Theory of Computation
More Number Theory Factoids, Pairing
Functions, Gödel Numbers
Vladimir Kulyukin

www.vkedco.blogspot...
Outline
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●

Review
More Number Theory



●

Euclid's 1st Theorem



●

Bezout's Identity

Fundamental Theorem of Arith...
Review
Review: Well-Ordering Principle
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●

●

The well-ordering principle states that every nonempty set of natural numbers has...
Review: Euclid’s 2nd Theorem

There are infinitely many primes
Review: Euclid’s 2nd Theorem (Formulation 1)

Given a finite sequence of primes 2,3,5,..., pi ,
Ei  2  3  5  ...  pi ...
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, p...
Review: Lower and Upper Bounds for Next Prime
pi 1   pi  1, pi !1.
1) pi  1 is the lower bound, because, naturally,...
Review: Computing N-th Prime is P.R.

1. p 0

 0

2. p n 1  min Primet  & t  p n 
t  pn !1
More Number Theory Factoids
Bezout’s Identity

If a and b are integers whose greatest

common divisor is d , i.e., gcda,b   d ,
then there are inte...
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...
Euclid’s 1st Theorem (Book VII of Euclid’s
Elements)

If a prime divides the product of two integers, then the
prime divid...
Proof Technique Note 1
●

●
●

●

Suppose we want to prove a statement:
if A
then B or C
We assume A and not B and prove C...
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...
Fundamental Theorem of Arithmetic
Every positive integer greater than 1 is either a prime or
can be written as a product o...
FTA: Examples

6  23
8  222  2

3

10  2  5  5  2
1

1

1

1

12  2  3
2

1200  2  3  5
4

2
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...
FTA: Proof
1. We need to prove 2 statements:
1. Every natural number greater than 1 has a
prime factorization, i.e., can b...
FTA: Proof (Part 1)
1. Suppose not every natural number greater than 1 has a prime factorization.
2. By the well - orderin...
Proof Technique Note 2
●

●

●

Suppose that we want to prove that some
mathematical object A is unique
A common way of do...
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 num...
Pairing Functions
Pairing Functions
●

●

●

Pairing functions are coding devices for mapping pairs of
natural numbers into single natural n...
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
Equation 8.1 (Ch. 3)

If z is a a natural number, there is a unique
solution for x, y to x, y  z.
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...
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

Ther...
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
...
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 ...
Splitting Natural Numbers
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 ...
Lemma

l z  and r z  are primitive recursive.
Proof
If z  N , then z  x, y , x  z , y  z. Thus,
l ( z )  miny  z z  x, y



r ( z )  minx  z z  x,...
Theorem 8.1 (Ch. 03)
The functions x,y , l  z , r  z  have the following properties :
1. They are primitive recursive
...
Proof 8.1 (Ch. 03)

This theorem summarizes the properties of the pairing
function and the splitting functions l(z) and r(...
Gödel Numbers
Kurt Gödel

1906 - 1978
Background
●

●

●

Gödel investigated the use of logic and set theory
to understand the foundations of mathematics
Gödel ...
Coding Programs by Numbers
Programming languages are symbolic formalisms
●
If we can convert statements of symbolic formal...
Gödel Numbers
Let a1 ,..., an  be a sequence of numbers. We define
n

a1 ,..., an    p

ai
i

th

, where pi is the ...
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  ...
Reading Suggestions
●

●

Ch. 03, Computability, Complexity, and Languages,
2nd Edition, by Davis, Sigal, Weyuker, Academi...
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Theory of Computation: More Number Theory Factoids, Pairing Functions, Godel Numbers

  1. 1. Theory of Computation More Number Theory Factoids, Pairing Functions, Gödel Numbers Vladimir Kulyukin www.vkedco.blogspot.com
  2. 2. Outline ● ● Review More Number Theory   ● Euclid's 1st Theorem  ● Bezout's Identity Fundamental Theorem of Arithmetic (Unique Factorization Theorem) Pairing Functions Gödel Numbers
  3. 3. Review
  4. 4. Review: Well-Ordering Principle ● ● ● 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. 5. Review: Euclid’s 2nd Theorem There are infinitely many primes
  6. 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. 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. 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. 9. Review: Computing N-th Prime is P.R. 1. p 0  0 2. p n 1  min Primet  & t  p n  t  pn !1
  10. 10. More Number Theory Factoids
  11. 11. Bezout’s Identity If a and b are integers whose greatest common divisor is d , i.e., gcda,b   d , then there are integers x and y such that ax  by  d .
  12. 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. 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. 14. Proof Technique Note 1 ● ● ● ● 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. 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. 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
  17. 17. FTA: Examples 6  23 8  222  2 3 10  2  5  5  2 1 1 1 1 12  2  3 2 1200  2  3  5 4 2
  18. 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. 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. 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. 21. Proof Technique Note 2 ● ● ● 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. 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 .
  23. 23. Pairing Functions
  24. 24. Pairing Functions ● ● ● 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. 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. 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. 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. 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. 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. 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
  31. 31. Splitting Natural Numbers
  32. 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. 33. Lemma l z  and r z  are primitive recursive.
  34. 34. Proof If z  N , then z  x, y , x  z , y  z. Thus, l ( z )  miny  z z  x, y  r ( z )  minx  z z  x, y  x z y z
  35. 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. 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).
  37. 37. Gödel Numbers
  38. 38. Kurt Gödel 1906 - 1978
  39. 39. Background ● ● ● 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. 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) ● The gist of the theory is three-fold: ● Given a program P, there is a computable function C (Compiler) such that C(P) = #(P); this is the compiler ● Given #(P), there is a computable function RC (Reverse Compiler) such that RC(#(P))= P; this function is the reverse compiler ● 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 ●
  41. 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. 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. 43. Reading Suggestions ● ● 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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