boolean Functions

1967 ◽  
Vol 29 ◽  
pp. 201-210
Author(s):  
Toshio Umezawa
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
The Other ◽  
Product Form ◽  
Automatic Computation ◽  
Prime Implicants ◽  
Other Hand ◽  
Group 1

There are a number of methods to find minimal two-level forms for a given Boolean function, e g. Harvard’s group [1], Veitch [2], Quine [3], [4], Karnaugh [5], Nelson [6], [7] etc,. This paper presents an approach which is suitable for mechanical or automatic computation, as the Harvard method and the Quine method are so. On the other hand, it shares the same property as the Veitch method in the sense that some of essential prime implicants may be found before all prime implicants are computed. It also adopts the procedure to reduce the necessary steps for computation which is shown in Lawler [8]. The method described is applicable to the interval of Boolean functions f, g such that f implies g where for simplification of sum form the variables occurring in g also occur in f and for product form the variables in f also occur in g.

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