Boolean Function Simplification


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1 Universit of Wisconsin  Madison ECE/Comp Sci 352 Digital Sstems Fundamentals Charles R. Kime Section Fall 200 Chapter 2 Combinational Logic Circuits Part 5 Charles Kime & Thomas Kaminski Boolean Function Simplification S Reducing the literal cost of a Boolean Epression leads to simpler circuits. S Simpler circuits are less epensive to implement. S Boolean Algebra can help us minimie literal cost. S When do we stop tring to reduce the cost? S Do we know when we have a minimum? S We will introduce a sstematic wa to arrive a a minimum cost, twolevel POS or SOP circuit. Chapter 2Part 5 2
2 Karnaugh Maps (Kmap) S A Kmap is a collection of squares Each square represents a minterm The collection of squares is a graphical representation of a Boolean function Adjacent squares differ in the value (complement) of one variable Alternative algebraic epressions for the same function are derived b recogniing patterns of squares S The Kmap can be viewed as a reorganied version of the truth table S The Karnaugh Map can be viewed as a topologicallwarped Venn diagram as used to visualie sets Chapter 2Part 5 3 Some Uses of KMaps S Provide a means for finding optimum: (Simple) SOP and POS standard forms, and (Small) twolevel AND/OR and OR/AND circuits S Visualie concepts related to manipulating Boolean epressions S Demonstrate concepts used b computeraided design programs to simplif large circuits Chapter 2Part 5 4 2
3 Two Variable Maps S A Two variable Karnaugh Map: = 0 = Note that minterm m0 and minterm m are "adjacent = 0 m 0 = m = and differ in the value of the variable. = m 2 = m 3 = Similarl, minterm m0 and minterm m2 differ in the variable. Also, m and m3 differ in the variable as well. Finall, m2 and m3 differ in the value of the variable Chapter 2Part 5 5 KMap and Truth Tables S S The KMap is just a different form of the truth table. Eample Two Variables: We choose a,b,c and d from the set {0,} to implement a particular function, F(,). Function Table KMap Input Values (,) Function Value F(,) 0 0 a 0 b 0 c d = 0 = = 0 a b = c d Chapter 2Part 5 6 3
4 KMap Function Representation S Eample: F(,) = F = = 0 = = = S For function F(,), the two adjacent cells containing s can be combined using the Minimiation Theorem: F (, ) = + = Chapter 2Part 5 7 KMap Function Representation S Eample: G(,) = + G = + = 0 = = 0 0 = For G(,), two pairs of adjacent cells containing s can be combined using the Minimiation Theorem: ( + )( + + ) = G (, ) = + Duplicate Chapter 2Part 5 8 4
5 Three Variable Maps S A threevariable Kmap: =00 =0 = =0 S S Where each minterm corresponds to the product terms below: =00 =0 = =0 Note that if the binar value for an inde differs in one bit position, the minterms are adjacent on the Karnaugh Map =0 m 0 m m 3 m 2 = m 4 m 5 m 7 m 6 =0 = Chapter 2Part 5 9 Alternative Map Labeling S Map use largel involves: Entering values into the map, and Reading off product terms from the map. S Alternate labelings are useful: Chapter 2Part 5 0 5
6 Eample Functions S B convention, we represent the minterms of F b a "" in the map and leave the minterms of blank S Eample: F(,, ) = Σm(2,3,4,5) S Eample: G(a, b,c) = Σ m (3,4,6,7) F Chapter 2Part 5 Combining Squares S B combining squares, we reduce the representation for a term, reducing the number of literals in the Boolean equation. S On a threevariable KMap: One square represents a minterm with three variables Two adjacent squares represent a product term with two variables Four adjacent terms represent a product term with one variable Eight adjacent terms is the function of all ones (no variables) =. Chapter 2Part 5 2 6
7 Eample: Combining Squares Eample: Let Appling the Minimiation Theorem three times: F(,, ) = = + = Thus the four terms that form a 2 2 square correspond to the term "". F = Σm(2,3,6,7) Chapter 2Part 5 3 ThreeVariable Maps S Reduced literal product terms for SOP standard forms correspond to rectangles on Kmaps containing cell counts that are powers of 2. S Rectangles of 2 cells represent 2 adjacent minterms; of 4 cells represent 4 minterms that form a pairwise adjacent ring. S Rectangles can be in man different positions on the Kmap since adjacencies are not confined to cells trul net to each other. Chapter 2Part 5 4 7
8 ThreeVariable Maps S Topological warps of 3variable Kmaps that show all adjacencies: Venn Diagram T Clinder 0 Y 2 4 X Z Chapter 2Part 5 5 ThreeVariable Maps S Eample Shapes of Rectangles: Chapter 2Part 5 6 8
9 Three Variable Maps KMaps can be used to simplif Boolean functions b sstematic methods. Terms are selected to cover the "ones" in the map. Eample: Simplif F(,, ) m(,2,3,5,7) = Σ F(,, ) = + Chapter 2Part 5 7 ThreeVariable Map Simplification S F(X, Y, Z) = Σm(0,,2,4,6,7) Chapter 2Part 5 8 9
10 Four Variable Maps S Map and location of minterms: Y X W Z Chapter 2Part 5 9 Four Variable Terms S Four variable maps can have rectangles corresponding to: A single one = 4 variables, (i.e. Minterm) Two ones = 3 variables, Four ones = 2 variables Eight ones = variable, Siteen ones = ero variables (i.e. Constant "") Chapter 2Part
11 FourVariable Maps S Eample Shapes of Rectangles: Y X W Z Chapter 2Part 5 2 FourVariable Map Simplification S F(W, X, Y, Z) m(0, 2,4,5,6,7,8,0,3,9) = Σ Chapter 2Part 5 22
12 FourVariable Map Simplification S F(W, X, Y, Z) m(3,4,5,7,3,4,5,7) = Σ Chapter 2Part 5 23 Sstematic Simplification A Prime Implicant is a product term obtained b combining the maimum possible number of adjacent squares in the map. A prime implicant is called an Essential Prime Implicant if it is the onl prime implicant that covers (includes) one or more minterms. Prime Implicants and Essential Prime Implicants can be determined b inspection of a KMap. A set of prime implicants that "covers all minterms" means, for each minterm of the function, at least one prime implicant in the set of prime implicants includes the minterm. Chapter 2Part
13 Eample of Prime Implicants B D BD A AB S Find ALL Prime Implicants D AD CD C BC B ESSENTIAL Prime Implicants C B D BD A D B Chapter 2Part 5 25 Prime Implicant Practice S F(A, B,C, D) = Σm(0,2,3,8,9,0,,2, 3,4,5) Chapter 2Part
14 Sstematic Approach (Suppl. 2). Find all PIs of the function F. 2. Select all essential PIs, checking off included minterms. 3. Find all less than PIs and delete those that are less than, but not equivalent to, at least one other PI.* (As a result, some of the other unselected PIs ma become essential.) 4. Repeat 2 and 3 until no more less than PIs appear. 5. Find equivalent PIs and select arbitraril one PI from each set of equivalent PIs, checking off included minterms. 6. If minterms remain unchecked and no PI less than relations can be obtained, then a cclic structure eists. For a cclic structure, (a) arbitraril select a PI and repeat steps through 6, and (b) delete the same PI selected and repeat steps through 6. Compare literal cost of the solutions generated and select the minimum literal cost cover. 7. Discard an redundant (unused) PIs. *Correction to Supplement 2 Chapter 2Part 5 27 Other PI Selection S Once the Essential Prime Implicants are selected, we need to "prune" the solution set further. To do this, we determine which can be eliminated b finding Less Than PIs and Redundant PIs. Less Than PIs: PI i is said to be Less Than PI j if PI i contains at least as man literals as PI j and PI j covers at least all of the as et uncovered minterms that PIi covers. Equivalent PIs: A set of PIs which are pairwise less than each other. Secondar Essential PIs: Once the less than PIs are removed from consideration, new PIs become essential and the are called Secondar Essential PIs. Redundant PIs: These are PIs whose minterms have been completel covered b the PIs selected and are removed from consideration. Chapter 2Part
15 Eample 2 from Supplement Select Essential PIs: Eliminate Less Than Pis: w w Chapter 2Part 5 29 Eample 2 (Continued) Select Secondar Essential PIs: Eliminate Redundant PIs: w w Chapter 2Part
16 Another Eample S G(A, B,C, D) = Σm(0,2,3,4,7,2,3,4, 5) Chapter 2Part 5 3 Five Variable or More KMaps For five variable problems, we use two adjacent KMaps. It becomes harder to visualie adjacent minterms for selecting PIs. You can etend the problem to si variables b using four K Maps. v=0 v= w w Chapter 2Part
17 Don't Cares in KMaps S Sometimes a function table contains entries for which it is known the input values will never occur. In these cases, the output val ue need not be defined. B placing a don't care in the function table, it ma be possible to arrive at a lower cost logic circuit. S Don't cares are usuall denoted with an "" in the KMap or function table. S Eample of Don't Cares  A logic function defined on 4bit variables encoded as BCD digits where the fourbit input variables never eceed 9, base 2. Smbols 00, 0, 00, 0, 0, and will never occur. Thus, we DON'T CARE what the function value is for these combinations. S Don't cares are used in minimiation procedures in such a wa that the ma ultimatel take on either a 0 or value in the result. Chapter 2Part 5 33 Eample: BCD 5 or More w S A function F(w,,,) which is defined as "5 or greater" over a BCD input, as below. Don't cares are on nonbcd values. S F (w,,,) = w X X X X X X S This is slightl lower in cost than F2 where the don't cares are required to be "0". F 2 (w,,, ) = w + w + w S For this particular function, note that the literal cost of the complement of F (w,,,), meaning "4 or less", is not changed b using the don't cares. Chapter 2Part
18 Product of Sums Eample S F(A, B,C, D) = Σm (3,9,,2,3,4,5) + Σd (,4,6) S Use F and take complement of result: Chapter 2Part
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