IST 4 Information and Logic

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IST 4

Information and Logic

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mon tue wed thr fri

28 _M₁ 1

4 _M₁

11 1 2 ^M²

18 2

25 _M₂

2 3

9 3 4

16 4 5

23

30 5

x= hw#x out x= hw#x due

Mx= MQx out Mx= MQx due

midterms

oh oh

oh

oh oh

oh

oh oh oh

oh

oh oh

oh = office hours ^oh

T

= today

T oh

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from

physics

to

symbols

envelops abacus

writing numbers

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from

physics

to

symbols

envelops abacus

writing numbers

starting today:

symbols to physics-computation

(5)

Today’s most important slide

0 1 0 0 1 1 1 0

m-box

Algorizms to small

syntax boxes

Syntax boxes to Boolean algebra

Boolean algebra to Physics

From ideas to physical implementation

Physical implementation of syntax boxes

with relay circuits

Claude Shannon

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Boolean algebra

formulae and functions

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All the coefficients are 1

Feasible for an arbitrary (finite) Boolean algebra

Simple Boolean formula

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Boolean

Functions

A Boolean function is a mapping from

{B} à {B}

defined by a simple Boolean formula n

Given a Boolean algebra with a set of elements B

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Start with a simple Boolean formula

Assign all possible elements to the formula

ab ^OR(a,b)

00 01 10 11

0 1 1 1

Two element Boolean Algebra B= {0,1}

Boolean Functions – 2 Elements

0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 1

Mapping of a simple Boolean formula to a syntax box a b

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+

00 01 10 11 00

01 10 11

01 01 11 11

10 11 11 11 10 11 11 11 00

01 10 11

ab ^OR(a,b)

00 01 10 11

0 1 1 1

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Assign all possible elements to the formula

ab ^XOR(a,b)

00 01 10 11

0 1 1 0

Two element Boolean Algebra B= {0,1}

Start with a simple Boolean formula

a b

Mapping of a simple Boolean formula to a syntax box

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00

00 00 01 10 11 00

01 10 11

11 10

01

10

10 01 01

11

ab ^XOR(a,b)

00 01 10 11

0 1 1 0

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Boolean algebra

DNF

^|s|

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D

isjunctive

DNF N

^ormal

F

^orm

Idea:

Representing a Boolean function with a formula of a specific form

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Representation of Boolean Functions

Disjunctive (Additive) Normal Form (DNF):

Sum of terms, each term is Normal

Normal = contains all the variables or their complements

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Representation of Boolean Functions

Disjunctive (Additive) Normal Form (DNF):

Sum of terms, each term is Normal

Normal = contains all the variables or their complements

Is it a DNF?

No!

This term is not normal

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DNF E-Theorem

Every Boolean function can be Expressed in a DNF

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Every Boolean function can be Expressed in a DNF

Expressed the following in a DNF T4

A1 A2 A4 A3 L1 T3

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DNF E-Theorem

DNF E-Theorem:

Every Boolean function can be xpressed in a Proof:

Apply DeMorgan Theorem (T4) until each negation is applied to a single variable

Apply distributive axiom (A4) to get a sum of terms

Use (A3, T3) and self absorption (L1) to eliminate duplicate terms

Augment a missing variable a to a term using (A1,A2,A4) multiplying by

By the algorithm

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Being Normal is Boring....

For which 0-1 assignments a normal term = 1 ?

011 101 110 101

100

A normal term is 1 for a single 0-1 assignment otherwise it is 0

xy ^AND(x,y)

00 01 10 11

0 0 0 1

1 iff both x and y are 1

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Being Normal is Boring....

For which 0-1 assignments a normal term = 1 ?

011 101 110

A normal term is 1 for a single 0-1 assignment otherwise it is 0

Normal 0-1 assignment,

corresponds to a normal term

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10 01 ab ^XOR(a,b)

00 01 10 11

0 1 1 0

A Boolean function is

1 for the normal 0-1 assignments and 0 for the other 0-1 assignments

Normal 0-1 assignments DNF

Boolean function syntax table

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A Boolean function is 1 for the normal assignments and 0 for the other 0-1 assignments

0-1 assignments DNF

00

00 10 11

00 01 10 11

01 10 11

11 10

01

10

10 01 01

11

11 01

Other 0-1 assignments Normal 0-1 assignments

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DNF Representation Theorem

DNF is a representation: two Boolean functions

are equal if and only if their DNFs are identical.

DNF Representation Theorem:

5 + 2 + 1 + 111 = ?? = 7+ 3 + 82+ 15 + 12

100 + 10 + 9 100 + 10 + 9

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DNF Representation Theorem

DNF is a representation: two Boolean functions are equal if and only if their DNFs are identical.

Proof:

DNF Representation Theorem:

Easy direction: DNFs identical implies functions are equal Other direction: DNF are different implies

functions are not equal Idea: consider the value of the functions for the 0-1 assignment

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Proof: Idea: consider the value of the functions for the 0-1 assignments

If two DNFs are different, there is a term

that appears in one and does not appear in the other Other direction:

two different DNFs implies two different functions

The 0-1 normal assignment that ‘corresponds’ to this term results in a 1 for one DNF and in a 0 for the other.

Q

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DNF is a representation: Two Boolean functions are equal if and only if their DNFs are identical.

Q: How is the all-0 function represented?

Q: How is the all-1 function represented?

Answer:

Answer: everything....

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Counting Functions

DNF is a representation: Two Boolean functions are equal if and only if their DNFs are identical.

Q1: How many different Boolean functions of n variables?

Answer:

The same as the number of different DNFs of n variables

Number of different normal terms of n variables?

A normal term can appear/not appear in a DNF

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Boolean algebra

DNF from syntax boxes

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DNF is a way to express

a syntax box using a formula

Any syntax box?

Represent a syntax box with formula?

We will focus on binary

It works only for binary!

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ab ^XOR(a,b)

00 01 10 11

0 1 1 0

Idea: construct a DNF by adding the normal terms that correspond to the normal assignments

a syntax box using a formula normal assignment in the syntax table

=

a normal term in the DNF

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ab ^f(a,b)

00 01 10 11

1 1 1 0

a syntax box using a formula normal assignment in the syntax table

=

a normal term in the DNF

Idea: construct a DNF by adding the normal terms that correspond to the normal assignments

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Boolean algebra

DNF and magic boxes

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Q:Can we reason about magic boxes using

Boolean algebra?

A Boolean algebra is a language for reasoning about syntax boxes

A MAGIC BOX:

binary s-box that can

compute binary s-box?

YES

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a b m

0 0

1 0

1 1

1 1 1 0

A magic box

Prove that it is magical?

with Boolean algebra?

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a b m

0 0

1 0

1 1

1 1 1 0

00 01 10

A magic box

First idea: DNF of m(a,b)

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a b m

0 0

1 0

1 1

1 1 1 0

A magic box

Second idea?

Compute the operations of the Boolean algebra with m(a,b)

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a b m

0 0

1 0

1 1

1 1 1 0

Analyzing the Magic Box using Boolean Algebra

Can realize the operations of the algebra hence, can compute any DNF

magical for Binary

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The Magic Box Can Compute any Boolean Function

? ?

? output in DNF?

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0 0

The Magic Box Can Compute any Boolean Function

HW#3

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(42)

2 symbol adder

digit 1 digit 2

carry

sum

carry

Binary Adder

3 bits to 2 bits

Gottfried Leibniz 1646-1716

Represent the number

of 1s in the input as two bits in base 2

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s 0

1 1 0

1 0 0 1 0 0

0 0

0 1 1 0

c 0 0 0 1

d1 d2 ^c

0 1 0

1 0

1

0 1 1 1 1 0

0 1

1 1 1 1

parity majority

0 0 0 0

0 1 1 0

d1 d2 ^c

0 1 0

1 0

1 1 0

0 1

1 1 1 1

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s 0

1 1 0

1 0 0 1 0 0

0 0

0 1 1 0

c 0 0 0 1

d1 d2 ^c

0 1 0

1 0

1

0 1 1 1 1 0

0 1

1 1 1 1

parity majority

0 0 0 0

0 1 1 0

d1 d2 ^c

0 1 0

1 0

1 1 0

0 1

1 1 1 1

majority 011 101 110 111

parity 001 010 100 111

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2 symbol adder c

s

d1 d2

c

parity majority

In HW#3: Compute parity

and majority with magic boxes...

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Shannon 1916-2001

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Connection Between

Boolean Calculus and Physical Circuits Shannon 1938

Shannon 1916-2001

1 Relay on the edge

controlled by a 0-1 variable

0

~90 AB (After Boole)

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Connection Between

Boolean Calculus and Physical Circuits Shannon 1938

2 relays

a b

a

b What is it computing?

ab ^OR(a,b)

00 01 10 11

0 1 1 1

ab ^AND(a,b)

00 01 10 11

0 0 0 1

connected

disconnected

When is it connected?

(49)

Next week: Implementing

Boolean

functions

^with

relay circuits

Shannon 1916-2001

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