Theory of Computation

More Number Theory Factoids, Pairing Functions, Gödel Numbers

Vladimir Kulyukin

www.vkedco.blogspot.comwww.vkedco.blogspot.com

Outline● Review● More Number Theory

Bezout's Identity Euclid's 1st Theorem Fundamental Theorem of Arithmetic (Unique Factorization

Theorem)● Pairing Functions● Gödel Numbers

Review

Review: Well-Ordering Principle

● The well-ordering principle states that every non-empty 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

Review: Euclid’s 2nd Theorem

There are infinitely many primes

Review: Euclid’s 2nd Theorem (Formulation 1)

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Review: Euclid’s 2nd Theorem (Formulation 2)

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Review: Lower and Upper Bounds for Next Prime

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Review: Computing N-th Prime is P.R.

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More Number Theory Factoids

Bezout’s Identity

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Bezout’s Identity: Example

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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.

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

Euclid’s 1st Theorem: Proof

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Fundamental Theorem of Arithmetic

Every positive integer greater than 1 is either a prime or can be written as a product of primes. The factorizationis unique except for the order of factor primes.

This theorem is also known as Unique Factorization Theorem

FTA: Examples

24

2

1111

3

5321200

3212

255210

22228

326

FTA: Key Insight

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where,532

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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

FTA: Proof (Part 1)

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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

FTA: Proof (Part 2)

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Pairing Functions

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

Pairing Functions

1221,

01122

1122,

yyx

y

yyx

x

x

x

Equation 8.1 (Ch. 3)

., tofor solution

unique a is therenumber, natural a a is If

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Equation 8.1 (Ch. 3)

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., Hence,

.1,1 Therefore,

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,1122, Since

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x

Example 1

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Splitting Natural Numbers

Equation 8.1 (Ch. 03)

.22,219;22,219

.55,010;05,010

:Examples

.,

.,

. , and :functions twodefines 8.1Equation

rrll

rrll

yyxrzr

xyxlzl

Nzzrzl

Lemma

recursive. primitive are and zrzl

Proof

,min)(

,min)(

Thus, .,,, then , If

yxzxzr

yxzyzl

zyzxyxzNz

zzy

zzx

Theorem 8.1 (Ch. 03)

zzrzzl

zzrzl

yyxrxyxl

zrzlx,y

, 4.

, 3.

,,, 2.

recursive primitive areThey 1.

:properties following thehave ,, functions The

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).

Gödel Numbers

Kurt Gödel

1906 - 1978

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

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

Gödel Numbers

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ain

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define Wenumbers. of sequence a be ,...,Let

naa ,...,1is the Gödel number (G-number) of this sequence.

Example

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32

11 ppp

Reading Suggestions

● Ch. 03, Computability, Complexity, and Languages, 2nd Edition, by Davis, Sigal, Weyuker, Academic Press

● http://en.wikipedia.org/wiki/Kurt_Gödel