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SYEN 3330 Digital Systems Chapter 2 – Part 6 SYEN 3330 Digital Systems Jung H. Kim Chapter 2-6 1 Table Methods for PI Generation This method of simplifying a Boolean equation was improved by Quine and McCluskey and is sometimes known as the Q-M Method. The Tabular Method: 1. Starts with a table of minterms. 2. Compares each minterm with every other minterm in the list to find minterms which differ in exactly one variable. 3. Constructs a new list with terms of one fewer variables, keeping track of which minterms were covered. 4. Compares every element in the new list with each other to find terms that differ in one more variable. 5. Repeats Steps 3 and 4 until done. SYEN 3330 Digital Systems Chapter 2-6 2 An Example: F(x,y,z)= m(2,3,6,7) Initial work: Step 1 creates a table of minterms. Column (a) Column (b) Column(c) xyz xyz xyz 010 m2 011 m3 110 m6 111 m7 Step 2: Compare each minterm with all others. 010: (011 01-), (110 -10), (111Ø) 011: (110 Ø), (111-11) 110: (111 11-) SYEN 3330 Digital Systems Chapter 2-6 3 Results - Step 2 Column (a) xyz 010 m2 011 m3 110 m6 111 m7 Column (b) Column(c) xyz m2,m3 xyz (01-) (-10) m2,m6 (-11) m3,m7 (11-) m6,m7 Note that there can be terms which are duplicated -- that is, they appear more than once. The terms are duplicates if they have the same pattern of bits (including the -) and they came from the same minterms. Only one of each duplicate term should remain. SYEN 3330 Digital Systems Chapter 2-6 4 Step 3 Column (a) xyz 010 m2 011 m3 110 m6 111 m7 Column (b) Column(c) xyz m2,m3 xyz (01-) (-10) m2,m6 (-11) m3,m7 (11-) m6,m7 Step 3: Make a new list by comparing items in Column (b). 01-: (-10 Ø), (-11 Ø ), (11- -1-) -10: (01- Ø), (-11 -1-), (11- Ø ) -11: (01- -1-), (-10 -1-), (11- Ø ) 11-: (01- -1-), (-10 Ø ), (-11 Ø ) SYEN 3330 Digital Systems Chapter 2-6 5 The Results of Step 3 Column (a) xyz 010 m2 011 m3 110 m6 111 m7 Column (b) xyz (01-) m2,m3 (-10) m2,m6 (-11) m3,m7 (11-) m6,m7 Column(c) xyz (-1-) m2,m3,m6,m7 (-1-) m2,m6,m3,m7 (-1-) m3,m7,m6,m7 (-1-) m6,m7,m2,m3 Obviously, the Column (c) can be simplified by eliminating duplicates -- this leads to only one entry: (-1-) from m2,m3,m6,m7 This corresponds to the Prime Implicant "y". The algorithm terminates at this point. Since this is the only PI and it covers all minterms, the result is: F(x,y,z) = y SYEN 3330 Digital Systems Chapter 2-6 6 Computational Complexity Issues The table method generates a lot of work. For "n" minterms, there are on the order of n2 comparisons required. The Q-M Method simplifies the work by sorting the minterms into terms that can compare favorably. Terms that have no chance of combining are not even tried. It also adds some bookkeeping to simplify PI identification. Grouping: Use the number of "1"s in the minterm to group the minterms. Preserve groups derived from this grouping in adjacent columns. Bookkeeping: Use a "check mark" () next to terms that have been combined. Carry along lists of minterms used. SYEN 3330 Digital Systems Chapter 2-6 7 Q-M on F(x,y,z)= m(2,3,6,7) Initial Group One 1 Column (a) Column (b) Column(c) xyz xyz xyz 010 m2 ---------------Two 1's 011 m3 110 m6 ---------------Three 1's 111 m7 Step 2: Compare terms from adjacent groups. Group 1 => Group 2 010: (011 01-), (110 -10) Group 2 => Group 3 011: (111-11) NOTICE: Fewer steps! 110: (111 11-) SYEN 3330 Digital Systems Chapter 2-6 8 The Result of Step 3 Initial Group One 1 Column (a) xyz 010 m2 ---------------Two 1's 011 m3 110 m6 ---------------Three 1's 111 m7 Column (b) Column(c) xyz xyz (01-) m2,m3 (-10) m2,m6 -----------------(-11) m3,m7 (11-) m6,m7 Step 4: Repeat on Column (b) Group (1-2) => Group (2-3) (01-): (-11 Ø), (11- -1-) (-10): (-11 -1-), (11-Ø) SYEN 3330 Digital Systems Chapter 2-6 9 Result of Step 4 Initial Group One 1 Two 1's Column (a) xyz 010 m2 ---------------011 m3 110 m6 ---------------111 m7 Column (b) xyz (01-) m2,m3 (-10) m2,m6 -----------------(-11) m3,m7 (11-) m6,m7 Column(c) xyz (-1-)m2,m3,m6,m7 (-1-)m2m6,m7 ,m3,m6,m7 m3,m7 Three 1's that the resulting terms are duplicates and are unchecked at Note termination. Final Result: F(x,y,z) = y In general, when no new terms can be generated, the set of all unchecked terms give the result. SYEN 3330 Digital Systems Chapter 2-6 10 Review of Boolean Logic Given a Boolean Function F, We have learned to express F as: 1. Canonical Sum-of-Minterms 2. Canonical Product-of-Maxterms 3. Standard Sum-of-Products (SOP) Form 4. Standard Product-of-Sums (POS) Form We have also learned how to minimize the number of literals in F by using: 1. Algebraic Simplification. 2. Selection of Prime Implicants (Karnaugh Map and Quine-McCloskey Method) 3. Systematic K-Map to minimum literal SOP. SYEN 3330 Digital Systems Chapter 2-6 11 Canonical Forms F ( w, x, y, z ) 0,2,4,5,8,10,11,13,15 F( w, x, y, z) 1,3,6,7,9,12,14 y 1 1 0 4 0 w 12 1 8 0 1 1 5 1 13 0 9 0 0 3 7 1 0 1 0 1 1 15 11 2 6 14 10 z NOTE: there is one and only one way to write the equations in Canonical Sum-of-minterm and Canonical Product-of-Maxterm form. SYEN 3330 Digital Systems Chapter 2-6 12 x Minimum Literal SOP Form Example: Find a minimum literal Standard SOP form for F(w,x,y,z) F(w,x,y,z) = F(w,x,y,z) = x'z' + w'y'z' + xy'z + wyz x'z' + w'xy' + wxz + wx'y y 1 1 0 w 1 0 4 12 8 0 1 1 0 0 1 0 5 1 13 1 9 3 7 15 11 y 1 0 0 1 z 2 1 6 1 x 0 14 w 10 1 0 4 12 8 0 1 1 0 0 1 0 5 1 13 1 9 3 7 15 11 1 0 0 1 2 6 x 14 10 z NOTE: Both are minimum literal SOP! SYEN 3330 Digital Systems Chapter 2-6 13 Tabular Method to Find a Cover 1). Construct a table with: a). Columns for each minterm and b). Rows for each Prime Implicant 2). Select Essential Prime Implicants and check off each covered minterm. 3). Delete Less Than Prime Implicants 4). Select Secondary Essential Prime Implicants and check off each covered minterm. 5). Repeat 3 and 4 until a cover is generated. If cycles exist, pick a PI and generate a cover and then delete that same PI and generate an alternate cover. Select the minimum literal cover. SYEN 3330 Digital Systems Chapter 2-6 14 Table Method Example Function g(w,x,y,z) y 1 0 4 w 12 1 8 1 1 1 1 5 1 13 9 z 3 1 7 1 15 11 Step 1, Enter table: 2 6 1 14 x PI 1 x' z' 10 w' x' w' y' z x y' z wxz wx y w y z' SYEN 3330 Digital Systems 0 x x 1 x x 2 x x 3 5 8 10 13 14 15 x x Type x x x x x x x x x x Chapter 2-6 15 Select Essential Prime Implicants Step 2: Select Essential Prime Implicants and check them off along with minterms covered. PI x'z' w'x' w'y'z xy'z wxz wxy wyz' 0 x x 1 x x 2 x x 3 5 8 10 13 14 15 x x x x x x x x x x Type Essential Essential x x Note: w'y'z is less than xy'z since xy'z covers two as yet uncovered minterms (m5 and m13) while w'y'z covers one (m5) uncovered minterm. Similarly wyz' is less than wxy. Why? SYEN 3330 Digital Systems Chapter 2-6 16 Select Essential Prime Implicants Reduced Covered Table PI w'y'z xy'z wxz wxy wyz' 5 13 14 15 x x x x x x x x SYEN 3330 Digital Systems Chapter 2-6 17 Less Than Prime Implicants Step 3: Delete Less Than Prime Implicants PI 0 x'z' x w'x' x w'y'z 1 2 3 5 x x x 8 10 13 14 15 x x Essential x x Essential x xy'z Less Than x x wxz x wxy x x wyz' x Type x x Less Than Note that after deleting the Less Than PIs, the following occurs: Minterm 5 is only covered by xy'z and Minterm 14 by wxy. Thus wxy and xy'z are Secondary Essential PIs. SYEN 3330 Digital Systems Chapter 2-6 18 Secondary Essential PIs PI x'z' w'x' w'y'z xy'z wxz wxy wyz' 0 x x 1 x x 2 x x 3 5 8 10 13 14 15 x x x x x x x x x x x x Type Essential Essential Less Than Secondary Essential (Redundant) Secondary Essential Less Than Note that at this point, minterms 13 and 15 are already covered, so the PI wxz is redundant. The algorithm terminates at this point with the minimum literal, Standard Sum-of-Products form: F(w,x,y,z) = x'z' + w'x' + xy'z + wxy SYEN 3330 Digital Systems Chapter 2-6 19 Cyclic Structures y Let F(x,y,z) = m(0,1,2,5,6,7) 1 We build the following PI Table: 0 x 4 1 1 1 3 1 5 7 1 1 2 6 z PI 0 1 2 5 6 7 x' y' x x y' z x x xz x x xy x x y z' x x x' z' x x SYEN 3330 Digital Systems Type Chapter 2-6 20 Cyclic Structure: Pick One NOTE: All minterms are covered twice. No PIs are essential. A cyclic structure exists. Step 1: Pick a PI and mark off the covered minterms. PI 0 1 2 5 6 7 x' y' x x y' z x x xz x x xy x x y z' x' z' x x Type (Picked) x x SYEN 3330 Digital Systems Chapter 2-6 21 Less Thans Step 2: Eliminate Less Thans. PI 0 1 2 5 6 7 x' y' x x y' z x x xz x x xy x x y z' x x x' z' x x Type (Picked) Less Than Less Than Once the less thans are deleted, there are two secondary essential PIs. SYEN 3330 Digital Systems Chapter 2-6 22 Secondary Essential PIs Step 3: Pick Secondary Essential PIs. PI 0 1 2 5 6 7 Type (Picked) x' y' x x y' z x x Less Than x x Secondary xz xy x x (Redundant) EPI x x Secondary y z' x' z' x x Less Than EPI Picking the two secondary essential PIs generates a complete cover and we can state that the Standard Sum of Products form for F(x,y,z) is: F(x,y,z) = x'y' + xz + yz' SYEN 3330 Digital Systems Chapter 2-6 23 Now Go Back and Try Again Step 1': GO BACK and eliminate the PI you picked and generate a new cover. PI 0 1 2 x' y' x x x y' z xz xy y z' x x x' z' x 5 6 7 x x Type (Eliminated) Essential x x x x Essential Note that y'z and x'z' become essential PIs. Select y'z and x'z' and then notice that xz and yz' become less thans. SYEN 3330 Digital Systems Chapter 2-6 24 Finish Up Step 2': Eliminate the less thans. Step 3': Select Secondary Essential PIs. PI 0 1 2 5 6 7 Type (Eliminated) x' y' x x Essential PI x x y' z x x Less Than xz x x Secondary EPI xy x x Less Than y z' Essential PI x x' z' x Notice that after Step 2', xy is a secondary essential PI and that picking it completes the cover. Thus the Standard Product of Sums form for this selection is: F(x,y,z) = y'z + xy + x'z' Both Implementations are Minimum Literal. SYEN 3330 Digital Systems Chapter 2-6 25 Quine-McCluskey (tabular) method 1. Arrange all minterms in group such that all terms in the same group have the same # of 1’s in their binary representation. 2. Compare every term of the lowest-index group with each term in the successive group. Whenever possible, combine two terms being compared by means of gxi+gxi´=g(xi+xi´)=g. Two terms from adjacent groups are combinable if their binary representation differ by just a single digit in the same position. 3. The process continues until no further combinations are possible. The remaining unchecked terms constitute the set of PI. SYEN 3330 Digital Systems Chapter 2-6 26 Ex) f(x1,x2,x3,x4) = (0,1,2,5,6,7,8,9,10,13,15) # x1,x2,x3,x4 x1,x2,x3,x4 0 0 0 0 0 1 2 8 0 0 1 0 0 0 0 1 0 1 0 0 5 6 9 10 0 0 1 1 1 1 0 0 0 1 0 1 1 0 1 0 7 13 0 1 1 1 1 0 1 1 15 1 1 1 1 x1,x2,x3,x4 0 0 - 0 0 0 0 0 0 0 (0,1) (0,2) (0,8) 0 0 1 1 0 0 0 0 0 0 1 1 0 - 1 1 0 0 0 (1,5) (1,9) (2,6) (2,10) (8,9) (8,10) 0 0 1 1 1 1 - 0 1 0 1 1 1 1 1 1 SYEN 3330 Digital Systems 1 - 1 1 (5,7) (5,13) (6,7) (9,13) (7,15) (13,15) - 0 0 1 0 0 - 0 1 1 x1,x2,x3,x4 (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15) Using prime implicant chart, we can find essential PI 0 (2,6) (6,7) (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15) 1 2 5 6 7 9 10 13 15 8 Chapter 2-6 27 The reduced PI chart 1 (2,6) (6,7) (0,1,8,9) (1,5,9,13) 6 The essential PI’s are (0,2,8,10) and (5,7,13,15) . So, f(x1,x2,x3,x4) = (0,2,7,8) + (5,7,13,15) + PI’s 9 Here are 4 different choices (2,6) + (0,1,8,9), (2,6) + (1,5,9,13) (6,7) + (0,1,8,9), or (6,7) + (1,5,9,13) A PI pj dominates PI pk iff every minterm covered by pk is also covered by pj. m1 m2 m3 m4 pj pk (can remove) Branching method m1 m2 m3 m4 m5 p1 p2 p3 p4 p5 If we choose p1 first, then p3, p5 are next. p1 p3 p5 p4 p2 p3 Quine – McCluskey method (no limitation of the # of variables) SYEN 3330 Digital Systems Chapter 2-6 28 Quine-McCluskey example • F(A,B,C,D) = (3,9,11,12,13,14,15) + d (1,4,6) SYEN 3330 Digital Systems Chapter 2-6 29 Ex) f(A,B,C,D) = (3,9,11,12,13,14,15) + d (1,4,6) PI chart: 3 9 11 12 (1,3, 9, 11) (4, 6,12,14) (9,13,11,15) (12,13,14,15) 13 14 15 Reduced PI chart: 12 (4, 6,12,14) (9,13,11,15) (12,13,14,15) 13 14 15 Result: (1,3,9,11) + (12,13,14,15) SYEN 3330 Digital Systems Chapter 2-6 30 Minimum SOP to Minimum POS We can use the minimization techniques learned so far to implement a minimum literal, Standard POS form. We take the following steps: 1. Implement the COMPLEMENT of a function in Standard Sum of Product form. 2. Use DeMorgan's Law to complement the function and convert it to Standard Product of Sum form. NOTE: Implementing the complement of a function means using the ZEROS on the K-Map. SYEN 3330 Digital Systems Chapter 2-6 31 Minimum POS Example Given g(w,x,y,z): Form the Complement (Circle Zeros): y 1 0 0 w 1 0 4 12 8 1 1 1 0 1 1 0 5 1 13 0 9 3 7 15 11 z y 1 0 1 1 2 6 14 10 0 0 x 0 w 1 4 5 12 13 8 0 3 0 0 9 7 2 0 6 15 14 11 10 x z Next we minimize the function using a table method (or from the K-Map). SYEN 3330 Digital Systems Chapter 2-6 32 Table Method Minimum SOP PI 4 wx'z xy'z' x w'xy w'xz' x 6 7 9 11 12 Type x x Essential PI x Essential PI x x x Essential PI Redundant Thus: g'(w,x,y,z) = wx'z + xy'z' + w'xy This implements a Standard Sum of Products form for the complement of the desired function. Applying DeMorgan's Law we get: g = (w'+x+z')(x'+y+z)(w+x'+y') A minimum literal Standard POS form. SYEN 3330 Digital Systems Chapter 2-6 33