IST 4
Information and Logic
Office hours
moved from Moore 331 to SFL
MQ1 grades
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please let the TAs know if you submitted MQ1 and did not get the email
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31 M1 1
7 M1
14 1 2M2
21 2
28 M2
5 3
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19 4 5
26
2 5
IST 4: Planned Schedule – Spring 2014
x= hw#x out x= hw#x due
= today
T
Mx= MQx out Mx= MQx due
midterms
T
oh oh
oh oh
oh oh
oh
oh
oh oh
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oh = office hours oh
• You are invited to write short essay on the topic of the
Magenta Question.
• Recommended length is 3 pages (not more)
• Submit the essay in PDF format to ta4@paradise.caltech.edu file name lastname-firstname.pdf
•
No collaboration. No extensions
Memory
MQ2
Deadline Thursday 4/29/2014 at 10pmGrading of MQ:
3 points (out of 103)
50% for content quality, 50% for writing quality Some students will be given an opportunity
to give a short presentation for up to 3 additional points
Building Blocks
finite number of different parts ‘infinitely’ many structures Natural language: the composition of a small number
of parts (letters/words) results in infinitely many conceptual structures
Any language: from parts to systems
life, biology, music, art, physics, chemistry, computing, engineering, mathematics, literature...
The most important idea in Information
Finite Universality
of a Language
Can construct ‘everything’
from a finite set of
building blocks
There is a finite
universal
set of building blocks Can construct‘everything’
Finite Universality
a b
m
a b m
0 0
1 0
1 0
1 1
1 1 1 0
Magic Syntax Boxes
Magic Syntax Boxes
Proof by induction:
Base: Can generate all the s-boxes of size 2 – a single variable:
0 0
0 1
1 0
1 1
Assume true for all s-boxes with n-1 variables Prove true for any s-box with n variables
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 z= 0 then
z= 1 then
Sub-Boxes
Example: n=3
max(a,b)
min(a,b) min(a,b)
o
z 1-z
Are tables with n-1 variables
General composition of small boxes
z= 0 then z= 1 then
Given a set of building blocks:
What can/cannot be constructed?
Feasibility
Questions about building blocks?
solving equations
Can the solutions be expressed using
Leonardo Fibonacci 1170-1250
solving equations
Found an approximate solution base-60....
Considered:
Source: wikipedia
solving equations
Considered:
Scipione del Ferro 1465-1526
Found a general solution... Did not publish it...
The power of keeping a secret.?.?
Source: wikipedia
Gerolamo Cardano 1501 - 1576 Niccolò Tartaglia
1500-1557
Lodovico Ferrari 1522 –1565
solving equations
Will be posted on the class website
Source: wikipedia
Niels Abel 1802-1826
solving equations
This activity led to the invention of complex numbers, group theory, abstract algebra....
Source: wikipedia
Arithmetic Boxes
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 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18
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
2 symbol adder c
c
d1 d2
c
1891 + 8709 = ??
2 symbol adder c
c
d1 d2
c c 2 symbol adder c
d1 d2
c c 2 symbol adder c
d1 d2
c
2 symbol adder c
c
1 d2
c c 2 symbol adder c
8 d2
c c 2 symbol adder c
9 d2
c c 2 symbol adder c
1 d2
0
1891 + 8709 = ??
2 symbol adder c
c
1 8
c c 2 symbol adder c
8 7
c c 2 symbol adder c
9 0
c c 2 symbol adder c
1 9
0
1891 + 8709 = ??
2 symbol adder c
c
1 8
c c 2 symbol adder c
8 7
c c 2 symbol adder c
9 0
c 1 2 symbol adder 0
1 9
0
1891 + 8709 = ??
2 symbol adder c
c
1 8
c c 2 symbol adder c
8 7
c 1 2 symbol adder
0
1 9
0 2 symbol adder
1
0
9 0
1
1891 + 8709 = ??
2 symbol adder c
c
1 8
c 1 2 symbol adder 6
8 7
1 1 2 symbol adder 0
9 0
1 1 2 symbol adder 0
1 9
0
1891 + 8709 = ??
2 symbol adder 1
0
1 8
1 1 2 symbol adder 6
8 7
1 1 2 symbol adder 0
9 0
1 1 2 symbol adder 0
1 9
0
1891 + 8709 = ??
2 symbol adder 1
0
1 8
1 1 2 symbol adder 6
8 7
1 1 2 symbol adder 0
9 0
1 1 2 symbol adder 0
1 9
0
1891 + 8709 = 10,600
Algorithms – Syntax Manipulation
0
2 symbol adder c
s
d1 d2
c
How will it look for c=1?
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 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18
sum
Algorithms – Syntax Manipulation
1 0
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 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 9 0 1 2 3 4 5 6 7 8
0 10 1 2 3 4 5 6 7 8 9 1 11 2 3 4 5 6 7 8 9 10 2 12 3 4 5 6 7 8 9 10 11 3 13 4 5 6 7 8 9 10 11 12 4 14 5 6 7 8 9 10 11 12 13 5 15 6 7 8 9 10 11 12 13 14 6 16 7 8 9 10 11 12 13 14 15 7 17 8 9 10 11 12 13 14 15 16 8 18 9 10 11 12 13 14 15 16 17 9 19 10 11 12 13 14 15 16 17 18
sum
Algorithms – Syntax Manipulation
1 0
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 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 9 0 1 2 3 4 5 6 7 8
0 10 1 2 3 4 5 6 7 8 9 1 11 2 3 4 5 6 7 8 9 10 2 12 3 4 5 6 7 8 9 10 11 3 13 4 5 6 7 8 9 10 11 12 4 14 5 6 7 8 9 10 11 12 13 5 15 6 7 8 9 10 11 12 13 14 6 16 7 8 9 10 11 12 13 14 15 7 17 8 9 10 11 12 13 14 15 16 8 18 9 10 11 12 13 14 15 16 17 9 19 10 11 12 13 14 15 16 17 18
sum
The Sum Table
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
s
d1 d2
c
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
The Carry Table
2 symbol adder c
s
d1 d2
c
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8 0 0 1 2 3 4 5 6 7 8 9
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
The Carry Table
2 symbol adder c
s
d1 d2
c
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
1 0 1
0 0 0
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
The Carry Table
2 symbol adder c
s
d1 d2
c
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
s
7 8
0
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
1 2 symbol adder 5
7 8
0
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
c
d1 d2
c
1891 + 8709 = ??
2 symbol adder c
c
d1 d2
c c 2 symbol adder c
d1 d2
c c 2 symbol adder c
d1 d2
c
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
c
1 d2
c c 2 symbol adder c
8 d2
c c 2 symbol adder c
9 d2
c c 2 symbol adder c
1 d2
0
1891 + 8709 = ??
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
c
1 8
c c 2 symbol adder c
8 7
c c 2 symbol adder c
9 0
c c 2 symbol adder c
1 9
0
1891 + 8709 = ??
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
c
1 8
c c 2 symbol adder c
8 7
c c 2 symbol adder c
9 0
c 1 2 symbol adder 0
1 9
0
1891 + 8709 = ??
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
c
1 8
c c 2 symbol adder c
8 7
c 1 2 symbol adder
0
1 9
0
1891 + 8709 = ??
2 symbol adder 1
0
9 0
1
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder c
c
1 8
c 1 2 symbol adder 6
8 7
1 1 2 symbol adder 0
9 0
1 1 2 symbol adder 0
1 9
0
1891 + 8709 = ??
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder 1
0
1 8
1 1 2 symbol adder 6
8 7
1 1 2 symbol adder 0
9 0
1 1 2 symbol adder 0
1 9
0
1891 + 8709 = ??
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
0 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 0 2 2 3 4 5 6 7 8 9 0 1 3 3 4 5 6 7 8 9 0 1 2 4 4 5 6 7 8 9 0 1 2 3 5 5 6 7 8 9 0 1 2 3 4 6 6 7 8 9 0 1 2 3 4 5 7 7 8 9 0 1 2 3 4 5 6 8 8 9 0 1 2 3 4 5 6 7 9 9 0 1 2 3 4 5 6 7 8 1 9 0 1 2 3 4 5 6 7 8
2 symbol adder 1
0
1 8
1 1 2 symbol adder 6
8 7
1 1 2 symbol adder 0
9 0
1 1 2 symbol adder 0
1 9
0
1891 + 8709 = 10,600
0 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 1 1 3 0 0 0 0 0 0 0 1 1 1 4 0 0 0 0 0 0 1 1 1 1 5 0 0 0 0 0 1 1 1 1 1 6 0 0 0 0 1 1 1 1 1 1 7 0 0 0 1 1 1 1 1 1 1 8 0 0 1 1 1 1 1 1 1 1 9 0 1 1 1 1 1 1 1 1 1
Algorithm =
a procedure for syntax manipulation Dear Algo and Fibo,
we get confused with those large tables... can we use a
smaller syntax?
The Binary
When was the binary system “invented”?
(11010100111)
2(1703)
10Leibniz – Binary System
Gottfried Leibniz 1646-1716
Algorithm =
a procedure for syntax manipulation
Gottfried Leibniz 1646-1716
Algorithm =
a procedure for syntax manipulation
Gottfried Leibniz 1646-1716
Algorithm =
a procedure for syntax manipulation
Gottfried Leibniz 1646-1716
Algorithm =
a procedure for syntax manipulation
Use the smallest syntax possible Binary – 0 and 1
Gottfried Leibniz 1646-1716
8
??
Binary Addition
carry
Gottfried Leibniz 1646-1716
Binary Multiplication
Addition of shifted versions
3 6
5 5
10 20
Gottfried Leibniz 1646-1716
Leibniz:
No Need for Flash Cards!
Invented the Binary System?
Gottfried Leibniz 1646-1716
Leibniz – Binary System
Gottfried Leibniz 1646-1716
Wen Wang (who flourished in about 1150 BC) is traditionally thought to have been author of the present hexagrams
63
Leibniz – Binary System
Gottfried Leibniz 1646-1716
Arithmetic Boxes
binary
2 symbol adder
digit 1 digit 2
carry
sum
carry
Adding Bits
3 bits to 2 bits
Adding Bits
2 symbol adder
0 0
0
digit 1 digit 2
carry
sum
0 0 carry
3 bits to 2 bits
2 symbol adder
1 0
1
digit 1 digit 2
carry
sum
0 0 carry
Adding Bits
3 bits to 2 bits
2 symbol adder
1 1
0
digit 1 digit 2
carry
sum
1 0 carry
Adding Bits
3 bits to 2 bits
2 symbol adder
1 0
0
digit 1 digit 2
carry
sum
1 1 carry
Adding Bits
3 bits to 2 bits
2 symbol adder
0 1
0
digit 1 digit 2
carry
sum
1 1 carry
Adding Bits
3 bits to 2 bits
2 symbol adder
1 1
1
digit 1 digit 2
carry
sum
1 1 carry
Adding Bits
3 bits to 2 bits
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
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
Magic boxes do not know arithmetic!!
However...
Arithmetic Boxes
meet magic boxes
There is a finite
universal
set of building blocks Can construct‘everything’
Finite Universality
a b
m
a b m
0 0
1 0
1 0
1 1
1 1 1 0
Magic Syntax Boxes
a b
d1 d2 s
0 0
1 0
1 0
1 1
0 1 1 0
m
d1 d2
0 0
1 0
1 0
1 1
s 1 0 0 1
0 1 0 0 1 1 1 0
0 1 0 1 0 1 0 1
2 symbol adder c
s
d1 d2
c
0 1
