Simplification of Boolean ExpressionKarnaugh MapsQUINE McCLUSKEY METHODLearning by exampleStep 1Step 2Step 3Step 4Covering ProcedureCovering Procedure Cont.Cyclic PIProcedureINCOMPLETELY SPECIFIED FUNCTIONSlide 14SYSTEMS WITH MULTIPLE OUTPUTSSlide 16Slide 17Slide 18Slide 19PETRICK’S ALGORITHMPETRICK’S ALGORITHMHAZARDS–Static and DynamicHAZARDS–Static and DynamicSlide 24Slide 25Example of Static 0 HazardExample of Static 0 Hazard1Simplification of Boolean ExpressionKarnaugh Maps QUINE McCLUSKY METHODEPRESSO AlgorithmHAZARDS–Static and DynamicEE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 2Karnaugh MapsReviewEE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 3QUINE McCLUSKEY METHOD Form the PI chartRemove the EPIRe-form the PI chart ( with non-essential PIs only) Apply the following 2 rules to reduce the PI chart A row that is covered by another row may be eliminated A column that covers another column may be eliminatedNote: If the PI chart is cyclic ( no SPI and can’t be reduced using the above two rules), then arbitrarily select a PI for SPI and apply the rules for the remaining PI chart.EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 4Learning by exampleF (A,B,C,D) =  m(2,4,6,8,9,10,12,13,15)1 1 11 111 1 1ABCD00 01 11 1000011110EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 5Step 1Step 1: Grouping Minterms based on number of 1’sMinterms ABCD2 00104 0100 Group 1 (a single 1)8 10006 01109 1001 Group 2 (two 1’s)10 101012 110013 1101 Group 3 (three 1’s)15 1111 Group 4 (four 1’s)EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 6Step 2Step 2: Make a list of minterms that can be combined ( minterms differing by a single literal)List 1 List 2 List 3Minterm ABCD Minterm ABCD Minterm ABCD2 00102,6 0-10 PI 2 8,9,12,13 1-0- PI14 01002,10 -010 PI 38 10004,6 01-0 PI 46 01104,12 -100 PI 59 10018,9 100-10 10108,10 10-0 PI 612 11008,12 1-0013 11019,13 1-0115 111112,13 110-13,15 11-1 PI 7EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 7Step 3Step 3: Determine the minimum number of PI’s    2 4 6 8 9 10 12 13 15** PI 1 x x x xPI 2 x xPI 3 x xPI 4 x xPI 5 x xPI 6 x x** PI 7 x xEPIEE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 8Step 4Step 4: Select additional PI’s for example    2 4 6 10PI 2 x x*PI 3 x x*PI 4x xPI 5xPI 6xF(A,B,C,D) = PI1 + PI3 + PI4 + PI7EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 9Covering ProcedureRule 1: A row that is covered by another row may be eliminated from the chart. When identical rows are present, all but one of the rows may be eliminated.Rule 2: A column that covers another column may be eliminated. All but one column from a set of identical columns may be eliminated.EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 10Covering Procedure Cont.       0 1 5 6 7 8 9 10 11 13 14 15** PI 1 X X X XPI 2 X X X XPI 3 X X X XPI 4 X X X XPI 5 X X X XPI 6 X X X X** PI 7 X X X XF(A,B,C,D) = ?EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 11Cyclic PIA cyclic PI chart is a chart that contains no essential PI and that cannot be reduced by rules 1 and 2.  1 2 3 4 5 6* PI 1 X X PI 2 X X PI 3 X XPI 4 X XPI 5 X XPI 6 X XF (A,B,C,D) =  m(1,2,3,4,5,6)EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 12ProcedureStep 1: Identify any minterms covered by only one PI in the chart. Select these PIs for the cover. Note that this step identifies essential PIs on the first pass and nonessential PIs on the subsequent passes (from step 4)Step 2: Remove rows corresponding to the identified essential and nonessential PIs. Remove columns corresponding o minterms covered by the removed rows.Step 3: If a cyclic chart results after completing step 2, go to step 5. otherwise, apply the reduction procedure of rules 1 and 2Step 4: if a cyclic chart results from step 3, go to step 5. otherwise, return to step 1.Step 5: apply the cyclic chart procedure. Repeat step 5 until a void chart occurs or until a noncyclic chart is produced. In the latter case, return to step 1.EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 13INCOMPLETELY SPECIFIED FUNCTIONF(A,B,C,D) =  m(2,3,7,10,12,15,27) + d(5,18,19,21,23)List 1 List 2 List 3Minterm ABCDE Minterm ABCDE Minterm ABCDE2 000102,3 0001-2,3,18,19 -001- PI13 000112,10 0-010 PI4 3,7,19,23 -0-11 PI25 001012,18 -00105,7,21,23 -01-1 PI310 010103,7 00-1112 01100 PI7 3,19 -001118 100105,7 001-17 001115,21 -010119 1001118,19 1001-21 101017,15 0-111 PI515 011117,23 -011123 1011119,23 10-1127 1101119,27 1-011 PI621,23 101-1EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 14     2 3 7 10 12 15 27PI 1 X X PI 2 X XPI 3 X ** PI 4 X X** PI 5 X X ** PI 6 X** PI 7 X•Listing don’t cares as regular minterms•Do not list don’t cares in the PI chart•Don’t care terms do not need to be checked outF(A,B,C,D,E) = PI1 + PI4 + PI5 + PI6 + PI7EE 270 Simplification of Boolean Expressions Dr. Tri Caohuu © 2006 Andy Davis Lecture 2 15SYSTEMS WITH MULTIPLE OUTPUTSSimplification of switching functions by exploiting potential gate sharing to obtain a simpler overall design.Introduction of “Flags” columnMinterms can be combined only if they possess one or more common flags and the term that results from the combination carries only flags that are common to both mintermsMinterm(s) are checked off only if all flags are included in the resulting