Incompletely Specified Functions with Don t Cares 2Level Transformation Review Boolean Cube KarnaughMap Representation and Methods Examples


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1 Lecture B: Logic Minimization Incompletely Specified Functions with Don t Cares 2Level Transformation Review Boolean Cube KarnaughMap 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 Twolevel Logic Sumofproducts AND gates to form product terms (minterms) OR gate to form sum Productofsums OR gates to form sum terms (maxterms) AND gates to form product
6 Twolevel Logic using NAND Gates Replace minterm AND gates with NAND gates Place compensating inversion at inputs of OR gate
7 Twolevel Logic using NAND Gates (cont d) OR gate with inverted inputs is a NAND gate de Morgan's: A'+ B'= (A B)' Twolevel NANDNAND network Inverted inputs are not counted In a typical circuit, inversion is done once and signal distributed
8 Twolevel Logic using NOR Gates Replace maxterm OR gates with NOR gates Place compensating inversion at inputs of AND gate
9 Twolevel Logic using NOR Gates (cont d) AND gate with inverted inputs is a NOR gate de Morgan's: A' B'= (A + B)' Twolevel NORNOR 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 twolevel logic Find two element subsets of the ONset 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 = ndimensional "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 fulladder carryout logic +)*, +)*, +*), $ "# #+ 1,'## '% $ 2 3 " (**
15 Higher dimensional cubes Subcubes 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 mdimensional cubes in a n dimensional Boolean space In a 3cube (three variables): 0cube, ie, a single node, yields a term in 3 literals 1cube, ie, a line of two nodes, yields a term in 2 literals 2cube, ie, a plane of four nodes, yields a term in 1 literal 3cube, ie, a cube of eight nodes, yields a constant term "1" In general, msubcube within an ncube (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 truthtables to help visualize adjacencies Guide to applying the uniting theorem Onset elements with only one variable changing value are adjacent unlike the situation in a linear truthtable ; =
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 ncube 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 Kmap: 4variable interactive quiz F(A,B,C,D) = Σm(0,2,3,5,6,7,8,10,11,14,15) F = ' #'# ## " $ +'% % '%,
23 Karnaugh map: 4variable 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 ;$##' $#" ) # % "
26 Design example: twobit comparator ; E> F > #D # G ( H E> F > % )4$"#C ''='
27 Design example: twobit comparator (cont d) C$ 'E> C$ ' F C$ ' > E> ( F ( > ( ))*)*) ))))*))**)A ))*)*) (+<,I+<, Canonical PofS vs minimal? E> > +'JJ,
28 Design example: twobit comparator (cont d) % " ' F % % 1 1 % =
29 Design example: 2x2bit multiplier ; ; #D # ; 4? ; ;? 4 ; 4$"#C$ ''4 '
30 Design example: 2x2bit 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 twolevel simplification Implicant Single element of ONset or DCset 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 ONset Will participate in ALL possible covers of the ONset DCset used to form prime implicants but not to make implicant essential Objective: Grow implicant into prime implicants (minimize literals per term) Cover the ONset 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 twolevel simplification Algorithm: minimum sumofproducts expression from a Karnaugh map Step 1: choose an element of the ONset 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 Kmap» 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 twolevel 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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