# Simplification of two-level combinational logic

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1 ombinational logic optimization! lternate representations of oolean functions " cubes " karnaugh maps! Simplification " two-level simplification " exploiting don t cares " algorithm for simplification utumn 2000 SE370 - IV - ombinational Logic Optimization 1 Simplification of two-level combinational logic! Finding a minimal sum of products or product of sums realization " exploit don't care information in the process! lgebraic simplification " not an algorithmic/systematic procedure " how do you know when the minimum realization has been found?! omputer-aided design tools " precise solutions require very long computation times, especially for functions with many inputs (> 10) " heuristic methods employed "educated guesses" to reduce amount of computation and yield good if not best solutions! Hand methods still relevant " to understand automatic tools and their strengths and weaknesses " ability to check results (on small examples) utumn 2000 SE370 - IV - ombinational Logic Optimization 2

2 The uniting theorem! Key tool to simplification: (' + ) =! 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 F = ''+' = ('+)' = ' F has the same value in both on-set rows remains has a different value in the two rows is eliminated utumn 2000 SE370 - IV - ombinational Logic Optimization 3 oolean cubes! Visual technique for indentifying when the uniting theorem can be applied! n input variables = n-dimensional "cube" 1-cube Y cube cube 000 Y Z 101 Y 0000 Z W cube utumn 2000 SE370 - IV - ombinational Logic Optimization 4

3 Mapping truth tables onto oolean cubes! Uniting theorem combines two "faces" of a cube into a larger "face"! Example: F F two faces of size 0 (nodes) combine into a face of size 1(line) ON-set = solid nodes OFF-set = empty nodes -set = 'd nodes varies within face, does not this face represents the literal ' utumn 2000 SE370 - IV - ombinational Logic Optimization 5 Three variable example! inary full-adder carry-out logic in out ('+)in (in'+in) (+')in the on-set is completely covered by the combination (OR) of the subcubes of lower dimensionality - note that 111 is covered three times out = in++in utumn 2000 SE370 - IV - ombinational Logic Optimization 6

4 Higher dimensional cubes! Sub-cubes of higher dimension than 2 F(,,) = Σm(4,5,6,7) on-set forms a square i.e., a cube of dimension 2 represents an expression in one variable i.e., 3 dimensions 2 dimensions is asserted (true) and unchanged and vary This subcube represents the literal utumn 2000 SE370 - IV - ombinational Logic Optimization 7 m-dimensional cubes in a n-dimensional oolean space! In a 3-cube (three variables): " a 0-cube, i.e., a single node, yields a term in 3 literals " a 1-cube, i.e., a line of two nodes, yields a term in 2 literals " a 2-cube, i.e., a plane of four nodes, yields a term in 1 literal " a 3-cube, i.e., a cube of eight nodes, yields a constant term "1"! In general, " an m-subcube within an n-cube (m < n) yields a term with n m literals utumn 2000 SE370 - IV - ombinational Logic Optimization 8

5 Karnaugh maps! Flat map of oolean cube " wrap around at edges " hard to draw and visualize for more than 4 dimensions " virtually impossible for more than 6 dimensions! lternative 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 F utumn 2000 SE370 - IV - ombinational Logic Optimization 9 Karnaugh maps (cont d)! Numbering scheme based on Gray code " e.g., 00, 01, 11, 10 " only a single bit changes in code for adjacent map cells = 1101= utumn 2000 SE370 - IV - ombinational Logic Optimization 10

6 djacencies in Karnaugh maps! Wrap from first to last column! Wrap top row to bottom row utumn 2000 SE370 - IV - ombinational Logic Optimization 11 Karnaugh map examples! F =! out =! f(,,) = Σm(0,4,6,7) + in +in in + + obtain the complement of the function by covering 0s with subcubes utumn 2000 SE370 - IV - ombinational Logic Optimization 12

7 More Karnaugh map examples G(,,) = F(,,) = Σm(0,4,5,7) =+ F' simply replace 1's with 0's and vice versa F'(,,) = Σ m(1,2,3,6) = + utumn 2000 SE370 - IV - ombinational Logic Optimization 13 Karnaugh map: 4-variable example! F(,,,) = Σm(0,2,3,5,6,7,8,10,11,14,15) F = find the smallest number of the largest possible subcubes to cover the ON-set (fewer terms with fewer inputs per term) utumn 2000 SE370 - IV - ombinational Logic Optimization 14

8 Karnaugh maps: don t cares! f(,,,) = Σ m(1,3,5,7,9) + d(6,12,13) " without don't cares f = utumn 2000 SE370 - IV - ombinational Logic Optimization 15 Karnaugh maps: don t cares (cont d)! f(,,,) = Σ m(1,3,5,7,9) + d(6,12,13) " f = ' + '' without don't cares " f = ' + ' with don't cares by using don't care as a "1" a 2-cube can be formed rather than a 1-cube to cover this node don't cares can be treated as 1s or 0s depending on which is more advantageous utumn 2000 SE370 - IV - ombinational Logic Optimization 16

9 ctivity! Minimize the function F = Σ m(0, 2, 7, 8, 14, 15) + d(3, 6, 9, 12, 13) F = F = + + F = utumn 2000 SE370 - IV - ombinational Logic Optimization esign example: two-bit comparator N1 N2 LT EQ GT block diagram and truth table < = > LT EQ GT we'll need a 4-variable Karnaugh map for each of the 3 output functions utumn 2000 SE370 - IV - ombinational Logic Optimization 18

10 esign example: two-bit comparator (cont d) K-map for LT K-map for EQ K-map for GT LT = EQ = GT = ' ' + ' + ' ' ' ' ' + ' ' + + ' = ( xnor ) ( xnor ) ' ' + ' + ' LT and GT are similar (flip / and /) utumn 2000 SE370 - IV - ombinational Logic Optimization 19 esign example: two-bit comparator (cont d) two alternative implementations of EQ with and without OR EQ EQ NOR is implemented with at least 3 simple gates utumn 2000 SE370 - IV - ombinational Logic Optimization 20

11 esign example: 2x2-bit multiplier block diagram and truth table P1 P2 P4 P P8 P4 P2 P variable K-map for each of the 4 output functions utumn 2000 SE370 - IV - ombinational Logic Optimization 21 esign example: 2x2-bit multiplier (cont d) 2 K-map for P8 1 K-map for P4 P4 = 221' + 21' P8 = K-map for P2 1 K-map for P1 P1 = P2 = 2' ' + 22'1 + 21'1 2 1 utumn 2000 SE370 - IV - ombinational Logic Optimization 22

12 esign example: increment by 1 I1 I2 I4 I8 block diagram and truth table O1 O2 O4 O8 I8 I4 I2 I1 O8 O4 O2 O variable K-map for each of the 4 output functions utumn 2000 SE370 - IV - ombinational Logic Optimization 23 esign example: increment by 1 (cont d) I8 1 O8 O4 I8 0 I2 0 I4 I8 0 I1 O2 O8 = I4 I2 I1 + I8 I1' O4 = I4 I2' + I4 I1' + I4 I2 I1 I2 O2 = I8 I2 I1 + I2 I1' O1 = I1' O1 0 I4 I8 1 I1 0 I1 0 I1 I2 I4 I2 I4 utumn 2000 SE370 - IV - ombinational Logic Optimization 24

13 efinition of terms for two-level simplification! Implicant " single element of ON-set or -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 LL possible covers of the ON-set " -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) utumn 2000 SE370 - IV - ombinational Logic Optimization 25 Examples to illustrate terms 0 6 prime implicants: '', ',, '',, ' essential minimum cover: + ' + '' 5 prime implicants:, ',, ', '' essential minimum cover: 4 essential implicants utumn 2000 SE370 - IV - ombinational Logic Optimization 26

14 lgorithm for two-level simplification! lgorithm: 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 s 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 utumn 2000 SE370 - IV - ombinational Logic Optimization 27 lgorithm for two-level simplification (example) primes around ''' primes around ' primes around ''' essential primes 0 0 minimum cover (3 primes) utumn 2000 SE370 - IV - ombinational Logic Optimization 28

15 ctivity utumn 2000 SE370 - IV - ombinational Logic Optimization 29 ombinational logic optimization summary! lternate representations of oolean functions " cubes " karnaugh maps! Simplification " two-level simplification! Later (in SE 467) " automation of simplification " optimization of multi-level logic " verification/equivalence utumn 2000 SE370 - IV - ombinational Logic Optimization 30

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