IST 4
Information and Logic
mon tue wed thr fri
28 M1
1
4 M1
11
1 2
M218
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 oh
oh
oh oh
oh = office hours oh
T
= today
T oh
Memory MQ2
Due Tuesday 4/26/2016 by 10pm
Please email PDF
lastname-firstname.pdf
to [email protected]
MQ1 grades
wereemailed
please let the TAs know if you submitted MQ1 and did not get the email
HW1 will be returned today
average grade is A
The geography of innovation We need the whole world!
Source: Wikipedia
Last time...
Fibonacci 1170-1250AD Algorizmi
780-850
Leibniz 1646-1716
base 10 base 10
in Europe The binary
Arithmetic boxes decimal
Algorizmi 780-850
base 10
Fibonacci 1170-1250AD
base 10 in Europe
Algorithms – Syntax Manipulation
2 symbol adder
digit 1 digit 2
carry
sum
carry
Algorithms – Syntax Manipulation
2 symbol adder
8 6
digit 1 digit 2
carry
sum
0 carry
Algorithms – Syntax Manipulation
2 symbol adder
8 6
4
digit 1 digit 2
carry
sum
1 0 carry
Arithmetic Boxes binary
Leibniz 1646-1716
The binary
2 symbol adder
digit 1 digit 2
carry
sum
carry
Adding Bits
3 bits to 2 bits
Count the number
of 1s and represent
in binary
Adding Bits
2 symbol adder
0 0
0
digit 1 digit 2
carry
sum
0 0 carry
3 bits to 2 bits
Count the number
of 1s and represent
in binary
2 symbol adder
1 0
1
digit 1 digit 2
carry
sum
0 0 carry
Adding Bits
3 bits to 2 bits
Count the number
of 1s and represent
in binary
2 symbol adder
1 1
0
digit 1 digit 2
carry
sum
1 0 carry
Adding Bits
3 bits to 2 bits
Count the number
of 1s and represent
in binary
2 symbol adder
1 0
0
digit 1 digit 2
carry
sum
1 1 carry
Adding Bits
3 bits to 2 bits
Count the number
of 1s and represent
in binary
2 symbol adder
0 1
0
digit 1 digit 2
carry
sum
1 1 carry
Adding Bits
3 bits to 2 bits
Count the number
of 1s and represent
in binary
2 symbol adder
1 1
1
digit 1 digit 2
carry
sum
1 1 carry
Adding Bits
3 bits to 2 bits
Count the number
of 1s and represent
in binary
Algorithms – Syntax Manipulation
0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9 2 2 3 4 5 6 7 8 9 3 3 4 5 6 7 8 9 4 4 5 6 7 8 9 5 5 6 7 8 9 6 6 7 8 9 7 7 8 9 8 8 9 9 9
2 symbol adder c
s d1 d2
c
0
Algorithms – Syntax Manipulation
2 symbol adder c
s d1 d2
c
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
sum
carry
0 1 0 0 1 1 1 2
0 0 1
0 1 2 1 2 3
1
Algorithms – Syntax Manipulation
2 symbol adder 1
0
1 1
0
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
Algorithms – Syntax Manipulation
2 symbol adder 1
0
1 0
1
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
2 symbol adder c
c
1 1
c c 2 symbol adder c
1 0
c c 2 symbol adder c
0 0
c c 2 symbol adder c
1 1
0
1101 + 1001 = ??
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
2 symbol adder c
c
1 1
c c 2 symbol adder c
1 0
c c 2 symbol adder c
0 0
c 1 2 symbol adder 0
1 1
0
1101 + 1001 = ??
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
2 symbol adder c
c
1 1
c c 2 symbol adder c
1 0
c 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
1101 + 1001 = ??
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
2 symbol adder c
c
1 1
c 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
1101 + 1001 = ??
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
2 symbol adder 1
0
1 1
0 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
1101 + 1001 = ??
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
2 symbol adder 1
0
1 1
0 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
1101 + 1001 = 10110 13 + 9 = 22
2 symbol adder 1
0
1 1
0 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
1101 + 1001 = 10110 13 + 9 = 22
8 symbol adder
1 1 0 1 1 0 0 1
1 0 1 1 0
a deeper idea?
1 1 0 1 1 0 0 1
1 0 1 1 0
8 input bits
5 output bits
28 input combinations
Can be implemented with a table with 5 x 256 bits what are the trade-offs?
2 symbol adder 1
0
1 1
0 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
0 1 0 0 1 1 1 0
0
0 1 0 1 0 1 0 1
1
0 1 0 0 0 1 0 1
0
0 1 0 0 1 1 1 1
1
sum
carry what are the trade-offs?
2 symbol adder 1
0
1 1
0 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
1 0 0
1 1 1 0
1
a table with 8 bits
8 tables with 8 bits
a table with 8 bits
Memory-only requires 5 x 256 bits =
20 x 64
bits2 symbol adder 1
0
1 1
0 0 2 symbol adder 1
1 0
0 0 2 symbol adder 1
0 0
1 1 2 symbol adder 0
1 1
0
Memory and logic require 8 8-bit tables =
64
bits Idea: Logic (algorithms) isa language that helps in optimizing memory Idea: Understanding is
is a method for remebering with a small memory…
and much more...
a deeper idea?
We need to
remember
• The building blocks
• The algorithms
Algorithm
A procedure to build
‘everything’ from a
set of building blocks
There is a trade-off between
the size of building blocks: memories
and the complexity of the algorithm:
composition of the memories
Algorithm
A procedure to build
‘everything’ from a
set of building blocks
Syntax Boxes BIG from small
Remembering a large memory
with a composition of small memories
S-Box
a b
o inputs
outputs
a b o
Binary S yntax Box es (s-box)
0 0
0 inputs
outputs
a b o
0 0 0
Binary S yntax Box es (s-box)
0 1
1 inputs
outputs
a b o
0 0
1 0
0 1
Binary S yntax Box es (s-box)
1 0
1 inputs
outputs
a b o
0 0
1 0
1 0
0 1 1
Binary S yntax Box es (s-box)
1 1
1 inputs
outputs
a b o
0 0
1 0
1 0
1 1
0 1 1 1 Suggest a name for this memory box?
a b
o
a b o
0 0
1 0
1 0
1 1
0 1 1 1
o = max (a,b)
o = max (a,b) Suggest a name for this memory box?
Can we remember
max (a,b,c) with max(a,b)?
a b
c
Composition:
build
big
s-boxesfrom small s-boxes o = max (a,b)
o = max (a,b,c) Can we remember
max (a,b,c) with max(a,b)?
a b
o
o = max (a,b) Can we remember w
with the max s-box?
a b w
0 0
1 0
1 0
1 1
0 1 1 0
?
NO Why not?
a b o
0 0
1 0
1 0
1 1
0 1 1 1 o = max (a,b)
How to compute the following (XOR)?
Big
s-boxa b
w
a b
0 0
1 0
1 0
1 1
0 1 1 0
?
Composition:
build
big
s-boxesfrom small s-boxes The output of every small s-box is bigger or equal to its inputs
The output of the
big
s-box must be bigger or equal to its inputs NO Why not?
w
Can we remember w with the max s-box?
A Magic (Universal) Box
a b
a b m
0 0
1 0
1 0
1 1
1 1 1 0
A binary s-box that can
compute any binary s-box?
M
m
A Magic Box
a b
a b m
0 0
1 0
1 0
1 1
1 1 1 0
Can you compute the
following with the magic box?
M
m
x y
0 0
1 0
1 0
1 1
o 0 0 0 1 min(x,y)
A Magic Box
a b
a b m
0 0
1 0
1 0
1 1
1 1 1 0
Can you compute the
following with the magic box?
M
m
x y
0 0
1 0
1 0
1 1
o 0 0 0 1
Hint 1:
min(x,y)A Magic Box
a b
a b m
0 0
1 0
1 0
1 1
1 1 1 0
Can you compute the
following with the magic box?
M
m
x y
0 0
1 0
1 0
1 1
o 0 0 0 1
Hint 2:
min(x,y)A Magic Box
x y
a b m
0 0
1 0
1 0
1 1
1 1 1 0
M
x y
0 0
1 0
1 0
1 1
o 0 0 0 M 1
o 1
0
min(x,y)
A Magic Box
x y
a b m
0 0
1 0
1 0
1 1
1 1 1 0
M
x y
0 0
1 0
1 0
1 1
o 0 0 0 M 1
o 0
1
min(x,y)
a b
a b m
0 0
1 0
1 0
1 1
1 1 1 0
Can you compute the
following with the m-box?
M
m
x y
0 0
1 0
1 0
1 1
o 0
1 1 1
HINT
max(x,y)x
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y
0 0
1 0
1 0
1 1
o 0
1 1 M 1
o
y
M M
max(x,y)
x
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y
0 0
1 0
1 0
1 1
o 0
1 1 M 1
o
y
M M
0
1 1
0
0
max(x,y)
x
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y
0 0
1 0
1 0
1 1
o 0
1 1 M 1
o
y
M M
0
0 1
1
1
max(x,y)
x
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y
0 0
1 0
1 0
1 1
o 0
1 1 M 1
o
y
M M
1
0 0
1
1
max(x,y)
a b
a b m
0 0
1 0
1 0
1 1
1 1 1 0
M
m
m-Box: A two input binary syntax box that can compute any (two
input) binary syntax box?
x y
0 0
1 0
1 0
1 1
o
*
*
*
* How many different binary 2-input
s-boxes?
How will you prove it?
24 = 16
Syntax Boxes
proof of universality
x y o
0 0
1 0
1 0
1 1
1 1 1 1
x y o
0 0
1 0
1 0
1 1
1 0
1 0
min(x,y) 1-y 1
max(x,y)
4 Useful Boxes
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y o
0 0
1 0
1 0
1 1
1 0
1 0 1-y
M o y
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y o
0 0
1 0
1 0
1 1
1 1 1 1 M
o y
1
1-y
a b m
0 0
1 0
1 0
1 1
1 1 1 0
x y o
0 0
1 0
1 0
1 1
1 1 1 1 M
o y
1
1-y y
M
4 Useful Boxes
min(x,y) 1-y 1
max(x,y)
Need to prove:
any (two input) binary syntax box can be computed by
the 4 Useful Boxes
So what?
x y o
0 0
1 0
1 0
1 1
*
*
*
An Arbitrary Two Input Box
* Two 1-input boxes!
x= 0 then x= 1 then
What are the possible values of 0
0
0 1
1 0
1 1
An Arbitrary Two Input Box
What are the possible values of 0
0
0 1
1 0
1 1
min(x,y) 1-y 1
max(x,y) y
Can we compute it with the m-box?
x y o
0 0
1 0
1 0
1 1
*
*
*
*
x y o
0 0
1 0
1 0
1 1
*
*
*
A Selector Box
* x= 0 then
x= 1 then
selector
o
x
x y o
0 0
1 0
1 0
1 1
*
*
*
A Selector Box
* x= 0 then
x= 1 then
selector
o
0
x y o
0 0
1 0
1 0
1 1
*
*
*
A Selector Box
* x= 0 then
x= 1 then
selector
o
1
x y o
0 0
1 0
1 0
1 1
*
*
*
A Selector Box
* x= 0 then
x= 1 then
selector
o
x How can you compute this box?
max(a,b)
min(a,b) min(a,b)
o x= 0 then
x= 1 then
1-x x
A Selector Box
max(a,b)
min(a,b) min(a,b)
o x= 0 then
x= 1 then
0 1
0
A Selector Box
max(a,b)
min(a,b) min(a,b)
o x= 0 then
x= 1 then
1 0
0
QED
A Selector Box
x y o
0 0
1 0
1 0
1 1
0 1 1
How to compute the following (XOR)?
0
selector
o
x
?
? x= 0 then
x= 1 then
x y o
0 0
1 0
1 0
1 1
0 1 1 0
selector
o
x 1-y
y
How to compute the following (XOR)?
x= 0 then x= 1 then
x y o
0 0
1 0
1 0
1 1
0 1 1 0
selector
o
x 1-y
y x= 0 then
x= 1 then
y
M
Given a 2-input binary box that can compute
any
2-input binary boxCan it compute
any
3-input binary box?Does the magic continue?
a b
M
o
x z
o y
x y o
0 0
1 0
1 0
1 1
*
*
*
3-input binary s-box
*
0 0
1 0
1 0
1 1
*
*
*
* z
0 0 0 0 1 1 1 1 How many different
binary 3-input s-boxes?
28 = 256
x y o
0 0
1 0
1 0
1 1
*
*
*
3-input binary s-box
*
0 0
1 0
1 0
1 1
*
*
*
* z
0 0 0 0 1 1 1 1 Two 2-input boxes!
z= 0 then z= 1 then
x y o 0 0
1 0
1 0 1 1
*
*
*
* 0 0
1 0
1 0 1 1
*
*
*
* z
0 0 0 0 1 1 1 1 x
selector
o
y x y
z
z= 0 then z= 1 then All boxes can be computed with M
Given a magical box for any 2-input binary box
We proved that it is magical for any 3-input binary box!
a b
M
o o
....
Is it magical for any n-input binary box????
YES!!!!
Proof by induction on the number of inputs
Does the magic continue?
x
selector
o
(n-1) inputs (n-1) inputs
y x y
z
z= 0 then z= 1 then All boxes can be computed with M
....
....
There is a set of building blocks – one magic box!
Non-binary Binary
Questions about
algorithms
and building blocks ?
There are no simple magic boxes
Given a set of building blocks:
What can/cannot be constructed?
Efficiency and complexity Feasibility
Time: How long will it take to complete the construction?
Size: If feasible, how many blocks are needed?
Questions about
algorithms
and building blocks ?
Magic boxes do not know arithmetic…
Leibniz: A universal language
Instead of “thinking” let’s “calculate with magic boxes”
....
Gottfried Leibniz 1646-1716
Magic boxes do not know arithmetic…
Instead of “thinking” let’s “calculate with magic boxes”
smile recognition 0
....
1
Gottfried Leibniz 1646-1716
Leibniz was the first person to understand that computation
is not just arithmetic
Universal Language for Reasoning The founder of
Information Science
Game time
Game Time
Who can WIN the game
WIN
VI
Game Time
Who can WIN
WIN
the game
Game Time
Who can WIN
WIN
the game
can take as many sticks from a single group
Players take turns in removing sticks
the player that removes the last stick wins
Who can WIN
WIN
the game Example: Who will win?
Game Time
Who can WIN
WIN
the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
WIN
the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
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the game
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Who can WIN
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the game
Game Time
Who can WIN
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the game
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Who can WIN
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the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
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the game
Game Time
Who can WIN
WIN
the game
I CAN
Game Time
Who can WIN the game
WIN
VI=110
Game Time
Who can WIN
WIN
the game
011
100
110 001 An even number of 1s in every position
Game Time
Who can WIN
WIN
the game
An even number of 1s in every position
011
100
110 001
Game Time
Who can WIN
WIN
the game
010
100
110 000 An even number of 1s in every position
Game Time
Who can WIN
WIN
the game
010
100
110 000 An even number of 1s in every position
Game Time
Who can WIN
WIN
the game
010
100
001 111 An even number of 1s in every position
Game Time
Who can WIN
WIN
the game
010
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001 111 An even number of 1s in every position
Game Time
Who can WIN
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010
011
001 000 An even number of 1s in every position
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Who can WIN
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001 000 An even number of 1s in every position
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Who can WIN
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the game
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Who can WIN
WIN
the game
010
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001 011 An even number of 1s in every position
Game Time
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WIN
the game
010
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001 001 An even number of 1s in every position
Game Time
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the game
001
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001 000 An even number of 1s in every position
Game Time
Who can WIN
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the game
001
000
001 000 An even number of 1s in every position
Game Time
Who can WIN
WIN
the game
I CAN
000
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001 001 An even number of 1s in every position
Game Time
Who can WIN
WIN
the game
I CAN
000
000
000 000 An even number of 1s in every position
Winning starts with the choice of the right
language
Game Time
Who can WIN
WIN
the game
