# CS 135: Computer Architecture I. Boolean Algebra. Basic Logic Gates

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1 Bsic Logic Gtes : Computer Architecture I Boolen Algebr Instructor: Prof. Bhgi Nrhri Dept. of Computer Science Course URL: Digitl Logic Circuits We sw how we cn build the simple logic gtes using trnsistors N-type: send to gte to close switch P-type: send to gte to close switch Use these gtes s building blocks to build more complex combintionl circuits Decoder: bsed on vlue of n-bit input control signl, select one of 2 N outputs Multiplexer: bsed on vlue of N-bit input control signl, select one of 2 N inputs. Adder: dd two binry numbers ny boolen function Theory of Combintionl Logic Design? Is there well grounded theory behind design of boolen logic circuits/functions? Equivlent circuits? Efficient design? Fewest gtes used

2 Boolen Algebr To describe behvior of combintionl circuit Truth tble Boolen lgebric expressions Digitl logic circuit/digrm Algebric expression written ccording to lws of boolen lgebr specifies not only wht combintionl circuit does, but lso how it does it! Truth Tble to Digitl Circuit design Look t ll rows with in the output ALL conditions in the input must hold for the output to be = AND of ll the input conditions Any row cn be = do n OR of ll the row conditions When is x =? Look t ll rows where x = Wht re the vlues of inputs for ech of these rows? Cnonicl Boolen expressions with Minterms Ech row in truth tble specifies vlues of ll the input vribles Wht is the vlue of ech input vrible or i.e., y= or y = for ech input vrible y For the specific output, wht row(s) re we interested? When is the vlue of the output = When the vlue in the row of the truth tble = Wht re the vlues of the input vribles? Cnonicl boolen expression Any boolen expression cn be converted to n equivlent two level AND-OR expression n OR of AND terms, ech AND term corresponds to in the truth tble row, ech AND term contins ll input vribles exctly once i.e., minterm Cnonicl expression: OR of minterms Boolen Algebr Definitions..recll from CS23 Boolen lgebr hs three opertions defined over boolen vribles: OR (+), AND (.) nd complement ( ) Recll fundmentl properties of Boolen lgebr These pply to nything tht is boolen lgebr Sets, digitl logic circuits, 2

3 Boolen Algebr Fundmentl Properties Lws of Boolen lgebr Commuttive: x+y = y+x Associtive (x+y)+z = x+(y+z) Distributive x+(y.z) = (x+y).(x+z) Identity x.y = y.x x= = x x. = x Complement (x.y).z = x.(y.z) x + (x ) = x.(x ) = x.(y+z)=(x.y)+(x.z) Dulity property: ech boolen property hs dul property Exchnge + nd. Exchnge nd Mny useful properties/theorems cn be proved from the fundmentl properties Exmple: Idempotent Property Some useful properties... Prove: x + x = x Proof: use only the fundmentl lws x+x = (x+x). ; From identity property (x+x). = (x+x).(x+x ) ; complement (x+x).(x+x )= (x.x) + (x.x ) ; distributive x + (x.x )= x + ; complement x+ = x ; identity property QED The dulity property is: x.x =x Zero theorem x+ = x. = Absorption property x + x.y = x x.(x+y) = x De Morgn s lw (.b) = + b (+b) =. b (.b.c) = +b +c (+b+c) =.b.c Complement (x ) = x 3

4 Two Level Circuits Every boolen expression cn be trnsformed to n AND-OR expression Resulting in 2 level circuit Advntge of 2 level circuit? Gte delys Go through only two levels/lyers of gtes Why this discussion of Boolen Algebr Every boolen expression hs corresponding logic circuit digrm; nd every logic circuit digrm hs corresponding boolen expression One to one correspondence But given truth tble cn hve severl corresponding implementtions How to mp from truth tble to boolen expression? How to pick the best boolen expression? Simplifiction of boolen expressions The boolen expression/function x(,b,c,d) = bd + c d + d Cn be simplified using bsorption property to bd + ( c d )+( c d )b = bd + c d Combintionl Circuit Design: Truth Tbles nd Boolen expressions Given truth tble we wnt to find n efficient implementtion (i.e., circuit) Efficient in speed Efficient in number of gtes Simplicity of design Cnonicl boolen expression Any boolen expression cn be converted to n equivlent two level AND-OR expression n OR of AND terms, ech AND term corresponds to in the truth tble row, ech AND term contins ll input vribles exctly once i.e., minterm Cnonicl expression: OR of minterms Grphicl method for designing two level circuits with 3 or 4 vribles using minimum possible number of gtes Wht is this method???? 4

5 Cnonicl Expressions Exmple Consider boolen expression x, where x(,b,c)= + +b First two re minterms since they contin ll three input vribles + +b = + + b(c+c ) = = + + Truth tble? b c x Trnsformtion of the Boolen expression + + = ( + ) + = = + b(c + c ) = + b + + = = ( +) + b (c+c ) = + b Truth tble in 2-dimensions b (b=) ( (=) c (c=) x = x 2 =b Therefore, x = + + = + b 5

6 Distnce between minterms Krnugh Mps Concept of distnce between two minterms (Hmming distnce): Number of vribles tht re different Distnce(, )= only c nd c different Distnce(, )=2 both nd c re different Arrnge 2-d truth tble so tht vlues in consecutive columns(rows) differ in one bit position Grphicl wy to represent boolen functions Bsed on concept of distnce Recognizing djcent minterms is key to minimiztion of AND-OR expression K-mp is tool to minimize two level circuit tht it mkes it esy to spot djcent minterms Krnugh Mp is truth tble rrnged so tht djcent entries represent minterms tht differ by one. Grouping minterms in K-Mp Minimiztion using K-Mps Group cells in K-mp tht re djcent nd hve vlue of in the cell Group of 2 cells in 3 vrible K-mp: is n AND of two vribles Group of 4 cells in 3 vrible K-mp: is single vrible Minimiztion procedure : determine best set of groups tht will cover ll the s in the K-mp best mens the set tht corresponds to two-level circuit with the lest number of gtes nd the lest number of inputs per gte. The number of groups equls number of AND gtes We wnt the smllest number of groups with ech group s lrge s possible such tht the groups cover ll the s 6

7 Exmple Exmple b (b=) ( (=) b (b=) ( (=) c (c=) c (c=) Exmple Exmple b (b=) ( (=) b (b=) ( (=) c (c=) c (c=) b + b c + c b + c 7

8 4 vrible Krnugh mp cd b b d c Summry of Combintionl Logic Combintionl vs. Sequentil Combintionl device/circuit: ny circuit built using the bsic gtes Expressed s Truth tble Digitl circuit Boolen function Any boolen function cn be expressed s two level function Minimiztion procedure: Krnugh Mp Try to minimize the number of gtes, nd inputs to gtes, in two level circuit Combintionl Circuit lwys gives the sme output for given set of inputs ex: dder lwys genertes sum nd crry, regrdless of previous inputs Sequentil Circuit stores informtion output depends on stored informtion (stte) plus input so given input might produce different outputs, depending on the stored informtion exmple: vending mchine Current totl increses when you insert coins output depends on previous stte useful for building memory elements nd stte mchines 8

9 Vending Mchine Next... Sequentil Logic Problem: Red input coins, Keep trck of current totl. If totl equl to (or greter thn) 75 cents then process request Wht do we need to keep trck of? Current totl (of coins fed into the mchine) Wht is stte of the mchine Current totl! Assume only 5c, c, 25cent coins trnsition between sttes? Current stte goes to next stte 5 if input = 5c Next stte 2 if input =c Next stte 35 if input=25c Build device, using combintionl logic devices, to store vlue RS Ltch (lso clled SR Ltch) Implement concept of memory Methodology behind design of sequentil logic circuits Finite Stte Mchines Combine sequentil nd combintionl logic devices to ssemble simple processor! Sequentil Circuits Combintionl logic circuits re perfect for situtions when we require the immedite ppliction of Boolen function to set of inputs. There re other times, however, when we need circuit to chnge its vlue with considertion to its current stte s well s its inputs. These circuits hve to remember their current stte. Sequentil logic circuits provide this functionlity for us. 9

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