Minimierung. Wintersemester 2018/19. Folien basierend auf Material von F. Vahid und S. Werner Vorlesender: Dr. Ing.

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1 Minimierung Grundlagen der technischen Informatik Wintersemester 28/9 Folien basierend auf Material von F. Vahid und S. Werner Wintersemester 28/9

2 Review - Boolean Algebra Properties Commutative (Kommutativgesetz) a + b = b + a a * b = b * a Distributive (Distributivgesetz) a * (b + c) = a * b + a * c Can write as: a(b+c) = ab + ac a + (b * c) = (a + b) * (a + c) Can write as: a+(bc) = (ab)(ac) Associative (Assoziativgesetz) (a + b) + c = a + (b + c) (a * b) * c = a * (b * c) Identity (Neutrale Elemente) + a = a + = a * a = a * = a Complement (Komplement) a + a = a * a = To prove, just evaluate all possibilities Wintersemester 28/9 2

3 Review - Representations of Boolean Functions English : F outputs when a is and b is, or when a is and b is. English 2: F outputs when a is, regardless of b s value Equation : F(a,b) = a b + a b Equation 2: F(a,b) = a a b F a b F Circuit a F Truth table Circuit 2 Function F A function can be represented in different ways Above shows seven representations of the same functions F(a,b), using four different methods: English, Equation, Circuit, and Truth Dr. Ing. Table Frank Sill Torres Wintersemester 28/9 3

4 Review - Standard Representation: Truth Table How can we determine if two functions are the same? Recall automatic door eample Same as f = hc + h pc? Used algebraic methods But if we failed, does that prove not equal? No. Solution: Convert to truth tables Only ONE truth table representation of a given function Standard representation for given function, only one version in standard form eists f = c hp + c hp + c h f = c h(p + p ) + c h p f = c h() + c h p f = c h + c h p (what if we stopped here?) f = hc + h pc Q: Determine if F=ab+a is same function as F=a b +a b+ab, by converting each to truth table first F = ab + a' a b F Wintersemester 28/9 F = a b + a b + ab a b F 4

5 Review - Canonical Form Sum of Minterms Truth tables too big for numerous inputs Use standard form of equation instead Known as canonical form Regular algebra: group terms of polynomial by power a 2 + b + c ( > ) Boolean algebra: create sum of minterms (also: Disjunctive normal form - DNF) Minterm: product term with every function literal appearing eactly once, in true or complemented form Just multiply-out equation until sum of product terms Then epand each term until all terms are minterms Q: Determine if F(a,b)=ab+a is equivalent to F(a,b)=a b +a b+ab, by converting first equation to canonical form (second already is) F = ab + a (already sum of products) F = ab + a (b+b ) (epanding term) F = ab + a b + a b (Equivalent Grundlagen der Technische same three Informatikterms as other equation) 5 Wintersemester 28/9

6 Alternatives Original Circuit Disjunctive Form Conjunctive Form AND XOR OR Which circuit is the best implementation? What is the BEST? Wintersemester 28/9 6

7 Circuit select Original Circuit Disjunctive Form Conjunctive Form AND XOR OR C A = Area C S = Stages Result: chose circuit or 3 for a compact solution chose circuit 2 or 3 for a fast solution Wintersemester 28/9 7

8 Karnaugh-Veich Maps (KV-Maps) Graphical method for finding terms that can be combined to eliminate variables in order to obtain the minimum Minterms differing from one variable are adjacent (neighbors) on the map We can clearly visualize the possible cases of combination of terms Note that it is not sorted binary F z = F = z = y z y z y z y z KV-map y = Reminder: y = Consider left and right neighbors too `, y`, y Wintersemester 28/9 z`, z 8

9 KV-maps - Styles A line indicates where a variable is Map for more variables: mirroring of basic structure a = b = c = Different versions for a KV-map Important for optimization: Neighboring fields can only differ in eactly one variable Wintersemester 28/9 9

10 KV-maps - Styles Compact representation of KV-maps via numbers KV-map with 2 variables A B A B = Nº 2 A B = Nº AB = Nº 2 AB = Nº 3 KV-map with 3 variables 3 A B C = Nº, A B C = Nº, A BC = Nº 2, A BC = Nº 3, AB C = Nº 4, AB C = Nº 5, ABC = Nº 6, ABC = Nº 7 AB C 2 3 B = A= Wintersemester 28/9

11 KV-maps - Styles KV-map with 4 variables Nº A B C D A = AB CD C= B = D = Wintersemester 28/9

12 Review - Compact DNF Representation List each minterm as a number Number determined from the binary representation of its variables values a'bcde corresponds to, or 5 abcde' corresponds to, or 3 abcde corresponds to, or 3 Thus, H = a'bcde + abcde' + abcde can be written as: H = m(5,3,3) "H is the sum of minterms 5, 3, and 3" Wintersemester 28/9 2

13 C AB CD C F(A,B,C)=A BC + AB C + ABC + ABC = Σm(3,5,6,7) A B A B A B D F(A,B,C,D)=Σm(,2,3,5,6,7,8,,,4,5) Wintersemester 28/9 A B C 3 F

14 KV-Maps Application: Draw circle, write term that has all the literals ecept the one that changes in the circle Circle y, = & y= in both cells of the circle, but z changes (z= in one cell, in the other) Top-left circle generates reduction of y z + y z = y (z +z) = y () = y Minimized function: OR of found terms Easier than the algebraic method: F = + + y z + y z F = y(z + z ) + y (z + z ) F = y* + y * F = y + y Wintersemester 28/9 F = y z y z F F y y F = y + y 4

15 KV-Maps Four adjacent s means two variables can be eliminated Makes intuitive sense those two variables appear in all combinations, so one must be true Draw one big circle shorthand for the algebraic transformations above G = y z + y z + + G = (y z + y z + + ) (must be true) G = (y (z +z) + y(z+z )) G = (y +y) G = G Draw the biggest circle possible, or you ll have more terms than really needed G y Wintersemester 28/9 y 5

16 KV-maps Four adjacent cells can be in shape of a square OK to cover a twice Just like duplicating a term Remember, c + d = c + d + d No need to cover s more than once Yields etra terms not minimized H H = y z + + y z + (y appears in all combinations) I y z z J y y z z The two circles are shorthand for: I = y z + y z + y z + + I = y z + y z + y z + y z + + I = ( y z + y z) + (y z + y z + + ) I = (y z) + () Wintersemester 28/9 6

17 KV-maps Circles can cross left/right sides K y z Remember, edges are adjacent Minterms differ in one variable only Circles must have, 2, 4, or 8 cells 3, 5, or 7 not allowed 3/5/7 doesn t correspond to algebraic transformations that combine terms to eliminate a variable Circling all the cells is OK Function just equals L E z Wintersemester 28/9 7

18 KV-maps for Four Variables Four-variable K-map follows same principle Adjacent cells differ in one variable Left/right adjacent Top/bottom also adjacent 5 and 6 variable maps eist But hard to use (see net slide) Two-variable maps eist But not very useful easy to do algebraically by hand F y z F w w y G w Grundlagen der Technische Informatik Wintersemester 28/9 z F=w y + G=z 8

19 KV-maps with 5 variables F(A,B,C,D)=Σm(2,5,7,8,,3,5,7,9,2,23,24,29,3) A= DE BC 3 2 DE BC A= BC DE A= DE BC A= Wintersemester 28/9 9

20 KV-maps with 6 variables AB= AB = AB = AB = CD EF CD EF CD EF CD EF AB= AB = AB = AB = CD EF CD EF CD EF CD EF Wintersemester 28/9 2

21 Two-Level Size Minimization Using KV-maps General KV-map method. Convert the function s equation into sum-of-products form* 2. Place s in the appropriate KVmap cells for each term 3. Cover all s by drawing the fewest largest circles, with every included at least once; write the corresponding term for each circle 4. OR all the resulting terms to create the minimized function. Eample: Minimize: G = a + a b c + b*(c + bc ). Convert to sum-of-products G = a + a b c + bc + bc 2. Place s in appropriate cells G bc bc a b c 3. Cover s a G a bc a c * hint: could also be DNF 4. OR terms: G = a + c 2 Wintersemester 28/9 a

22 Two-Level Size Minimization Using KV-maps Four Variable Eample Minimize: H = a b (cd + c d ) + ab c d + ab cd + a bd + a bcd. Convert to sum-of-products: H = a b cd + a b c d + ab c d + ab cd + a bd + a bcd 2. Place s in K-map cells 3. Cover s 4. OR resulting terms H ab a b c d a b cd ab c d a bd a bcd cd b d a bc a bd ab cd H = b d + a bc + a bd Funny-looking circle, but remember that left/right adjacent, and top/bottom adjacent Wintersemester 28/9 22

23 KV-Maps from Thruth Table Fill the map s fields with a where the function s value equals Wintersemester 28/9 23

24 Don t Care Input Combinations What if particular input combinations can never occur? e.g., Minimize F = y z, given that y z (=) can never be true, and that y z (=) can never be true So it doesn t matter what F outputs when y z or y z is true, because those cases will never occur Thus, make F be or for those cases in a way that best minimizes the equation On KV-map Draw Xs for don t care combinations Include X in circle ONLY if minimizes equation Don t include other Xs F y z X X Good use of don t cares F y z unneeded X X y Unnecessary use of don t cares; results in etra term Wintersemester 28/9 24

25 Minimizization Eample using Don t Cares Minimize: F = a bc + abc + a b c Given don t cares: a bc, abc F a bc a c b X Note: Use don t cares with caution Must be sure that we really don t care what the function outputs for that input combination If we do care, even the slightest, then it s probably safer to set the output to X F = a c + b Wintersemester 28/9 25

26 Minimization with Don t Cares Eample: Sliding Switch Switch with 5 positions 3-bit value gives position in binary Want circuit that y z 2,3,4, detector G Outputs when switch is in position 2, 3, or 4 Outputs when switch is in position or 5 Note that the 3-bit input can never output binary, 6, or 7 Treat as don t care input combinations Without don t cares: F = y + y z G G Wintersemester 28/9 y X X X z y With don t cares: F = y + z y z 26

27 Automating Two-Level Logic Size Minimization Minimizing by hand Is hard for functions with 5 or more variables May not yield minimum cover depending on order we choose Is error prone Minimization thus typically done by automated tools Eact algorithm: finds optimal solution Heuristic: finds good solution, but not necessarily optimal I (a) I (b) Wintersemester 28/9 y z y y z 4 terms z y y Only 3 terms 27

28 Basic Concepts Underlying Automated Two-Level Logic Minimization Definitions On-set: All minterms that define when F= Off-set: All minterms that define when F= Implicant: Any product term (minterm or other) that when causes F= On KV-map: any legal (but not necessarily largest) circle Cover: Implicant y covers minterms and Epanding a term: removing a variable (like larger KV-map circle) y is an epansion of F Wintersemester 28/9 y z 4 implicants of F y Note: We use KV-maps here just for intuitive illustration of concepts; automated tools do not use K-maps. Prime implicant (PI): Maimally epanded implicant any epansion would cover s not in on-set y z, and y, above But not or they can be epanded 28

29 Basic Concepts Underlying Automated Two-Level Logic Minimization Definitions (cont) Essential prime implicant: The only prime implicant that covers a particular minterm in a function s on-set Importance: We must include all essential PIs in a function s cover In contrast, some, but not all, non-essential PIs will be included G y essential z not essential not essential y z y essential Wintersemester 28/9 29

30 Eample of Automated Two-Level Minimization. Determine all prime implicants 2. Add essential PIs to cover Italicized s are thus already covered Only one uncovered remains 3. Cover remaining minterms with non-essential PIs z Pick among the two possible PIs I (a) I (b) I (c) z y z z y z y y z z y z Wintersemester 28/9 z 3

31 Was haben Sie heute gelernt? Karnaugh-Veich Diagramme sind eine graphische Methode zur Minimierung digitaler Schaltungen Indentifizierung von Literalen, die entfernt werden können Anwendung von don t care -Ausdrücken Grundlagen automatischer Lösungen Wintersemester 28/9 3