Theory of Computation

Gödel Numbers

Vladimir Kulyukin

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

Outline

● Pairing Functions● Gödel Numbers● Gödel Coding & Decoding

Just a Thought on Why Natural Numbers are Natural

Every natural number in the universe is a program awaiting its computer. The program runs if and when its computer appears.

Theory of Program Compilation & Execution

● Programming languages are symbolic formalisms● If we can convert statements of symbolic formalisms into numbers (i.e., any

program P is associated with a unique number), 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

The Big Picture

GÖDEL Coder

GÖDEL Decoder

L Program Number Universal Program

Number

???

GÖDEL Coder = CompilerGÖDEL Decoder = Reverse CompilerUniversal Program = Operating System/VM/Interpreter

Pairing Functions

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

zyxx, y

z

Equation 8.1 (Ch. 3)

2

12

1

2

112 .5

.1|2such that number largest theis s,other wordIn

.1z| 2max .4

).12(21z .3

).12(21, .2

, .1

x

x

x

d

d

x

x

z

yz

y

zx

x

y

yyx

zyx

Equation 8.1 (Ch. 3): Upper Bound for x & y

., Hence,

.1,1 Therefore,

.12211,

,1122, Since

zyzx

zyzx

yzyx

yzyxx

x

Example 1

.10111115225,0 :Check

.51112

011|2max

11122

10112210,

.10, Solve

0

yy

xx

y

yyx

yx

d

d

x

x

Example 2

.19120112222,2:Check

.251220122

220|2max

20122

191122,

.19, Solve

2

2

yyy

x

y

yyx

yx

d

d

x

x

Splitting Natural Numbers into

Left & Right Constituents

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

Gödel Numbers

Background

● Kurt Gödel used of logic and set theory to formulate and understand the foundations of mathematics

● Gödel developed a technique to convert formal symbolic statements into natural numbers

● The technique was later called Gödel numbering

Kurt Gödel, 1906 - 1978

Gödel Numbers

n

ii

ain

n

ippaa

aa

i

1

th1

1

prime. theis where,,...,

as defined is sequence thisof number)-(Gnumber

Gödel The numbers. of sequence a be ,...,Let

Example

.5321,3,2

is sequence thisofnumber -G The

.2,3,1 sequence following thehave weSuppose

23123

32

11 ppp

Lemma

recursive. primitive is ,..., ,each For 1 naan

Proof

3. Ch. 6.2,Corollary

by recursive, primitive isfunction recursive primitive a of

product bounded thebecause recursive, primitive is

recursive. primitive is so recursive primitive is

recursive. primitive is

1

n

i

ai

ai

y

i

i

i

p

px

p

Note 1 on Theorem 8.2 (Ch. 3)

....11......

,..., because,0,0,,...,,...,

example,For number.-G its changing without sequence the

ofright the toszero' ofnumber any addcan wegeneral,In

02

01111

111

111

nnan

aan

aan

a

nnn

pppppppp

aaaaaannn

Note 2 on Theorem 8.2 (Ch. 3)

...75325323221

because 1, be to

defined is () sequenceempty theofnumber -G The

0000000000

Note 2 on Theorem 8.2 (Ch. 3)

.755322,1,018322,1

because,2,1,02,1

example,For

.......

because ,,...,,0,...,

general,In number.-G its changingwithout

sequence theofleft the toszero' addcannot We

21021

12011

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11

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naa

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aaaa

Accessing Individual Elements of

Gödel Numbers

Motivation● We can map every natural number n into its Gödel number g ● Every n has a unique prime factorization, thus its g can be

construed as a sequence of powers of the prime factors of n● We can think of n as an array of numbers and, as with arrays

in other programming languages, we need access to their individual elements

● In our context, we would like to access those individual prime powers

Example

101

20

3

1000

11

100

2

10

1`

532]1,0,1[10

32]2,0[9

.238

7532[0,0,0,1]7

32[1,1]6

532[0,0,1]5

2[2]4

32[0,1] 3

2[1]2

Access Function for Gödel Numbers

.3;4;2

then,3,4,2 if example,For .1 ,function

recursive primitive a define want to We.,...,Let

321

1

xxx

xniax

aax

ii

n

Access Function for Gödel Numbers

. allfor ,00; allfor ,0 that Note

|min

0

1

ixx

xpx

i

ti

xti

Example 01

.2 so ,18|3 but 18;|3 ;18|3 ;18|3

and ;3 because ,218| min18

.1 so,18|2 but ;18|2 ;18|2

and 2; because ,118| min18

Then .322,118Let

3210

21

218

2

210

11

118

1

2122

11

t

pp

t

pp

pp

tttt

t

t

ttt

t

t

Example 02

3108

2108

Then .274323,2108Let

2

1

3222

11

pp

Access Function for G-Numbers: Implementation

// x is a natural number, compute (x)i

long GAccess(long x, long i) {

long p = computePrime(i); // compute i-th prime pi

for t from 0 upto x {

if ( pt does not divide x )

return t – 1;

}

}

Access Function for G-Numbers: Implementation

● The previous slide assumes that we represent G-Numbers as integers● This is a sensible choice, but only for small numbers● If we want to be generic, we can represent G-numbers as sequences of prime

powers, e.g. arrays, lists, etc. ● In this case, we can represent much larger values without actually having to

compute them● Access function becomes much easier, because we do not have to compute

primes or their powers

Lengths of Natural Numbers

Length of a Natural Number

● Since Gödel numbering and access functions give us techniques to split any natural number into its constituents

● Next question: how many constituents does a given natural number have if looked at as a G-number? In other words, how long is the number?

● How long is 20? How long is 7?

Length of a Natural Number: Definition

.010 ,definitionBy

number.-G ain power prime

zero-nonlast theofindex theis s,other wordIn

.0&0min

LtLt

i

xijjxxLt jxixi

How Long is Natural Number 20?

.320

.0...

.1

.1,0,253220

.?20

2054

3

102

Lt

xxx

x

x

Lt

How Long is Natural Number 7?

.47

.0

.1

.1,0,0,075327

.?20

765

4

1000

Lt

xxx

x

x

Lt

Just In Case You Are Not Confused Yet

20 > 7? Sure, but 7 is longer!

Gödel Coding & Decoding

Gödel Coder & Decoder● We will demonstrate that compilation (coding) and reverse

compilation (decoding) are both primitive recursive ● In other words:

Given an L program P, there is a primitive recursive function that maps P to its number #(P)

Given #(P), there is a primitive recursive function that maps #(P) to its source code P

Gödel Coder

● We need to solve three problems:● Problem 1: Mapping variables and labels to numbers● Problem 2: Mapping instructions to numbers● Problem 3: Mapping programs to numbers

Problem 1: Mapping Variables & Labels● We order the variables as follows: Y, X1, Z1, X2, Z2, X3, Z3, … ● We order the labels as follows: A1, B1, C1, D1, E1, A2, B2, C2,

D2, E2, A3, …● Let #(V) be the position of a variable in the variable ordering ● Let #(L) be the position of a label in the label ordering ● Note that the chosen orderings are arbitrary in the sense that

other orderings are possible

Examples of #(V) & #(L)

● We order the variables as follows: Y, X1, Z1, X2, Z2, X3, Z3, …

● We order the labels as follows: A1, B1, C1, D1, E1, A2, B2, C2, D2, E2, A3, …

● #(Y)=1, #(X1)=2, #(X3)=6, #(Z2)=5● #(B1)=2, #(A2)=6, #(C2)=8

Problem 2: Mapping Instructions to Numbers● There are four primitive instruction types in L: addition, subtraction,

self-assignment (no-op), conditional dispatch● There are three pieces of information we have to known in order to

map an instruction to a number: label or no label variable used in the instruction instruction type

Sub-Task 2: Assigning Numbers to Instruction Types

2# n typeinstructio be willThis // 0

2 n typeinstructio be willThis // 1

1 n typeinstructio be willThis // 1

0 n typeinstructio be willThis //

L GOTO LIF V

-VV

VV

VV

Problem 2: Mapping Instructions to Numbers

ninstructio in the used variable theofnumber theis .3

number n typeinstructio theis 2.

number label theis 1.

where,,,#Then n.instructio primitive a be Let

c

b

a

cbaILI

Problem 2: Mapping Instructions to Numbers

.22

case in this that Note .2then ,0! is if 7.

;2 then ,1 is if 6.

;1 then ,1 is if 5.

;0 then , is if .4

;1 then ,in used is variablea if .3

; then , label some with labeled is if 2.

;0 then unlabeled, is if 1.

where,,,#Then n.instructio primitive a beLet

L#b

L#b GOTO LIF VI

b-VVI

bVVI

bVV I

V#cIV

L#aLI

aI

cbaILI

Instruction Coding Example 01

Let I be X1 ← X1 + 1. What is #(I)?

Instruction Coding Example 01

.101111113221

111112222

111,1221,1,0#

Thus, 4.

;21 because ,111 3.

addition; is because ,1 .2

unlabeled; is because ,0 .1

where,,,# so ;111 is

10

0

I

X#X#c

Ib

Ia

cbaIXXI

Instruction Coding Example 02

Let I be [A1] X1 ← X1+1. What is #(I)?

Instruction Coding Example 02

.211112111112222

111,1221,1,1

11#,1,1##

,111 1 is Since

111

1

XAI

XXAI

Problem 3: Definition of #(P)

.1#,...,#,##

thenns,instructio primitive are ,...,, and

0 where,,...,, of consists that program a be Let

21

21

21

k

k

k

IIIP

LIII

kIIIP

Example of #(P)

1 GOTO 0 1 IF

111 1

program? thisofnumber theisWhat

AX

XXA

Example of #(P)

1#,##

Thus,

1 GOTO 0 1 IF is

111 1 is

21

2

1

IIP

AXI

XXAI

Example of #(P)

21111,1221,1,1 So,

11211# .3

1 2.

11# .1

where

,,,111 1## example, previous aBy

1

1

Xc

b

Aa

cbaXXAI

Example of #(P)

.46147112321

111112222

111,3221,3,0# Thus,

.111# .3

;32121# .2

;0 .1

where,,,1 GOTO 0! 1 IF##

30

02

2

I

Xc

Ab

a

cbaAXI

Example of #(P)

.1321]46,21[#

1 GOTO 0 1 IF

111 1

? program thisofnumber theisWhat

4621

P

AX

XXA

P

Back to The Big Picture

GÖDEL Coder

GÖDEL Decoder

L Program Number Universal Program

Number

???

The red arrows show the implemented tasks (program compilation): we now know how to map L programs to numbers.

Gödel Decoder

Gödel Decoder

● Problem 1: Obtain the source code number● Problem 2: Extract from the source code number the

numbers for individual instructions● Problem 3: Extract the source from each individual

instruction number

GÖDEL Decoder: Problem 1

.1##,...,#,#

isnumber code source theThus,

.1#,...,#,##

21

21

PIII

IIIP

k

k

Side Note: A Prime Factorization Algorithm// n is a natural number > 0, i is the prime counter, initially// set to 1. We need this algorithm to solve problem 2 of // Gödel DecodingFactors(n, i)

If n is prime then n is the factorization;If n is composite thenIf pi divides n then

add pi to the list of factors;Factors((n / pi), i);

If pi does not divide n ThenFactors(n, i+1)

Prime Factorization Algorithm: Example

2] 2, [2,return

2] [2,return

[2]return so prime a is 2

1) factors(2, to2 add so

4|2

1) factors(4, to2 add so

8|2

1)factors(8,

1

1

p

p

GÖDEL Decoder: Problem 2

number-G the

i.e., powers, prime oflist obtain the factors, prime oflist theFrom .3

factors prime oflist obtain the to,1,1nfactorsRun 2.

#,...,#,#1 isnumber code source theSo,

1#,...,#,# be Let 1.

21

21

k

k

IIIn

IIIP#n

Examples of Source Code Extraction

lyrespective 1, and 1,0,0, into compile that nsinstructiofour of consists P .3

1475321] 0, 0, [1,number -G the toconverted is [2,7] 2.

[2,7] returns ,1)factors(14 then 13, into compiles P If .1

3number into compilesn that instructio one of consists code source sP' .5

82[3]number -G the toconverted is ]2,2,2[ 4.

2] 2, [2, returns 1)factors(8, 3.

8 is Pfor number code source The 2.

7number into compiled is P program some Suppose 1.

1001

3

GÖDEL Decoder: Problem 3

ordering. variablein the 1 isposition

whose variableis in used variable theThus, . 3.

ordering. label in the 2 isposition

whoselabel is L whereL GOTO 0 ! V IF typeof is 2.4.

;2 if 1 - V V typeof is 2.3.

;1 if 1 V V typeof is 2.2.

;0 if V V typeof is 2.1.

Then . 2.

ordering. label in the

number label has Otherwise, unlabeled. is ,0 If . 1.

.,,#Let

number.-G in then instructioeach of code

source eextract th toneed wenumber,-G thehave that weNow

c

Iqrrc

b-

I

b I

b I

b I

qrlb

ql

IIaqla

cbaqI

#(P) P: GÖDEL Decoding Algorithm

recursive primitive are steps all that Note

3 stepin obtained

number-G theofelement each from code source eExtract th 4.

powers prime of sequence the

get i.e. factors, prime oflist thefromnumber -G Obtain the .3

1,1nFactorsRun 2.

#,...,#,#1 isnumber code source theSo,

;1#,...,#,# be Let 1.

21

21

k

k

IIIn

IIIP#n

GÖDEL Decoding Algorithm: Example1. Let #(P) = 199.2. The source code number = 199+1=200.3. The prime factorization of 200 is 233052. Thus, the G-number = [3,0,2].4. 3 = <2, <0, 0>>. So the label is B1, the instruction variable is Y and the type of instruction is self-assignment. Thus, first instruction is [B1] Y ← Y.5. 0 = <0, <0, 0>>. Thus, second instruction is Y ← Y.6. 2 = <0, <1, 0>>. Thus, third instruction is Y ← Y + 1.

GÖDEL Decoding Algorithm: Example

Let #(P) = 199. The source code number = 199+1=200. So, the source code of P is

[B1] Y ← Y

Y ← Y

Y ← Y + 1

Back To The Big Picture

GÖDEL Coder

GÖDEL Decoder

L Program Number Universal Program

Number

???

GÖDEL Coder = CompilerGÖDEL Decoder = Reverse CompilerUniversal Program = Operating System/VM/Interpreter

Reading Suggestions

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

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