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Boolean Minimizer

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Boolean function minimization using M-terms is a modi?ed Quine-McCluskey [4] [6] method; it is a very simple and systematic technique for minimizing Boolean functions. Why do we want to minimize a Boolean expression? By simplifying the logic function we can reduce the original number of digital components (gates) required to implement digital circuits. Therefore by reducing the number of gates, the chip size and the cost will be reduced, and the speed will be increased. Logic minimization uses a variety of techniques to obtain the simplest gate-level implementation of a logic function.

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A The heart of digital logic design is the Boolean algebra (Boole, 1954)[2]. A few decades later C. E. Shannon showed how the Boolean algebra can be used in the design of digital circuits(Shannon,1938)[7]. Using Boolean laws It is possible to minimize digital logic circuits(Huntington,1904). Since minimization with the use of Boolean laws is not systematic nor suitable for computer implementation, a number of algorithms were proposed in order to overcome the implementation issue.

Karnaugh [3] proposed a technique for simplifying Boolean expressions using an elegant visual technique, which is actually a modi?ed truth table intended to allow minimal sum-of- @ 2011 http://www. ijaest. iserp. org. All rights Reserved. ES II. Abstract— In this paper a simplification technique is introduced to minimize a Boolean function. Karnaugh map (K-map) and Quine-McCluskey methods are well established methods to simplify a logic function. Using this method the simplification is fast, also the number of gates required to realize a function gets reduced to greater extent with a minimum effort.

In present discussion a new method is introduced in which new terms called M-terms obtained in order to improve the performance of conventional methods. The given method can be implemented to any number of variables. The Quine-McCluskey (Q-M) method is a computerbased technique for simpli?cation and has mainly two advantages over the K-Map method. Firstly, it is systematic for producing a minimal function that is less dependent on visual patterns. Secondly, it is available scheme for handling a large number of variables.

A number of methods have been developed that can generate optimal solutions directly at the expense of additional computation time. Another algorithm was reported by Petrick(1959)[5]. This algorithm uses an algebraic approach to generate all possible covers of a function. A popular tool for simplifying Boolean expressions is the Espresso, but it is not guaranteed to ?nd the best two-level expression (Katz,1994)[1]. The main advantage of proposed method over the conventional methods is the entire simplification is based on the decimal values (E. g.

M-terms) which are well known to each and every designer, so in practice it is easy to implement this method to improve the performance. RELATEDWORK The digital gates (Logic gates) are basic electronic components of any digital circuit. A logic gate performs a logical operation based on one or more inputs and produces a single output voltage value (i. e. voltage levels high and low). Logically these voltage values can be referred to as 1s and 0s and are used in designing and analyzing the operations of logic gates. A logic gate represents a Boolean function.

A Boolean function is an algebraic expression formed with Boolean variables (having values true or 1 and false or 0) and the logical operators (i. e. OR, AND, and NOT). These may be a large number of Boolean algebraic expressions that specify a T products(SOP)and product-of-sums(POS) expressions to be obtained(Karnaugh,1953)[3]. The Karnaugh Map (KMap) based technique breaks down beyond six variables. Quine[6] and McCluskey[4] proposed an algorithmicbased technique for simplifying Boolean logic functions (McCluskey, 1956;Quine,1952). Page 12

R . Mohana Ranga Rao – (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 3, Issue No. 1, 012 – 014 given Boolean function. It is therefore important to find the simplest one. A Boolean function can be represented by a truth table. A truth table is a tabular arrangement of representing all the input-output relationships of a digital circuit. It displays all possible input values combinations with their respective output values. Normally a Boolean expression can be given using two forms: SOP and POS.

SOP Boolean expressions may be generated from truth tables quite easily by forming an OR of the ANDs of all input variables (standard products or min terms) for which the output is 1. POS expressions are based on the 0s, in a truth table and generated oppositely as SOP by taking an AND of the ORs of all input variables (standard sums or max terms). III. DEFINITIONS easy to implement this method to improve the performance. The procedure for M-terms based simplification is presented using the proposed algorithm: 1. 2. Enter the min terms which are to be simplified and obtain the binary notations for each min term.

Now obtain the face values of all the given min terms. (Usually it would be same for all the min terms). Calculate the M-terms for the given min terms. Compare each M-term with provided input (min terms), to identify the adjacent terms. If a M-term is Equivalent to any min term of provided input then the min term and the M-term has to be grouped together to make a pair (prime implicant). While grouping the two terms follow ascending order. Repeat the above step until it is impossible to compare the terms, and exclude all the repeated pairs.

Compare Prime implicants with each other such that the difference between the elements of a prime implicant should be same and it should be equal to the 2L , L=0,1,2….. N. While comparing make sure that the comparison is done between like elements i. e. the first elements of two pairs and the second elements of two pairs has to be compared. Once there are any two terms that differ by 2 L ,then group the two pairs as a single Quad. Now repeat the same procedure among all pairs. Swap all the combined terms, and terms that were not combined at all and the prime implicants which are not a part of any single quad.

Now repeat the VI & VII steps to minimize the given Boolean function/min terms until it is impossible to combine the terms. Now eliminate the redundant prime implicants by using Prime implicant chart as in Quine-McCluskey method 3. 4. B. Product of sums(POS): This is less commonly used form of Boolean expressions. The expressions are implemented as OR gates (sums) feeding into a single AND gate (product). C. Face value: List the powers of two from right to left. Start at 20, evaluating it as “1”. Increment the exponent by one for each power.

Stop when the amount of elements in the list is equal to the amount of digits in the binary number. A 8 4 2 1 The example number, 10011011, has eight digits, so the list, to eight elements, would look like this: 128, 64, 32, 16, 8, 4, 2, 1 these elements are called face values of corresponding elements of respective positions. IJ Minterm 5 0101 IV. ISSN: 2230-7818 D. M-terms:M-terms are generated by addition/subtraction of corresponding min term with its face values. [addition (if the Binary value is a ‘0’) or subtraction (if the Binary value is a ‘1’].

For example: 5 is the given min term Binary notation face value M-terms 13 1 7 4 1. ES 7. 8. 9. V. Min term 0 1 2 3 4 5 A. Sum of products(SOP): This is the more common form of Boolean expressions. The expressions are implemented as AND gates (products) feeding a single OR gate (sum). 5. 6. M-TERM BASED SIMPLIFICATION PROCEDURE The main advantage of proposed method over the conventional methods is, the entire simplification is based on the decimal values (E. g. M-terms) which are well known to each and every designer, so in practice it is @ 2011 http://www. ijaest. iserp. org. All rights Reserved. T

EXAMPLE 0000 0001 0010 0011 0100 0101 Simplify ?m (0,1,2,3,4,5) Binary notation Page 13 R . Mohana Ranga Rao – (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 3, Issue No. 1, 012 – 014 2. Min term 0 1 2 3 4 5 Binary notation 0000 0001 0010 0011 0100 0101 Face Value 8421 8421 8421 8421 8421 8421 9. By using Prime Implicant chart we can eliminate the redundant pair (1, 5) VI. CONCLUSION In this paper a new method called M-term based Boolean Simplification is introduced by using which performance of a digital circuit can be increased by reducing the number of literals in a Boolean function.

As we are using M-terms instead of comparing the binary values as in Quine McCluskey method we can avoid the possibility of errors such that we can improve the performance of conventional methods by using the present method. VII. REFERENCES SP Tomaszewski(2003):WWW-based Boolean function minimization-International journal of applied mathematics and Computer Sciences, 2003, vol. 13, No. 1,577-583 Boole G. (1954): An Investigation of the Laws of Thought. Newyork Karnaugh M. (1953):The map method for synthesis of combinatorial logic circuits. Trans. AIEEComm. Electron. ,Vol. 72, No. 4, pp. 593– 598. McCluskeyE. J. (1956): Minimization of Boolean functions. —BellSystemTech. J. , Vol. 35,No. 5,pp. 1417–1444 PetrickS. K. (1959): On the minimization of Boolean functions Proc. Int. Conf. InformationProcessing,Paris:Unesco,pp. 422–423. QuineW. V. (1952): The problem of simplifying truthtables. — Amer. Math. Month. ,Vol. 59,No. 8,pp. 521–531. ShannonC. E. (1938): symbolic analysis of relay and switchingcircuits. —Trans. AIEE, Vol. 57,No. 6,pp. 713–723. 3.

Min term 0 1 2 3 4 5 Binary notation 0000 0001 0010 0011 0100 0101 Face Value 8421 8421 8421 8421 8421 8421 M-terms 8 4 2 1 9 5 3 0 10 6 0 3 11 7 1 2 12 0 6 5 13 1 7 4 [1] [2] [3] 4. Comparing M-terms with Input ?m(0,1,2,3,4,5) Comparing the M-terms of the first min term i. e. ‘0’ with input the possible prime implicants are (0, 1), (0, 2), (0, 4) Similarly by comparing the other terms we will have the following prime implicants A terms remained (1,0),(1,3),(1,5),(2,0),(2,3),(3,1),(3,2),(4,0),(4,5),(5,1), (5,4) 5. Excluding the repeated terms which satisfies associative law [(x,y)==(y,x)] and making the elements in ascending order

IJ 7. On swapping, the (0,1,2,3),(1,5) and (4,5) 8. (0,1,2,3),(1,5) and (4,5) ISSN: 2230-7818 (0,1),(0,2),(0,4),(1,3),(1,5),(2,3),(4,5) 6. Now compare (0,1) with all other prime implicants (such that the difference between the corresponding elements should be same and it should be equal to 2L) (0, 1) and (0,2) — 0-0=0 & 1-2=1(ignore the negative values) here the difference is not same (0, 1) and (2, 3) — 0-2=2 & 1-3=2(ignore the negative values) here the difference is same and is equal to21 So they can be combined as a group (0, 1, 2, 3) are 2011 http://www. ijaest. iserp. org. All rights Reserved. ES [4] [5] [6] [7] About Author: R. Mohana Ranga Rao is an Asst professor in KITE college of Professional engineering sciences, JNTUH & Pursuing M. Tech from St. Mary’s college of engineering and technology, Hyderabad & completed his B. Tech from sree kavitha engineering college, khammam

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