A Prelude

1
A Prelude

• Practice yourself, for heavens sake, in little
  things and thence proceed to greater.
  Epictetus
• Today we are going to build simple systems
  satisfying binary input/output tables

2
(No Transcript)
3
Combinational Logic Units / Prelude
Input/output Spec Table

• Let us look at the pencil-and-paper assignment
  which was due today
• Describe a physical implementation of gadget
  which takes two inputs and one output.Each of the
  inputs can either take the value YES or the value
  NO.The same goes for the output. The gadget is
  specified by the input-output table to the left.
  Describe how the values NO and YES are
  represented in your physical system for the
  inputs as well as for the output. Moreover
  describe the implementation details of your
  gadget(system).

In1 In1 Out
No No No
No Yes No
Yes No No
Yes Yes Yes
4
Combinational Logic Units/Prelude
Input/output Spec Table

• Discussion of some answers

In1 In1 Out
No No No
No Yes No
Yes No No
Yes Yes Yes
5
Combinational Logic Units / Prelude
Input/output Spec Table

• Same question but for a different Inpu/Output
  Spec Table
• Describe a physical implementation of gadget
  which takes two inputs and one output.Each of the
  inputs can either take the value YES or the value
  NO.The same goes for the output. The gadget is
  specified by the input-output table to the left.
  Describe how the values NO and YES are
  represented in your physical system for the
  inputs as well as for the output. Moreover
  describe the implementation details of your
  gadget(system).

In1 In1 Out
No No No
No Yes Yes
Yes No Yes
Yes Yes Yes
6
Combinational Logic Units / Prelude
Input/output Spec Table

• Disscussion of some answers

In1 In1 Out
No No No
No Yes Yes
Yes No Yes
Yes Yes Yes
7
Combinational Logic Units /Prelude
Input/output Spec Table

• Let us look at the pencil-and-paper assignment
  which was due today
• Describe a physical implementation of gadget
  which takes two inputs and one output.Each of the
  inputs can either take the value YES or the value
  NO.The same goes for the output. The gadget is
  specified by the input-output table to the left.
  Describe how the values NO and YES are
  represented in your physical system for the
  inputs as well as for the output. Moreover
  describe the implementation details of your
  gadget(system).

In Out
No Yes
Yes No
8
Combinational Logic Units /Prelude
Input/output Spec Table

• Discussion of some answers

In Out
No Yes
Yes No
9
Combinational Logic Units / Prelude
How do implement the following Input/output Spec
Table
In1 In2 In3 Out
No No No No
No No Yes No
No Yes No No
No Yes Yes No
Yes No No No
Yes No Yes No
Yes Yes No No
Yes Yes Yes Yes
10
Combinational Logic Units

• After the previous prelude we will discuss the
  following points
• Combinational Logic Units
• Specifying a CLU with a Input/Output Tables
• Reading off a Logic Diagram of a CLU from its 0-1
  Specification Tables
• Specifying a CLU with an Algebraic Expression of
  the Switching Algebra

11
Combinational Logic UnitsSome Definitions

• Combinational logic a digital logic circuit in
  which logical decisions are made based only on
  combinations of the inputs. e.g. an adder.
• Sequential logic a circuit in which decisions
  are made based on combinations of the current
  inputs as well as the past history of inputs.
  e.g. a memory unit.
• Finite state machine a circuit which has an
  internal state, and whose outputs are functions
  of both current inputs and its internal state.
  e.g. a vending machine controller.

12
Combinational Logic Units

• Combinational logic a digital logic circuit in
  which logical decisions are made based only on
  combinations of the inputs. e.g. an adder.

13
Combinational Logic Unit

• Translates a set of inputs into a set of outputs
  according to one or more mapping functions.
• Inputs and outputs for a CLU normally have two
  distinct (binary) values high and low, 1 and 0,
  0 and 1, or 5 v. and 0 v. for example.
• The outputs of a CLU are strictly functions of
  the inputs, and the outputs are updated
  immediately after the inputs change. A set of
  inputs i0 in are presented to the CLU, which
  produces a set of outputs according to mapping
  functions f0 fm

14
Combinational Logic Unit
15
Specifying Combinational Logic Unit
Enough if you know how to handle the case of ONE
output and arbitrary number of inputs
16
SpecifyingCombinational Logic Unit

• CLU (with one output )is completely specified by
  an input/output table if you have more than one
  output you will have an input/output table for
  each output
• Since the inputs and outputs assume only two
  values say 0 or 1 (low voltage or high voltage
  etc) a CLU can be specified by 0-1 Table or Truth
  Table

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SpecifyingCombinational Logic Unit
0-1 Table/Truth Table
Truth Table
A B Out
F F F
0 T F
T F F
T T T
A B Out
0 0 0
0 1 0
1 0 0
1 1 1
A B Out
No No No
No Yes No
Yes No No
Yes Yes Yes
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Truth Tables Showing All Possible Functions of
Two Binary Variables

• The more frequently used functions AND, XOR,
  OR, NOR, XOR, and NAND. (Always use upper case
  spelling.) See also handout

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Combinational Logic Unit

• We have constructed an implementation of an AND,
  OR and NOT gate (waterworks ? )

A B OR
0 0 0
0 1 1
1 0 1
1 1 1
20
Combinational Logic Unit

• We have constructed an implementation of an AND,
  OR and NOT gate (waterworks ? )

A B AND
0 0 0
0 1 1
1 0 1
1 1 1
21
Combinational Logic Unit

• We have constructed an implementation of an AND,
  OR and NOT gate (waterworks ? )

A NOT
0 1
1 0
22
SpecifyingCombinational Logic Unit Example One
Bit Full Adder
1 1 1 ----- 11
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from spec table to Logic Diagram (LD) for a CLU
Output 2
inputs
Output 1
S
0
1
1
0
1
0
0
1
Cout
0
0
0
1
0
1
1
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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from spec table to (LD) for a CLU
inputs
Output 1
S
0
1
1
0
1
0
0
1
Let us concentrate on the First output the sum
of the two input bits and the the carry_in.
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
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from spec table to (LD) for a CLU

• Given the Truth table we can implement a circuit
  as follows
• For each input provide a line(wire) and a negated
  wire (by using inverters)
• Draw a multi-AND gate (that is, n-AND gate where
  n equals the number of inputs) for each 1 in the
  output column
• Wire the AND gates to the appropriate inputs (see
  next slides)
• Feed the output of all the AND gates into an OR
  gate
• In the next slides we show in detail how the
  above algorithm works.

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from spec table to LD Step 1
For each input we introduce a wire and a negated
wire
inputs
Output 1
A
Cin
B
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Digital Techniques Fall 2007 André Deutz, Leiden
University
27
from spec table to LD Step 2
Cin
A
B
For each 1 in the output Column introduce a
multi-way AND gate
inputs
Output 1
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Digital Techniques Fall 2007 André Deutz, Leiden
University
28
From spec table to LD Step 3
Consider each (input) row of which the output is
1 connect the input lines to the AND gate
appropriately if 0 appears connect to neg line
for a 1 to the line itself
Cin
A
B
inputs
Output 1
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Digital Techniques Fall 2007 André Deutz, Leiden
University
29
spec table to LD Step 3 (contd)
Consider each (input) row of which the output is
1 connect the input lines to the AND gate
appropriately if 0 appears connect to neg line
for a 1 to the line itself
Cin
A
B
inputs
Output 1
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Digital Techniques Fall 2007 André Deutz, Leiden
University
30
spec table to LD Step 3 (contd)
Consider each (input) row of which the output is
1 connect the input lines to the AND gate
appropriately if 0 appears connect to neg line
for a 1 to the line itself
Cin
A
B
inputs
Output 1
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Digital Techniques Fall 2007 André Deutz, Leiden
University
31
spec table to LD Step 3 (contd)
Consider each (input) row of which the output is
1 connect the input lines to the AND gate
appropriately if 0 appears connect to neg line
for a 1 to the line itself
Cin
A
B
inputs
Output 1
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Digital Techniques Fall 2007 André Deutz, Leiden
University
32
spec table to LD Step 4
Cin
Wire the AND gates to a multi OR gate (for this
example a 4-OR gate)
A
B
inputs
Output 1
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
S
Digital Techniques Fall 2007 André Deutz, Leiden
University
33
from spec table to LD for (CLU)

• Does our 4-step algorithm produce the right Logic
  Diagram for the Combinational Logic Unit? Why?
• This algorithm will generally not produce the
  most frugal combinational logic unit.
• Despite the previous remark this algorithm is
  still very important! Why?
• This algorithm convinced us that for each
  input/output table (0/1-table) there exists an LD
  ( or for that matter a CLU) satisfying it.
  Besides the LD (or for that matter also the CLU)
  produced by the algorithm there are, of course,
  many more LDs satisfying the 1/0-table (some of
  them will in general be more efficient)
• Secondly once you can translate your task into
  an input/output table you are home free (we can
  worry later about optimizing the circuit)

34
Two more examples a 4x1 Multiplexer and 1x4
Demultiplexer and the Music the pair can make
35
Boolean Algebra
36
Boolean Algebra

• Actually a special kind of Boolean Algebra
  namely, Switching Algebra

37
Intro to Boolean Algebra Specifying

• Sofar we can specify a CLU either by a Logic
  Diagram of NOT, AND and OR gates or by a
  1/0-table
• There is at least a third way of specifying CLU
  namely via Algebra (algebraic equations/expression
  s)
• What is the advantage of Algebraic
  specifications? Among others can simplify
  circuits by manipulating expressions with pencil
  and paper or computer programs

38
Boolean Algebra

• In order to analyze and design digital
  combinational logic units we need a mathematical
  system.
• Binary logic system called Boolean Algebra is
  used.
• George Boole (1815-1864) An investigation of
  the laws of thought a book published in 1854
  introducing the mathematical theory of logic.
• Boolean Algebra deals with binary variables that
  take 2 discrete values (0 and 1), and with logic
  operations.
• Binary/logic variables are typically represented
  as letters A,B,C,,X,Y,Z or a,b,c,,x,y,z.
• Three basic logic operations
• AND, OR, NOT (complementation or negation).

39
Basic Logic Operations

• AND operation is represented by operators or
  or by the absence of an operator.
• Z XY or Z X Y, or Z XY is read Z is
  equal to X AND Y meaning that
• Z 1 if and only if X 1 and Y 1 otherwise Z
  0.
• AND resembles binary multiplication
• 0 0 0, 0 1 0,
• 1 0 0, 1 1 1
• OR operation is represented by operators or
  .
• Z XY or Z X Y is read Z is equal to X
  OR Y meaning that
• Z 1 if X 1 or Y 1, or if both X 1 and Y
  1. Z 0 if and only if X 0 and Y 0.
• OR resembles binary addition, except in one case
• 0 0 0, 0 1 1,
• 1 0 1, 1 1 1 (? 102)
• NOT operation is represented by operator or
  by a bar over a variable.
• Z X or Z X is read Z is equal to NOT X
  meaning that
• Z 1 if X 0 but Z 0 if X 1
• NOT operation is also referred to as complement
  operation.

40
Basic Identities of Boolean Algebra

• Let X be a boolean variable and 0,1 constants
• 1. X 0 X -- Zero Axiom
• 2. X 1 X -- Unit Axiom
• 3. X 1 1 -- Unit Property
• 4. X 0 0 -- Zero Property
• 5. X X X -- Idempotence
• 6. X X X -- Idempotence
• 7. X X 1 -- Complement
• 8. X X 0 -- Complement
• 9. (X) X -- Involution

41
Boolean Algebra Properties

• Let X,Y, and Z be boolean variables
• Commutative
• 10. X Y Y X 11.
  X Y Y X
• Associative
• 12. X (YZ) (XY) Z 13.
  X(YZ) (XY)Z
• Distributive
• 14. X(YZ) XY XZ 15.
  X(YZ) (XY) (XZ)
• DeMorgans Theorem
• 16. (X Y) X Y 17.
  (X Y) X Y
• In general for DeMorgan,
• ( X1 X2 Xn ) X1 X2 Xn,
• ( X1 X2 Xn ) X1 X2 Xn

42
The Duality Principle

• The dual of an expression is obtained by
  exchanging ( and ), and (1 and 0) in it,
  provided that the precedence of operations is not
  changed.
• Cannot exchange x with x.
• Example
• Find the dual of expression xyz xyz
• Answer (x y z ) (x y z)
• Dual expression does not always equal the
  original expression.
• If a Boolean equation/equality is valid, its dual
  is also valid.

43
The Duality Principle (cont.)
With respect to duality, Identities 1 8 and
Properties 10 17 have the following
relationship 1. X 0 X
2. X 1 X (dual of 1) 3. X 1
1 4. X 0 0 (dual of
3) 5. X X X 6. X X X
(dual of 5) 7. X X 1 8.
X X 0 (dual of 7) 10. X Y Y X
11. X Y Y X
(dual of 10) 12. X (YZ) (XY) Z 13.
X(YZ) (XY)Z (dual of 12) 14.
X(YZ) XY XZ 15. X(YZ) (XY)
(XZ) (dual of14) 16. (X Y) X Y
17. (X Y) X Y (dual
of16)
44
Absorption Property (Covering)

• X XY X -- (absorption property)
• X(XY) X -- (dual absorption property)
• ProofX XY X1 XY X (1 Y)
  X 1 X
• Can you prove the dual absorption property?

45
Principle of dualityThe dual of a Boolean
function is gotten by replacing AND with OR and
OR with AND, constant 1s by 0s,and 0s by 1s
Occams Razor for Boolean Algebras
A(AB) AB AABAB elimination of
complement
46
A more formal definition Boolean Algebras

• A Boolean algebra is a set B on which two binary
  operations and and one unary operation ' are
  defined and in which there are two distinct
  elements 0 and 1 such that the following
  properties hold for all x, y, z ? B

47
A more formal definition of Boolean Algebras
(continued)

• 1a. x y y x 1b. x y y x (commutative
  properties)
• 2a. (x y) z x (y z) 2b. (x y) z x
  (y z) (associative properties)
• 3a. x (y z) (x y ) (x z)
• 3b. x (y z) (x y) (x z)
  (distributive properties)
• 4a. x 0 x 4b. x 1 x (identity
  properties)
• 5a. x x 1 5b. x x 0 (complement
  properties)

48
The specific Boolean Algebra we use

• Example Let B 0,1 ( the set of integers 0
  and 1) define binary operations and on B by
• x y max(x,y ),
• x y min(x,y )
• Then we can illustrate the operations of and
  by the tables
• 0 1 0 1
• 0 0 0 0 0 1
• 1 0 1 1 1 1
• A unary operation ' can be defined as follows
• '
• 0 1
• 1 0 Thus 0' 1 and 1' 0. B, , , ', 0, 1
  is a Boolean algebra We can verify the 10
  properties by checking all possible cases

49
Remark on Boolean Algebras

• idempotent property x x x
• This holds in any Boolean algebra since
• x x (x x) 1 (4b)
• (x x) (x x' ) (5a)
• x (x x') (3a)
• x 0 (5b)
• x (4a)

50
Non-Example on Boolean Algebra

• The operations on integers share many of the
  properties of the operations for boolean
  algebras.
• Idempotency is a property which is not shared.
• Thus the set of integers is not a boolean algebra
  (aside from needing a complementation operation)

51
Transition from 1/0 table to LD to Boolean
expression

• Combinational Logic Units with one output can be
  specified by
• A Boolean equation
• A Logical Diagram
• 1/0 table
• We already saw how to get a Logic Diagram (LD)
  from a 1/0 table it is clear that from the Logic
  Diagram you can read off a 1/0 table
• It is also clear how you get a Boolean expression
  from a 1/0 table

52
From 1/0 table to Boolean expression
S
0
1
1
0
1
0
0
1
A B Cin
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
S A ' B ' C A ' B C ' AB ' C ' ABC
53
From Boolean expression to 1/0 table?
F
0
1
1
1
0
0
1
1
A B C
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1
1 1 0
1 1 1
F A ' B ' C B
54
To remember

• With Boolean expressions (BE), or Logic
  Diagrams(LD) or Truth tables(TT) you can specify
  CLU with one output.
• The transition between any of the two of ( BEs,
  TTs or LDs) can be readily done