# Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples

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1 Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2-Level Transformation Review Boolean Cube Karnaugh-Map Representation and Methods Examples

2 Incompletely specified functions Example: Fruit Detector at Fruit Factory To distinguish Round Fruits (X = 1) from Odd Shaped Fruits (X = 0) Three input signals! Intuition tells us that X = C!! Check this out algebraically

3 Review of DeMorgan s Laws DeMorgan s Law: (a + b) = a b (a b) = a + b a + b = (a b ) (a b) = (a + b ) = = = = push bubbles or introduce in pairs or remove pairs

4 Transformation to Simple Gates Sum of Products Involution: x = (x ) = De Morgans

5 Implementations of Two-level Logic Sum-of-products AND gates to form product terms (minterms) OR gate to form sum Product-of-sums OR gates to form sum terms (maxterms) AND gates to form product

6 Two-level Logic using NAND Gates Replace minterm AND gates with NAND gates Place compensating inversion at inputs of OR gate

7 Two-level Logic using NAND Gates (cont d) OR gate with inverted inputs is a NAND gate de Morgan's: A'+ B'= (A B)' Two-level NAND-NAND network Inverted inputs are not counted In a typical circuit, inversion is done once and signal distributed

8 Two-level Logic using NOR Gates Replace maxterm OR gates with NOR gates Place compensating inversion at inputs of AND gate

9 Two-level Logic using NOR Gates (cont d) AND gate with inverted inputs is a NOR gate de Morgan's: A' B'= (A + B)' Two-level NOR-NOR network Inverted inputs are not counted In a typical circuit, inversion is done once and signal distributed

10 Another look at simplification Key tool to simplification: A (B'+ B) = A Essence of simplification of two-level logic Find two element subsets of the ON-set where only one variable changes its value this single varying variable can be eliminated and a single product term used to represent both elements ())*)(+)*,)() "#\$% & ''"% % &

11 Boolean cubes Visual technique for identifying when the uniting theorem can be applied n input variables = n-dimensional "cube Neighbors address differs by one bit flip

12 Hamming Distance Number of Bit Differences Note Hamming distance between the numbers in the figures

13 Mapping truth tables onto Boolean cubes Uniting theorem combines two "faces" of a cube into a larger "face" Example: % ''/+, #''/ +, "% '0 ') \$( \$( - \$( )

14 Three variable example Binary full-adder carry-out logic +)*, +)*, +*), \$ -"#- #+ 1,'## '% -\$ 2 3 " (**

15 Higher dimensional cubes Sub-cubes of higher dimension than 2 +00,(Σ , % 8 9 \$' : 0#' ; <"# 0= & ; +, "- > ## 9+00,= Σm(Decimal list of min terms) = Π M(Decimal list of MAX terms)

16 m-dimensional cubes in a n- dimensional Boolean space In a 3-cube (three variables): 0-cube, ie, a single node, yields a term in 3 literals 1-cube, ie, a line of two nodes, yields a term in 2 literals 2-cube, ie, a plane of four nodes, yields a term in 1 literal 3-cube, ie, a cube of eight nodes, yields a constant term "1" In general, m-subcube within an n-cube (m < n) yields a term with n m literals

17 Karnaugh maps Flat map of Boolean cube Wrap around at edges Hard to draw and visualize for more than 4 dimensions Virtually impossible for more than 6 dimensions Alternative to truth-tables to help visualize adjacencies Guide to applying the uniting theorem On-set elements with only one variable changing value are adjacent unlike the situation in a linear truth-table ; =

18 Karnaugh maps (cont d) Numbering scheme based on Gray code eg, 00, 01, 11, 10 2 n values of n bits where each differs from next by one bit flip» Hamiltonian circuit through n-cube Only a single bit changes in code for adjacent map cells ; = ; = = 7 ; 6 ;? 5 4 =( (A

19 Adjacencies in Karnaugh maps Wrap from first to last column Wrap top row to bottom row

20 Karnaugh map examples F = Cout = f(a,b,c) = Σm(0,4,6,7) A ** *AA *A # '' #-" % # #

21 More Karnaugh map examples +00,( +00,(Σ , (*AA ) - )% )"" )+00,(Σ + 0;0=06, (A*A

22 K-map: 4-variable interactive quiz F(A,B,C,D) = Σm(0,2,3,5,6,7,8,10,11,14,15) F = ' #'# ## " \$ +'% % '%,

23 Karnaugh map: 4-variable example F(A,B,C,D) = Σm(0,2,3,5,6,7,8,10,11,14,15) F = *A *AA ' #'# ## " \$ +'% % '%,

24 Karnaugh maps: don t cares f(a,b,c,d) = Σ m(1,3,5,7,9) + d(6,12,13) without don't cares» f = A *AA

25 Karnaugh maps: don t cares (cont d) f(a,b,c,d) = Σ m(1,3,5,7,9) + d(6,12,13) f = A'D + B'C'D without don't cares f = with don't cares #-)B B ;\$##' \$#" ) # % "

26 Design example: two-bit comparator ; E> F > #D # G ( H E> F > % )4\$"#C ''='

27 Design example: two-bit comparator (cont d) C\$ 'E> C\$ ' F C\$ ' > E> ( F ( > ( ))*)*) ))))*))**)A ))*)*) (+<,I+<, Canonical PofS vs minimal? E> > +'JJ,

28 Design example: two-bit comparator (cont d) % " ' F % % 1 1 % =

29 Design example: 2x2-bit multiplier ; ; #D # ; 4? ; ;? 4 ; 4\$"#C\$ ''4 '

30 Design example: 2x2-bit multiplier (cont d) ; C\$ '? C\$ '4 4(;; ) *; ); ; ;?(; ; ; ; C\$ '; C\$ ' ( ; ; ; (;) ; * ; ) *;;) *; ) ;

31 Design example: BCD increment by 1 K K; K4 K? #D # ; 4? K? K4 K; K? 4 ; 4\$"#C\$ '' 4'

32 Design example: BCD increment by 1 (cont d) K?? 4 K? K; K4 K? K ;?(K4K;K *K?K ) 4(K4K;)*K4K )*K4AK;K ;(K?AK;AK *K;K ) (K ) K; K4 K? K K K K; K; K4 K4

33 Definition of terms for two-level simplification Implicant Single element of ON-set or DC-set or any group of these elements that can be combined to form a subcube Prime implicant Implicant that can't be combined with another to form a larger subcube Essential prime implicant Prime implicant is essential if it alone covers an element of ON-set Will participate in ALL possible covers of the ON-set DC-set used to form prime implicants but not to make implicant essential Objective: Grow implicant into prime implicants (minimize literals per term) Cover the ON-set with as few prime implicants as possible (minimize number of product terms)

34 Examples to illustrate terms 6 L "L* * 5 L 0000 "L4

35 Algorithm for two-level simplification Algorithm: minimum sum-of-products expression from a Karnaugh map Step 1: choose an element of the ON-set Step 2: find "maximal" groupings of 1s and Xs adjacent to that element» consider top/bottom row, left/right column, and corner adjacencies» this forms prime implicants (number of elements always a power of 2) Repeat Steps 1 and 2 to find all prime implicants Step 3: revisit the 1s in the K-map» if covered by single prime implicant, it is essential, and participates in final cover» 1s covered by essential prime implicant do not need to be revisited Step 4: if there remain 1s not covered by essential prime implicants» select the smallest number of prime implicants that cover the remaining 1s

36 Algorithm for two-level simplification (example)

37 Summary Boolean Algebra provides framework for logic simplification Binary Cubes and Karnaugh maps provide visual notion of simplifications Algorithm for producing reduced form

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