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# MODIFIED QUINE-MCCLUSKEY METHOD

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PRESENTS OPTIMIZATION TO QUINE-MCCLUSKEY METHOD ...

PRESENTS OPTIMIZATION TO QUINE-MCCLUSKEY METHOD
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1) http://arxiv.org/abs/1203.2289
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2)http://www.amazon.com/Modern-Approach-Speed-Math-Secret-ebook/dp/B00D2UE1AA
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## MODIFIED QUINE-MCCLUSKEY METHOD Presentation Transcript

• RESEARCH ON Modified Quine-McCluskey Method PRESENTED BY: Mathematician Vitthal Jadhav Guided By Prof . Amar Buchade
• Overview 2 1.Introduction to Boolean Algebra 2.Minimization Techniques 2.1 Algebraic Minimization 2.2 Karnaugh Map Minimization 2.3 Quine-McCluskey method 2.4 Modified Quine-McCluskey method(MQM) 3.Application of MQM 4.Conclusion and Future Enhancement 5.References
• Introduction to Boolean Algebra 3  Invented by Gaorge Bool in 1854.  Only two outcome exist in boolean algebra. - Logic 0 (Low) -Logic 1 (High)  Main Logic Operator - AND operator (.) - OR Operator (+) - NOT oprator  Mainly used in design and analysis of digital circuit.  Boolean laws are used to minimize Boolean expression.
• Goal of minimization 4  Goal is to find an equivalent of an original logic expression that:  has fewer variables per term  has fewer terms  needs less logic to implement .  Main Minimization methods  Algebraic minimization  Karnaugh Map minimization  Quine-McCluskey (tabular) minimization
• Term Related to Minimization 5  Sum of Product ( SOP) - implemeted as AND gate feeding single OR gate. E.g. Y= AB+ABC+ ĀB̄  Product of Sum(POS) -implemeted as AND gate feeding single OR gate. E.g. Y=(A+B).(A+B+C). (Ā+B̄)  Canonical or standard form - Boolean expression containing all available input variable. E.g Y= AB + ABC + ĀB̄ Y = AB(C+C̄) + ABC + ĀB̄(C+ C̄) ( Since C+C̄=1) Y = ABC +ABC̄ + ĀB̄C + ĀB̄C̄ (standard SOP)
• Term Related to Minimization 6  Minterm - individual term in standard SOP form .  Mintermlist - set of minterm which can be combined.  Prime implicant - minterm which can’t be combined with other minterm.  E-sum - Sum of eliminated variable’s positional weight in Mintermlist. E.g Let mintermlist - AB - - (simplification of ABCD,ABCD ,ABC ) E-sum = 1+2 = 3.
• Algebraic Minimization 7  Boolean Laws and theorem are used to minimize Boolean functions.  E.g Y= AB + ABC + ĀB̄ Y = AB(C+C̄) + ABC + ĀB̄(C+ C̄) ( Since C+C̄=1) Y = ABC +ABC̄ + ĀB̄C + ĀB̄C̄ (standard SOP)  Disadvantages  Hard to recognize when a particular law can be applied  Difficult to know if resulting expression is truly minimal or not. E.g Y= AB + ABC + ĀB̄ Y = AB(C+C̄) + ABC + ĀB̄(C+ C̄) ( Since C+C̄=1) Y = ABC +ABC̄ + ĀB̄C + ĀB̄C̄ (standard SOP)
• 8 Karnaugh Map Minimization  Karnaugh Map (or K-map) minimization is a visual minimization technique  Is an application of adjacency  Procedure guarantees a minimal expression  Easy to use; fast  Problems include:  Applicable to limited number of variables (4 ~ 6)  Errors in translation from TT to K-map  Errors in reading final expression
• 9 K-map Minimization Method Two Basic steps  Find all prime implicants (largest groups of 1s or 0s in order of largest to smallest)  Identify minimal set of PI that cover given minterm The resulting expression is minimal.
• 10 Grouping - Applying Adjacency CD 00 01 11 10 00 01 11 10 A AB C B D 1 00 0 1 0 1 1 1 1 10 0 0 0 0 E.g.  If two cells have the same value and are next to each other, the terms are adjacent. This adjacency is shown by enclosing them. Groups can have common cells. Group size is a powerof 2 and groups are rectangular. You can group 0s or1s. ∑== 9,12,13)m(2,5,6,8,D)C,B,F(A,Y 0 1 3 4 5 7 8 15 14 9 12 1011 CADCBDCAY ++= DCBA DBCA DCA
• Quine-McCluskey Minimization Method 11  Advantages Over K-maps - less dependant on designer’s ability to recognize the pattern. - Can handle Boolean functions of more than six variables - can be computerized.
• QM-Algorithm 12  Two basic stages  Finding all prime implicants  Partition minterms into groups according to the number of 1’s  Exhaustively search for prime implicants  Selecting a minimum prime implicant cover  Construct a prime implicant chart  Select the minimum number of prime implicants
• E.g  Stage-I :Finding all Prime Implicants  Transform the given Boolean function into a canonical SOP function  Partition all minterms into groups according to number of 1’s in their binary representation.  All the minterms in one group contain the same number of 1’s. 13 )15,7,0(9,10,13)m(4,5,6,8,E)D,C,B,F(A, d+= ∑
• Grouping of minterms 14 Group Mintermlist Binary Representation A B C D 0 (No one’s) 0 0 0 0 0 1 (1 one’s) 4 8 0 1 0 0 1 0 0 0 2 (Two one’s) 5 6 9 10 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 3 (Three one’s) 7 13 0 1 1 1 1 1 0 1 4 (Four one’s) 15 1 1 1 1 )15,7,0(9,10,13)m(4,5,6,8,E)D,C,B,F(A, d+= ∑
• 15  Combine minterm which differ in single bit from adjacent groups  Check (√√) the minterms being combined – The checked minterms are “covered” by the combined new minterm  Keep doing this untill it is impossible to combine minterms in adjacent groups
• 16 A B C D √ √ (0) 0 0 0 0 (4) 0 1 0 0 (8) 1 0 0 0 √ √ √ (5) 0 1 0 1 √ (6) 0 1 1 0 (9) 1 0 0 1 (10) 1 0 1 0 (7) 0 1 1 1 (13) 1 1 0 1 (15) 1 1 1 1 A B C D (4,5,6,7) 0 1 - – (4,6,5,7) 0 1 - – √ √ (5,7,13,15) - 1 - 1 √ (5,13,7,15) - 1 - 1 √ A B C D √ (0,4) 0 - 0 0 (4,6) 0 1 - 0 √ √ √ (5,13) - 1 0 1 (13,15) 1 1 1 1 √ √ √ (0,8) - 0 0 0 (4,5) 0 1 0 - (8,9) 1 0 0 - (8,10)1 0 - 0 (5,7) 0 1 - 1 (9,13)1 - 0 1 (7,15) - 1 1 1 (6,7) 0 1 1 -
• Prime Implicants 17 (0,4) 0 - 0 0 (0,8) - 0 0 0 (8,9) 1 0 0 - (8,10) 1 0 - 0 (9,13) 1 - 0 1 (4,5,6,7) 0 1 - - (5,7,13,15) - 1 - 1 A B C D • Unchecked terms are prime implicants
• Prime Implicants DCA= DCB= CBA= 18 • Unchecked terms are prime implicants DBA= DCA= BA= BD= (0,4) 0 - 0 0 (0,8) - 0 0 0 (8,9) 1 0 0 - (8,10) 1 0 - 0 (9,13) 1 - 0 1 (4,5,6,7) 0 1 - - (5,7,13,15) - 1 - 1 A B C D
• Stage-II: Selection of Minimal Cover 19  Form a Prime Implicant Table – Top horizontal Row : real minterm – Left-Vertical Column : prime implicants  An ×× is placed at the intersection of a row and column if the corresponding prime implicant includes the corresponding product (term)
• Prime Implicant Chart 20 Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X X ACD̄ 9,13 X X AB̄ 4,5,6,7 X X
• 21  Locate the essential row from the table  These are essential prime implicants  The row consists of minterms covered by a single “×”  Mark all minterms covered by the essential prime implicants  Find non-essential prime implicants to cover the rest of minterms  Form the SOP function with the prime implicants selected, which is the minimal representation
• Minimal Cover Selection Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X
• Minimal Cover Selection Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X  Select PI A’B(4,5,6,7)
• Minimal Cover Selection 24 Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X  Select PI A’B(4,5,6,7) , AB’D’ (8,10)
• Minimal Cover Selection 25 Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X  Select PI A’B(4,5,6,7) , AB’D’ (8,10) , AC’D (9,13)  Now all real minterms are covered by selected prime implicants !
• Example Result DCADBABAY (9,13)(8,10)(4,5,6,7) d(0,7,15)10,13)8,9,5,6,4,m(E)D,C,B,F(A,Y ++= ++= +== ∑ 26
• Disadvantages of QM Method  Slower method for large no. of variable .  Difficult to handle repetition.  Total no. of comparison required to check combinability between two mintermlist in adjacent group is too large. 27 Mintermlist Binary Representation A B C D 0 0 0 0 0 4 8 0 1 0 0 1 0 0 0 5 6 9 10 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 7 13 0 1 1 1 1 1 0 1 15 1 1 1 1 Mintermlist (Quad of minterm) Binary Representation ABCD 4,5,6,7 4,6,5,7 01– – 01– – 5,13,7,15 –1–1 –1–1
• Modified Quine-McCluskey Method (MQM)  Matching Principal Two basic condition :  (least minterm in a mintermlist of (n+1)th group) (least minterm in a mintermlist of nth group) = 2n ( n>=0) .  Elimination sum of both mintermlist must be equal . 28
• MQM Algorithm 29 29  Stage-I :Finding Prime Implicants • Transform the given Boolean function into a canonical SOP function. • Arrange all minterm in groups according to number of 1’s in their binary representation. – All the minterms in one group contain the same number of “1”. )15,7,0(9,10,13)m(4,5,6,8,E)D,C,B,F(A, d+= ∑
• Grouping of Minterms 30 Group No. Minterm Binary Representation A B C D 0 (No one’s) 0 0 0 0 0 1 (1 one’s) 4 8 0 1 0 0 1 0 0 0 2 (Two one’s) 5 6 9 10 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 3 (Three one’s) 7 13 0 1 1 1 1 1 0 1 4 (Four one’s) 15 1 1 1 1
• 31 31 Combine mintermlist in adjacent groups according to MQM matching principal. - use algebraic approach to reduce comparison. Check (√√) the mintermlist being combined – The checked mintermlist are “covered” by the combined new mintermlist. Eliminate repetition from combined mintermlist. - Two mintermlist in same group becomes identical if their E-sum,least minterm and largest minterm are equal. Keep doing this until it is impossible to combine mintermlist in adjacent group.
• 32 minterm E-sum √ √ 0 0 4 0 8 0 √ √ √ 5 0 √ 6 0 9 0 10 0 7 0 13 0 15 0 (4,5,6,7) 1+2=3 (4,6,5,7) 2+1=3 √ √ (5,7,13,15) 2+8=10 √ (5,13,7,15) 8+2=10 √ minterm E-sum √ (0,4) 0+4=4 (4,6) 0+2=2 √ √ √ (5,13) 0+8=8 (13,15) 0+2=2 √ √ √ (0,8) 0+8=8 (4,5) 0+1=1 (8,9) 0+1=1 (8,10) 0+2=2 (5,7) 0+2=2 (9,13) 0+4=4 (7,15) 0+8=8 (6,7) 0+1=1 minterm E-sum
• Prime Implicants DCA= 33 DCB= • Unchecked terms are prime implicants CBA= DBA= DCA= BA= BD= Prime Implicant E-sum Binary Representation A B C D (0,4) 4 0 – 0 0 (0,8) 8 – 0 0 0 (8,9) 1 1 0 0 – (8,10) 2 1 0 – 0 (9,13) 4 1 – 0 1 (4,5,6,7) 3 0 1 – – (5,7,13,15) 10 –1– 1
• selection of minimal cover 34 34  Form a Prime Implicant Table – Top horizontal Row : real minterm – Left-Vertical Column : prime implicants  An ×× is placed at the intersection of a row and column if the corresponding prime implicant includes the corresponding product (term).
• Prime Implicant Chart 35 35 Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X X ACD̄ 9,13 X X AB̄ 4,5,6,7 X X
• 36 36 Locate the essential row from the table – These are essential prime implicants – The row consists of minterms covered by a single “×” Mark all minterms covered by the essential prime implicants Find non-essential prime implicants to cover the rest of minterms Form the SOP function with the prime implicants selected, which is the minimal representation
• Minimal Cover Selection Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X XACD̄ 9,13 ⨂ X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X
• Minimal Cover Selection Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X  Select PI A’B(4,5,6,7)
• Minimal Cover Selection 39 Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X  Select PI A’B(4,5,6,7) , AB’D’ (8,10)
• Minimal Cover Selection 40 Prime Implicant Minterms covered by PI . Given Minterm(Excluding Don’t cares) 4 5 6 8 9 10 13 ACD̄ ̄ ̄ 0,4 X BCD̄ ̄ ̄ 0,8 X ABC̄ ̄ 8,9 X X ABD̄ ̄ 8,10 X ⨂ ACD̄ 9,13 X X AB̄ 4,5,6,7 X X ⨂ BD 5,7,13,15 X X  Select PI A’B(4,5,6,7) , AB’D’ (8,10) , AC’D (9,13)  Now all real minterms are covered by selected prime implicants !
• Example Result DCADBABAY (9,13)(8,10)(4,5,6,7) d(0,7,15)10,13)8,9,5,6,4,m(E)D,C,B,F(A,Y ++= ++= +== ∑ 41
• Disadvantages of MQM  Input Boolean Function must be in SOP form. - adds redundant minterms in input. E.g. 42 DABCCBACDBACABABCD )DABC(D)DC)(CBA(B ABCADY ++++= ++++= += DD
• Application Of MQM- Data Compression 43
• CONCLUSION In this paper a new method called modified Quine-McCluskey method (MQM) is introduced by using which performance of digital circuit can be increased. Thus MQM is efficient method than conventional Quine-McCluskey method. 44
• FUTURE ENHANCEMENT  Programmer can use this method due to its lower time complexity.  Student can use this method to achieve speed in minimizing Boolean function manually. 45
• REFERENCES [1] McCluskey E.J.(1956), “minimization of Boolean function” Bell system Tech. Journal , vol.35 , No.5 , pp. 1417-1444.   [2] SP Tomaszeweski, “WWW Based Boolean function minimization”- in International journal of applied mathematics and computer sciences,2003,vol. 13,No. 1, 577- 583.   [3] Eyas El-Qawasmeh , Ahmed Kattan “New Technique for Data Compression” in International Conference on Applied Computing (IADIS),2005. [4] Jain T.K., Kushwaha D.S., Misra A.K., “ Optimization of the Quine-McCluskey Method for the Minimization of the Boolean Expressions” in International Conference on Autonomic and Autonomous Systems, 2008. ICAS 2008. [5] R. Mohan Ranga Rao ,”An Innovative procedure to minimize Boolean function ” in International Journal of Advanced Engineering Sciences and Technologies 46
• Question ? mailto : jadhavvitthal1989@gmail.com 47 47