4.
Review: Well-Ordering Principle
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The well-ordering principle states that every nonempty set of natural numbers has a smallest
element
In axiomatic set theory, the set of natural numbers
is defined as the set that contains 0 and is closed
under the successor operation
The set of natural numbers {n | {0, …, n}} is
well-ordered contains all natural numbers
5.
Review: Euclid’s 2nd Theorem
There are infinitely many primes
6.
Review: Euclid’s 2nd Theorem (Formulation 1)
Given a finite sequence of primes 2,3,5,..., pi ,
Ei 2 3 5 ... pi 1 is either a new prime or
has a prime divisor greater than pi .
7.
Review: Euclid’s 2nd Theorem (Formulation 2)
We set p0 0 for the base case. Then we define pi
to be the i prime. Thus, p1 2, p2 3, p3 5, p4 7,...
th
Consider En pn !1. Then En is either a prime or is
divisible by a prime greater than pn .
8.
Review: Lower and Upper Bounds for Next Prime
pi 1 pi 1, pi !1.
1) pi 1 is the lower bound, because, naturally, the next
prime, pi 1 , can be right after pi , (e.g., 2, 3).
2) The upper bound is pi !1, because pi 1 | pi !1, by
nd
Euclid' s 2 theorem.
9.
Review: Computing N-th Prime is P.R.
1. p 0
0
2. p n 1 min Primet & t p n
t pn !1
11.
Bezout’s Identity
If a and b are integers whose greatest
common divisor is d , i.e., gcda,b d ,
then there are integers x and y such that
ax by d .
12.
Bezout’s Identity: Example
gcd(12,42) 6
12 x 42 y 6
x 4, y 1
12 4 42 (1) 6
x 3, y 1
(3) 12 1 42 6
13.
Euclid’s 1st Theorem (Book VII of Euclid’s
Elements)
If a prime divides the product of two integers, then the
prime divides at least one of the two integers. Formally, if
p|ab, then p|a or p|b, where p is a prime and a and b
are integers.
14.
Proof Technique Note 1
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●
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Suppose we want to prove a statement:
if A
then B or C
We assume A and not B and prove C
In other words, if A and not B are true, then C
must be true, because otherwise, B or C cannot
be true
We can also assume A and not C and prove B
15.
Euclid’s 1st Theorem: Proof
Assume that p | ab, p is prime, ( p | a ). Since p | ab, rp ab,
for some number r. Then gcd (p,a) 1. By Bezout' s Identity,
there are integers x and y such that px ay 1. Now,
b b( px ay ) bpx bay bpx rpy p (bx ry ).
So p is a factor of b.
We could have also assumed that ( p | b) and shown
that p is a factor of a.
16.
Fundamental Theorem of Arithmetic
Every positive integer greater than 1 is either a prime or
can be written as a product of primes. The factorization
is unique except for the order of factor primes.
This theorem is also known as Unique Factorization
Theorem
18.
FTA: Key Insight
1200 2 3 5
Any divisor of 1200 is of the form
4
2
2 3 5 , where
x [0,4], y [0,1], z [0,2]
x
y
z
19.
FTA: Proof
1. We need to prove 2 statements:
1. Every natural number greater than 1 has a
prime factorization, i.e., can be written as a
product of primes
2. The prime factorization is unique
20.
FTA: Proof (Part 1)
1. Suppose not every natural number greater than 1 has a prime factorization.
2. By the well - ordering principle, there must be the smallest such number.
Call this number n.
3. n is not a prime, because, if it were, it would have itself as its factorization.
4. So n is a composite.
5. Since n is a composite, n ab, where 1 a n and 1 b n.
6. Since a and b are positive numbers less than n and n is the
smallest number that does not have a prime factorization,
a and b both have prime factorizations.
7. But then n has a prime factorization that consists ofthe prime
factorizations of a and b.
21.
Proof Technique Note 2
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●
●
Suppose that we want to prove that some
mathematical object A is unique
A common way of doing this is to postulate the
existence of another mathematical object B with
A’s properties and then show that A and B are the
same
In other words, A is unique, because any other
object that has the same properties is A itself
22.
FTA: Proof (Part 2)
1. Recall Euclid' s 1st Theorem : if p is a prime and p|ab, then p|a or p|b.
2. Let n be a natural number greater than 1 that has two prime factorizations
F1 and F2 .
3. n F1 p1...pn F2 q1...qm n.
4. Take p1. We know that p1|n. Thus, p1|q1...qm .
5. By Euclid' s 1st Theorem, p1|q1 or p1|q2 ...qm . Since p1 is a prime, it must be
the case that p1 q1 or p1 qi , 2 i m. Thus, p1 can be taken out of F1 and
F2 by dividing them both by p1.
6. The same trick can be repeated for p2 , p3 , ..., pn .
7. But, since F1 F2 , it must be the case that n m and for 1 i, j n, pi q j .
24.
Pairing Functions
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●
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Pairing functions are coding devices for mapping pairs of
natural numbers into single natural numbers and vice
versa
Once we have pairing functions we will be able to map
lists of numbers into single numbers and vice versa
Remember that our end objective is to compile L
programs into natural numbers
25.
Pairing Functions
x, y 2 2 y 1 1
x
2 2 y 1 1 0
x
x, y 1 2 2 y 1
x
26.
Equation 8.1 (Ch. 3)
If z is a a natural number, there is a unique
solution for x, y to x, y z.
27.
Equation 8.1 (Ch. 3)
1. x, y z
2. x, y 1 2 (2 y 1).
x
3. z 1 2 x (2 y 1).
4. x max 2 | z 1 .
d
d
In other words, x is the largest number such that 2 x | z 1.
z 1
1
x
z 1
5. 2 y 1 x y 2
2
2
28.
Equation 8.1 (Ch. 3): Upper Bounds on x & y
Since x, y z 2 2 y 1 1,
x
x, y 1 z 1 2 2 y 1.
x
Therefore, x z 1, y z 1.
Hence, x z , y z.
29.
Example 1
Solve x, y 10.
x, y 10 2 2 y 1 1 10
x
2 2 y 1 11
x
x max 2 | 11 x 0
d
d
2 y 1 11 y 5.
Check : 0,5 2 2 5 1 1 11 1 10.
0
30.
Example 2
Solve x, y 19.
x, y 2 2 y 1 1 19
x
2 2 y 1 20
x
x max 2 | 20 2
d
d
2 2 y 1 20 2 y 1 5 y 2.
2
Check : 2,2 2 2 2 1 1 20 1 19.
2
32.
Equation 8.1 (Ch. 03)
Equation 8.1 defines two functions : l z and r z , z N .
l z l x, y x.
r z r x, y y.
Examples :
l 10 l 0,5 0; r 10 r 0,5 5.
l 19 l 2,2 2; r 19 r 2,2 2.
33.
Lemma
l z and r z are primitive recursive.
34.
Proof
If z N , then z x, y , x z , y z. Thus,
l ( z ) miny z z x, y
r ( z ) minx z z x, y
x z
y z
35.
Theorem 8.1 (Ch. 03)
The functions x,y , l z , r z have the following properties :
1. They are primitive recursive
2. l x, y x, r x, y y
3. l z , r z z
4. l z z , r z z
36.
Proof 8.1 (Ch. 03)
This theorem summarizes the properties of the pairing
function and the splitting functions l(z) and r(z). Properties
2, 3, 4 follow from Equation 8.1. Property 1 follows from
Equation 8.1 and the definitions for l(z) and r(z).
39.
Background
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Gödel investigated the use of logic and set theory
to understand the foundations of mathematics
Gödel developed a technique to convert formal
symbolic statements into natural numbers
The technique was later called Gödel numbering
40.
Coding Programs by Numbers
Programming languages are symbolic formalisms
●
If we can convert statements of symbolic formalisms into numbers
[in other words, any program P is associated with a unique number
#(P)], we have the foundations of a mathematical theory of program
compilation and program execution (interpretation)
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The gist of the theory is three-fold:
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Given a program P, there is a computable function C (Compiler)
such that C(P) = #(P); this is the compiler
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Given #(P), there is a computable function RC (Reverse Compiler)
such that RC(#(P))= P; this function is the reverse compiler
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Given a program P, there is a partially computable function VM
that can execute C(P); this is the operating system or the virtual
machine
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41.
Gödel Numbers
Let a1 ,..., an be a sequence of numbers. We define
n
a1 ,..., an p
ai
i
th
, where pi is the i prime.
i 1
a1 ,..., an
is the Gödel number (G-number) of this sequence.
42.
Example
Suppose we have the following sequence 1,3,2 .
The G - number of this sequence is
1,3,2 p
1
1
p p 2 3 5 .
3
2
2
3
1
3
2
43.
Reading Suggestions
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Ch. 03, Computability, Complexity, and Languages,
2nd Edition, by Davis, Sigal, Weyuker, Academic Press
http://en.wikipedia.org/wiki/Kurt_Gödel
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