EE260: Digital Design, Spring n Binary logic and Gates n Boolean Algebra. n Basic Properties n Algebraic Manipulation

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1 EE26: Digital Desig, Sprig 28 2-Ja-8 EE 26: Itroductio to Digital Desig Boolea lgebra: Logic Represetatios ad alysis Yao Zheg Departmet of Electrical Egieerig Uiversity of Hawaiʻi at Māoa Overview Biary logic ad Gates Boolea lgebra Basic Properties lgebraic Maipulatio Stadard ad Caoical Forms Miterms ad Materms (Caoical forms) SOP ad POS (Stadard forms) Karaugh Maps (K-Maps) 2, 3, 4, ad 5 variable maps Simplificatio usig K-Maps K-Map Maipulatio Implicats: Prime, Essetial Do t Cares 2 Biary Logic Deals with biary variables that take 2 discrete values ( ad ), ad with logic operatios Three basic logic operatios: ND, OR, NOT Biary/logic variables are typically represeted as letters:,b,c,,x,y,z Biary Logic Fuctio F(vars) = epressio set of biary variables Eample: F(a,b) = a b + b G(,y,z) = (y+z ) Operators ( +,, ) Variables Costats (, ) Groupigs (parethesis) 3 4 Basic Logic Operators ND (also, ) Biary OR (also +, ) NOT (also, ) Uary F(a,b) = a b, reads F is if ad oly if a=b= G(a,b) = a+b, reads G is if either a= or b= H(a) = a, reads H is if a= Basic Logic Operators (cot.) -bit logic ND resembles biary multiplicatio: =, =, =, = -bit logic OR resembles biary additio, ecept for oe operatio: + =, + =, + =, + = ( 2 ) 5 6 Chapter 3: Switchig lgebra ad Combiatioal Logic

2 EE26: Digital Desig, Sprig 28 2-Ja-8 Truth Tables for logic operators Truth table: tabular form that uiquely represets the relatioship betwee the iput variables of a fuctio ad its output 2-Iput ND B F= B 2-Iput OR B F=+B NOT F= Truth Tables (cot.) Q: Let a fuctio F() deped o variables. How may rows are there i the truth table of F()? : 2 rows, sice there are 2 possible biary patters/combiatios for the variables 7 8 Logic Gates Timig Diagram Logic gates are abstractios of electroic circuit compoets that operate o oe or more iput sigals to produce a output sigal. Iput sigals B t t t 2 t 3 t 4 t 5 t 6 Trasitios B 2-Iput ND 2-Iput OR NOT (Iverter) F = B F G H B G = +B H = Gate Output Sigals F= B G=+B H= Basic ssumptio: Zero time for sigals to propagate Through gates 9 Combiatioal Logic Circuit from Logic Fuctio Cosider fuctio F = + B C + B combiatioal logic circuit ca be costructed to implemet F, by appropriately coectig iput sigals ad logic gates: Circuit iput sigals à from fuctio variables (, B, C) Circuit output sigal à fuctio output (F) Logic gates à from logic operatios C B F Combiatioal Logic Circuit from Logic Fuctio (cot.) I order to desig a cost-effective ad efficiet circuit, we must miimize the circuit s size (area) ad propagatio delay (time required for a iput sigal chage to be observed at the output lie) Observe the truth table of F= + B C + B ad G= + B C Truth tables for F ad G are idetical à same fuctio Use G to implemet the logic circuit (less compoets) B C F G 2 Chapter 3: Switchig lgebra ad Combiatioal Logic 2

3 EE26: Digital Desig, Sprig 28 2-Ja-8 Combiatioal Logic Circuit from Logic Fuctio (cot.) Boolea lgebra C B C B F G VERY ice machiery used to maipulate (simplify) Boolea fuctios George Boole (85-864): ivestigatio of the laws of thought Termiology: Literal: variable or its complemet Product term: literals coected by Sum term: literals coected by Boolea lgebra Properties Let X: boolea variable,,: costats Boolea lgebra Properties (cot.) Let X: boolea variable,,: costats. X + = X -- Zero iom 2. X = X -- Uit iom 3. X + = -- Uit Property 4. X = -- Zero Property 5. X + X = X -- Idepotece 6. X X = X -- Idepotece 7. X + X = -- Complemet 8. X X = -- Complemet 9. (X ) = X -- Ivolutio 5 6 The Duality Priciple The dual of a epressio is obtaied by echagig ( ad +), ad ( ad ) i it, provided that the precedece of operatios is ot chaged. Caot echage with Eample: Fid H(,y,z), the dual of F(,y,z) = yz + y z H = ( +y+z ) ( +y + z) The Duality Priciple (cot.) With respect to duality, Idetities 8 have the followig relatioship:. X + = X 2. X = X (dual of ) 3. X + = 4. X = (dual of 3) 5. X + X = X 6. X X = X (dual of 5) 7. X + X = 8. X X = (dual of 8) 7 8 Chapter 3: Switchig lgebra ad Combiatioal Logic 3

4 EE26: Digital Desig, Sprig 28 2-Ja-8 More Boolea lgebra Properties Let X,Y, ad Z: boolea variables. X + Y = Y + X. X Y = Y X -- Commutative 2. X + (Y+Z) = (X+Y) + Z 3. X (Y Z ) = (X Y) Z -- ssociative 4. X (Y+Z) = X Y + X Z 5. X+(Y Z) = (X+Y) (X+Z) -- Distributive 6. (X + Y) = X Y 7. (X Y) = X + Y -- DeMorga s I geeral, ( X + X X ) = X X 2 X, ad ( X X 2 X ) = X + X X bsorptio Property (Coverig). + y = 2. (+y) = (dual) Proof: + y = + y = (+y) = = QED (2 true by duality, why?) 9 2 Power of Duality. + y = is true, so ( + y) = 2. ( + y) = ( +y ) 3. ( +y ) = 4. Let X=, Y=y 5. X (X+Y) =X, which is the dual of + y =. 6. The above process ca be applied to ay formula. So if a formula is valid, the its dual must also be valid. 7. Provig oe formula also proves its dual Cosesus Theorem. y + z + yz = y + z 2. (+y) ( +z) (y+z) = (+y) ( +z) -- (dual) Proof: y + z + yz = y + z + (+ )yz = y + z + yz + yz = (y + yz) + ( z + zy) = y + z QED (2 true by duality) Truth Tables (revisited) Eumerates all possible combiatios of variable values ad the correspodig fuctio value Truth tables for some arbitrary fuctios F (,y,z), F 2 (,y,z), ad F 3 (,y,z) are show to the right. y z F F 2 F 3 Truth Tables (cot.) Truth table: a uique represetatio of a Boolea fuctio If two fuctios have idetical truth tables, the fuctios are equivalet (ad vice-versa). Truth tables ca be used to prove equality theorems. However, the size of a truth table grows epoetially with the umber of variables ivolved, hece uwieldy. This motivates the use of Boolea lgebra Chapter 3: Switchig lgebra ad Combiatioal Logic 4

5 EE26: Digital Desig, Sprig 28 2-Ja-8 Boolea epressios-not uique lgebraic Maipulatio Ulike truth tables, epressios represetig a Boolea fuctio are NOT uique. Eample: F(,y,z) = y z + y z + y z G(,y,z) = y z + y z The correspodig truth tables for F() ad G() are to the right. They are idetical! Thus, F() = G() y z F G Boolea algebra is a useful tool for simplifyig digital circuits. Why do it? Simpler ca mea cheaper, smaller, faster (revisit slides -3). Eample: Simplify F = yz + yz + z. F = yz + yz + z = y(z+z ) + z = y + z = y + z lgebraic Maipulatio (cot.) Eample: Prove y z + yz + yz = z + yz Proof: y z + yz + yz = y z + yz + yz + yz = z (y +y) + yz ( +) = z + yz = z + yz QED. Complemet of a Fuctio The complemet of a fuctio is derived by iterchagig ( ad +), ad ( ad ), ad complemetig each variable. Otherwise, iterchage s to s i the truth table colum showig F. The complemet of a fuctio IS NOT THE SME as the dual of a fuctio Complemetatio: Eample Fid G(,y,z), the complemet of F(,y,z) = y z + yz G = F = (y z + yz) = (y z ) ( yz) DeMorga = ( +y+z) (+y +z ) DeMorga agai Note: The complemet of a fuctio ca also be derived by fidig the fuctio s dual, ad the complemetig all of the literals Caoical ad Stadard Forms We eed to cosider formal techiques for the simplificatio of Boolea fuctios. Miterms ad Materms Sum-of-Miterms ad Product-of- Materms Product ad Sum terms Sum-of-Products (SOP) ad Product-of-Sums (POS) 29 3 Chapter 3: Switchig lgebra ad Combiatioal Logic 5

6 EE26: Digital Desig, Sprig 28 2-Ja-8 Defiitios Literal: variable or its complemet Product term: literals coected by Sum term: literals coected by + Miterm: a product term i which all the variables appear eactly oce, either complemeted or ucomplemeted Materm: a sum term i which all the variables appear eactly oce, either complemeted or ucomplemeted Miterm Represets eactly oe combiatio i the truth table. Deoted by m j, where j is the decimal equivalet of the miterm s correspodig biary combiatio (b j ). variable i m j is complemeted if its value i b j is, otherwise is ucomplemeted. Eample: ssume 3 variables (,B,C), ad j=3. The, b j = ad its correspodig miterm is deoted by m j = BC 3 32 Materm Represets eactly oe combiatio i the truth table. Deoted by M j, where j is the decimal equivalet of the materm s correspodig biary combiatio (b j ). variable i M j is complemeted if its value i b j is, otherwise is ucomplemeted. Eample: ssume 3 variables (,B,C), ad j=3. The, b j = ad its correspodig materm is deoted by M j = +B +C Truth Table otatio for Miterms ad Materms Miterms ad Materms are easy to deote usig a truth table. Eample: ssume 3 variables,y,z (order is fied) y z Miterm Materm y z = m +y+z = M y z = m +y+z = M yz = m 2 +y +z = M 2 yz = m 3 +y +z = M 3 y z = m 4 +y+z = M 4 y z = m 5 +y+z = M 5 yz = m 6 +y +z = M 6 yz = m 7 +y +z = M Caoical Forms (Uique) y Boolea fuctio F( ) ca be epressed as a uique sum of miterms ad a uique product of materms (uder a fied variable orderig). I other words, every fuctio F() has two caoical forms: Caoical Sum-Of-Products (sum of miterms) Caoical Product-Of-Sums (product of materms) Caoical Forms (cot.) Caoical Sum-Of-Products: The miterms icluded are those m j such that F( ) = i row j of the truth table for F( ). Caoical Product-Of-Sums: The materms icluded are those M j such that F( ) = i row j of the truth table for F( ) Chapter 3: Switchig lgebra ad Combiatioal Logic 6

7 EE26: Digital Desig, Sprig 28 2-Ja-8 Eample Truth table for f (a,b,c) at right The caoical sum-of-products form for f is f (a,b,c) = m + m 2 + m 4 + m 6 = a b c + a bc + ab c + abc The caoical product-of-sums form for f is f (a,b,c) = M M 3 M 5 M 7 = (a+b+c) (a+b +c ) (a +b+c ) (a +b +c ). Observe that: m j = M j a b c f 37 Shorthad: ad f (a,b,c) = m(,2,4,6), where idicates that this is a sum-of-products form, ad m(,2,4,6) idicates that the miterms to be icluded are m, m 2, m 4, ad m 6. f (a,b,c) = M(,3,5,7), where idicates that this is a product-of-sums form, ad M(,3,5,7) idicates that the materms to be icluded are M, M 3, M 5, ad M 7. Sice m j = M j for ay j, m(,2,4,6) = M(,3,5,7) = f (a,b,c) 38 Coversio Betwee Caoical Forms Stadard Forms (NOT Uique) Replace with (or vice versa) ad replace those j s that appeared i the origial form with those that do ot. Eample: f (a,b,c) = a b c + a bc + ab c + abc = m + m 2 + m 4 + m 6 = (,2,4,6) = (,3,5,7) = (a+b+c) (a+b +c ) (a +b+c ) (a +b +c ) Stadard forms are like caoical forms, ecept that ot all variables eed appear i the idividual product (SOP) or sum (POS) terms. Eample: f (a,b,c) = a b c + bc + ac is a stadard sum-of-products form f (a,b,c) = (a+b+c) (b +c ) (a +c ) is a stadard product-of-sums form Coversio of SOP from stadard to caoical form Epad o-caoical terms by isertig equivalet of i each missig variable : ( + ) = Remove duplicate miterms f (a,b,c) = a b c + bc + ac = a b c + (a+a )bc + a(b+b )c = a b c + abc + a bc + abc + ab c = a b c + abc + a bc + ab c 4 Coversio of POS from stadard to caoical form Epad ocaoical terms by addig i terms of missig variables (e.g., = ) ad usig the distributive law Remove duplicate materms f (a,b,c) = (a+b+c) (b +c ) (a +c ) = (a+b+c) (aa +b +c ) (a +bb +c ) = (a+b+c) (a+b +c ) (a +b +c ) (a +b+c ) (a +b +c ) = (a+b+c) (a+b +c ) (a +b +c ) (a +b+c ) 42 Chapter 3: Switchig lgebra ad Combiatioal Logic 7

8 EE26: Digital Desig, Sprig 28 2-Ja-8 Karaugh Maps Karaugh maps (K-maps) are graphical represetatios of boolea fuctios. Oe map cell correspods to a row i the truth table. lso, oe map cell correspods to a miterm or a materm i the boolea epressio Multiple-cell areas of the map correspod to stadard terms. 43 Two-Variable Map 2 m 2 3 m 2 m m 3 OR 2 2 NOTE: orderig of variables is IMPORTNT for f(, 2 ), is the row, 2 is the colum. Cell represets 2 ; Cell represets 2 ; etc. If a miterm is preset i the fuctio, the a is placed i the correspodig cell. m 3 m m 2 m3 44 Two-Variable Map (cot.) 2-Variable Map -- Eample y two adjacet cells i the map differ by ONLY oe variable, which appears complemeted i oe cell ad ucomplemeted i the other. Eample: m (= 2 ) is adjacet to m (= 2 ) ad m 2 (= 2 ) but NOT m 3 (= 2 ) f(, 2 ) = = m + m + m 2 = + 2 s placed i K-map for specified miterms m, m, m 2 Groupig (ORig) of s allows simplificatio What (simpler) fuctio is represeted by each dashed rectagle? = m + m 2 = m + m 2 Note m covered twice Miimizatio as SOP usig K-map Eter s i the K-map for each product term i the fuctio Group adjacet K-map cells cotaiig s to obtai a product with fewer variables. Groups must be i power of 2 (2, 4, 8, ) Hadle boudary wrap for K-maps of 3 or more variables. Realize that aswer may ot be uique yz Three-Variable Map m m 4 m m m m 7 -Note: variable orderig is (,y,z); yz specifies colum, specifies row. -Each cell is adjacet to three other cells (left or right or top or bottom or edge wrap) m 2 m Chapter 3: Switchig lgebra ad Combiatioal Logic 8

9 EE26: Digital Desig, Sprig 28 2-Ja-8 Three-Variable Map (cot.) The types of structures that are either miterms or are geerated by repeated applicatio of the miimizatio theorem o a three variable map are show at right. Groups of, 2, 4, 8 are possible. group of 2 terms group of 4 terms miterm Simplificatio Eter miterms of the Boolea fuctio ito the map, the group terms Eample: f(a,b,c) = a c + abc + bc Result: f(a,b,c) = a c + b a bc 49 5 More Eamples X yz f (, y, z) = m(2,3,5,7) f (, y, z) = y + z YZ WX m m 4 m 2 Four-Variable Maps m m 3 m 2 m 5 m 7 m 6 m 3 m 5 m 4 f 2 (, y, z) = m (,,2,3,6) f 2 (, y, z) = +yz m 8 m 9 m m Top cells are adjacet to bottom cells. Leftedge cells are adjacet to right-edge cells. Note variable orderig (WXYZ) Four-variable Map Simplificatio Oe square represets a miterm of 4 literals. rectagle of 2 adjacet squares represets a product term of 3 literals. rectagle of 4 squares represets a product term of 2 literals. rectagle of 8 squares represets a product term of literal. rectagle of 6 squares produces a fuctio that is equal to logic. Eample Simplify the followig Boolea fuctio (,B,C,D) = m(,,2,4,5,7,8,9,,2,3). First put the fuctio g( ) ito the map, ad the group as may s as possible. cd ab g(,b,c,d) = c +b d +a bd Chapter 3: Switchig lgebra ad Combiatioal Logic 9

10 EE26: Digital Desig, Sprig 28 2-Ja-8 DE BC 5-Variable K-Map DE BC = = BCDE BCDE Implicats ad Prime Implicats (PIs) Implicat (P) of a fuctio F is a product term which implies F, i.e., F(P) =. implicat (PI) of F is called a Prime Implicat of F if ay product term obtaied by deletig a literal of PI is NOT a implicat of F Thus, a prime implicat is ot cotaied i ay larger implicat Eample Essetial Prime Implicats (EPIs) Cosider fuctio f(a,b,c,d) whose K-map is show at right. a b is ot a prime implicat because it is cotaied i b. acd is ot a prime implicat because it is cotaied i ad. b, ad, ad a cd are prime implicats. cd ab a b a cd b ad acd If a miterm of a fuctio F is icluded i ONLY oe prime implicat p, the p is a essetial prime implicat of F. essetial prime implicat MUST appear i all possible SOP epressios of a fuctio To fid essetial prime implicats: Geerate all prime implicats of a fuctio Select those prime implicats that cotai at least oe that is ot covered by ay other prime implicat. For the previous eample, the PIs are b, ad, ad a cd ; all of these are essetial. a cd b ad The Other Eample Cosider f 2 (a,b,c,d), whose K-map is show below. The oly essetial PI is b d. cd ab 59 Systematic Procedure for Simplifyig Boolea Fuctios. Geerate all PIs of the fuctio. 2. Iclude all essetial PIs. 3. For remaiig miterms ot icluded i the essetial PIs, select a set of other PIs to cover them, with miimal overlap i the set. 4. The resultig simplified fuctio is the logical OR of the product terms selected above. 6 Chapter 3: Switchig lgebra ad Combiatioal Logic

11 EE26: Digital Desig, Sprig 28 2-Ja-8 Eample Product of Sums Simplificatio f(a,b,c,d) = m(,,2,3,4,5,7,4,5). Five grouped terms, ot all eeded. 3 shaded cells covered by oly oe term 3 EPIs, sice each shaded cell is covered by a differet term. F(a,b,c,d) = a b + a c + a d + abc ab cd Use sum-of-products simplificatio o the zeros of the fuctio i the K-map to get F. Fid the complemet of F, i.e. (F ) = F Recall that the complemet of a boolea fuctio ca be obtaied by () takig the dual ad (2) complemetig each literal. OR, usig DeMorga s Theorem POS Eample ab cd F (a,b,c,d) = ab + ac + a bcd Fid dual of F, dual(f ) = (a+b )(a+c )(a +b+c+d ) Complemet of literals i dual(f ) to get F F = (a +b)(a +c)(a+b +c +d) Do't Care Coditios There may be a combiatio of iput values which will ever occur if they do occur, the output is of o cocer. The fuctio value for such combiatios is called a do't care. They are deoted with or. Each may be arbitrarily assiged the value or i a implemetatio. Do t cares ca be used to further simplify a fuctio Miimizatio usig Do t Cares Treat do't cares as if they are s to geerate PIs. Delete PI's that cover oly do't care miterms. Treat the coverig of remaiig do't care miterms as optioal i the selectio process (i.e. they may be, but eed ot be, covered). 65 Eample Simplify the fuctio f(a,b,c,d) whose K-map is show at the right. f = a c d+ab +cd +a bc or f = a c d+ab +cd +a bd The middle two terms are EPIs, while the first ad last terms are selected to cover the miterms m, m 4, ad m 5. (There s a third solutio!) ab cd 66 Chapter 3: Switchig lgebra ad Combiatioal Logic

12 EE26: Digital Desig, Sprig 28 2-Ja-8 The Other Eample Simplify the fuctio g(a,b,c,d) whose K-map is show at right. g = a c + ab or g = a c +bd ab cd lgorithmic miimizatio What do we do for fuctios with more tha 4-5 variables? You ca code up a miimizer (Computer-ided Desig, CD) Quie-McCluskey algorithm Iterated cosesus We wo t discuss these techiques here Chapter 3: Switchig lgebra ad Combiatioal Logic 2