CSE 502N Fundamentals of Computer Science

1
CSE 502NFundamentals of Computer Science

• Summer 2005
• Lecture 17
• Introduction to Digital Logic and
• Logic Minimization

2
Elementary Binary Logic Functions

• Digital circuits represent information using 2
  voltage levels.
• binary variables (1 0) are used to denote these
  values
• Functions of binary variables are called logic
  functions.
• AND(A,B) 1 if A1 and B1, else it is zero.
• AND is generally written in the shorthand AB (or
  AB or AÙB)
• OR(A,B) 1 if A1 or B1, else it is zero.
• OR is generally written in the shorthand form AB
  (or AB or AÚB)
• NOT(A) 1 if A0 else it is zero.
• NOT is generally written in the shorthand form
  (or ØA or A?)
• AND, OR and NOT can be used to express all other
  logic functions.

3
Two Variable Binary Logic Functions

• Similar truth tables for 3 or 4 variable
  functions are too big
• How can we represent functions in terms of AND,
  OR, NOT?
• NAND? EXOR?

4
Basic Logic Gates

• Logic gates compute elementary binary
  functions.
• output of an AND gate is 1 when both of its
  inputs are 1, otherwise the output is zero
• similarly for OR gate and inverter

5
Multivariable Gates

• AND function on n variables is 1 if and only
  if ALL its arguments are 1.
• n input AND gate output is 1 if all inputs are
  1
• OR function on n variables is 1 if and only if
  at least one of its arguments is 1.
• n input OR gate output is 1 if any inputs are
  1
• Can construct large gates from 2 input gates.
• however, large gates can be less expensive than
  required number of 2 input gates

6
Elements of Boolean Algebra

• Boolean algebra defines rules for manipulating
  symbolic binary logic expressions.
• Can define circuit for expression by combining
  gates.
• a symbolic binary logic expression consists of
  binary variables and the operators AND, OR and
  NOT (e.g. ABC?)
• The possible values for any Boolean expression
  can be tabulated in a truth table.

7
Boolean Functions to Logic Circuits

• Any Boolean expression can be converted to a
  logic circuit made up of AND, OR and NOT gates.
• Add parentheses to expression to fully define
  order of operations A(B(C ?))
• Create gate for last operation in expression
• Continue inward
• Number of simple gates needed to implement
  expression equals number of operations in
  expression.
• so, simpler equivalent expression yields less
  expensive circuit
• Boolean algebra provides rules for simplifying
  expressions

8
Basic Identities of Boolean Algebra

• 1. X 0 X
• 3. X 1 1
• 5. X X X
• 7. X X 1
• 9. (X ) X
• 10. X Y Y X
• 12. X(YZ ) (XY )Z
• 14. X(YZ ) XY XZ
• 16. (X Y )? X ?Y ?

• 2. X1 X
• 4. X0 0
• 6. XX X
• 8. XX 0
• 11. XY YX
• 13. X(YZ ) (XY )Z
• 15. X(YZ ) (XY )(XZ )
• 17. (XY) X?Y ?

• Identities define intrinsic properties of Boolean
  algebra.
• Note 15-17 have no counterpart in ordinary
  algebra.
• Parallel columns illustrate duality principle.

9
DeMorgans Laws for n Variables

• As we saw earlier in the semester, we can extend
  DeMorgans laws to 3 variables by applying the
  laws for two variables.
• Generalization to n variables.
• (X1 X2 Xn)? X ?1X ?2 X ?n
• (X1X2 Xn)? X ?1 X ?2 X ?n

10
Simplification of Boolean Expressions
FX ?YZ X ?YZ ?XZ
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The Duality Principle

• The dual of a Boolean expression is obtained by
  interchanging all ANDs and ORs, and all 0s and
  1s.
• E.g., the dual of A(BC ?)0 is A(BC ?)1
• If E1 and E2 are Boolean expressions then
• E1 E2 ? dual (E1)dual (E2)
• A(BC ?)0 (B ?C )D ? A(BC ?)1 (B
  ?C )D
• The pairs of identities (1,2), (3,4), (5,6),
  (7,8), (10,11), (12,13), (14,15) and (16,17) all
  follow from each other through the duality
  principle.

12
The Consensus Theorem

• Theorem. XY YZ X ?Z XY X ?Z
• Proof. XY YZ X ?Z XY (X X ?)YZ X ?Z
  2,7
• XY XYZ X ?YZ X ?Z 14
• XY(1 Z ) X ?Z(Y 1) 2,11,14
• XY X ?Z
  3,2
• Example. (A B )(A? C ) AA? AC A?B BC
• AC A?B BC
• AC A?B
• Dual. (X Y )(Y Z )(X ? Z ) (X Y )(X ?
  Z )

13
Taking the Complement of a Function

• Any suggestions?
• Method 1. Apply DeMorgans Theorem repeatedly.
• (X(Y ?Z ? YZ ))? X ? (Y ?Z ? YZ )?
• X ? (Y ?Z ?)?(YZ )?
• X ? (Y Z )(Y ? Z ?)
• Method 2. Complement literals and take dual
• (X (Y ?Z ? YZ ))? dual (X ?(YZ Y ?Z ?))
• X ? (Y Z )(Y ? Z ?)

14
Sum of Products Form

• The sum of products is one of two standard forms
  for Boolean expressions.
• ?sum-of-products-expression? ?p-term?
  ?p-term? ... ?p-term?
• ?p-term? ?literal? ?literal?
  ?literal?
• example. X ?Y ?Z X ?Z XY XYZ
• A minterm is a term that contains every variable,
  in either complemented or uncomplemented form.
• example. in expression above, X ?Y ?Z is minterm,
  but X ?Z is not
• A sum of minterms expression is a sum of products
  expression in which every term is a minterm.
• X ?Y ?Z X ?YZ XYZ ? XYZ
• shorthand list minterms numerically, so X ?Y ?Z
  X ?YZ XYZ ? XYZ becomes 001011110111 or
  (1,3,6,7)

15
Simplifying Sum-of-Products Expressions

• Sum of products forms yield 2 level AND-OR
  circuits.
• Any expression can be put into sum of products
  form by applying distributive laws.
• The simplest sum of products expression yields
  simplest 2 level AND-OR circuit.
• Any Boolean expression can be viewed as a set of
  minterms.
• An expression F covers another expression G, if
  the minterms in G are a subset of the minterms in
  F.
• AC covers ABC, since AC contains minterms 5 and
  7 (from the set of 8 minterms on the variables A,
  B, and C ) and ABC contains only minterm 5.

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General Simplification Procedure

• Given an expression F (e.g. ABDA?BBC?D?B?CDB?C
  D?)
• Step 1. Let M be the set of minterms covered by
  F.
• A?B?CD A?B?CD?
• A?BC?D? A?BC?D A?BCD A?BCD?
• ABC?D? ABC?D ABCD
• AB?CD AB?CD?
• Step 2. For each minterm, m, find all maximal
  terms that cover m and also cover other minterms
  in M, but no minterms that are not in M. Let T be
  the resulting set of terms.
• (T A?B, BC?, BD, CD, A?C, B?C )
• Step 3. Select all terms in T that cover minterms
  covered by no other terms in T ( BC?, B?C )
• Step 4. Select additional terms in T until
  selected terms cover all minterms. At each step,
  select a term that covers the largest possible
  number of new minterms. ( A?B, CD )