Simplification of twolevel combinational logic


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1 ombinational logic optimization! lternate representations of oolean functions " cubes " karnaugh maps! Simplification " twolevel simplification " exploiting don t cares " algorithm for simplification utumn 2000 SE370  IV  ombinational Logic Optimization 1 Simplification of twolevel 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?! omputeraided 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 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 F = ''+' = ('+)' = ' F has the same value in both onset 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 = ndimensional "cube" 1cube 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) ONset = solid nodes OFFset = 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 fulladder carryout logic in out ('+)in (in'+in) (+')in the onset 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! Subcubes of higher dimension than 2 F(,,) = Σm(4,5,6,7) onset 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 mdimensional cubes in a ndimensional oolean space! In a 3cube (three variables): " a 0cube, i.e., a single node, yields a term in 3 literals " a 1cube, i.e., a line of two nodes, yields a term in 2 literals " a 2cube, i.e., a plane of four nodes, yields a term in 1 literal " a 3cube, i.e., a cube of eight nodes, yields a constant term "1"! In general, " an msubcube within an ncube (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 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 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: 4variable 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 ONset (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 2cube can be formed rather than a 1cube 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: twobit comparator N1 N2 LT EQ GT block diagram and truth table < = > LT EQ GT we'll need a 4variable Karnaugh map for each of the 3 output functions utumn 2000 SE370  IV  ombinational Logic Optimization 18
10 esign example: twobit comparator (cont d) Kmap for LT Kmap for EQ Kmap for GT LT = EQ = GT = ' ' + ' + ' ' ' ' ' + ' ' + + ' = ( xnor ) ( xnor ) ' ' + ' + ' LT and GT are similar (flip / and /) utumn 2000 SE370  IV  ombinational Logic Optimization 19 esign example: twobit 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: 2x2bit multiplier block diagram and truth table P1 P2 P4 P P8 P4 P2 P variable Kmap for each of the 4 output functions utumn 2000 SE370  IV  ombinational Logic Optimization 21 esign example: 2x2bit multiplier (cont d) 2 Kmap for P8 1 Kmap for P4 P4 = 221' + 21' P8 = Kmap for P2 1 Kmap 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 Kmap 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 twolevel simplification! Implicant " single element of ONset 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 ONset " will participate in LL possible covers of the ONset " set 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) 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 twolevel simplification! lgorithm: minimum sumofproducts expression from a Karnaugh map " Step 1: choose an element of the ONset " 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 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 utumn 2000 SE370  IV  ombinational Logic Optimization 27 lgorithm for twolevel 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 " twolevel simplification! Later (in SE 467) " automation of simplification " optimization of multilevel logic " verification/equivalence utumn 2000 SE370  IV  ombinational Logic Optimization 30
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