Chaitin. Algorithmic information theory

(eq.1)

total.input = input.A + input.B + input.C

(eq.2)

number.of.instructions = 16

(eq.3)

longest.label = 3

(eq.4)

q = 256

** ( total.input + time +

number.of.instructions + longest.label + 3 )

(eq.5)

q.minus.1 + 1 = q

Equation defining i, all of whose base q digits are 1's:

(eq.6)

1 + q *

i = i + q ** time

Basic Label

Variable Equations *******************************

(imp.1)

L1

=> i

(imp.2)

L2

=> i

(imp.3)

L3

=> i

(imp.4)

L4

=> i

(imp.5)

L5

=> i

(imp.6)

L6

=> i

(imp.7)

L7

=> i

(imp.8)

L8

=> i

(imp.9)

L9

=> i

(imp.10)

L10

=> i

(imp.11)

L11

=> i

(imp.12)

L12

=> i

(imp.13)

L13

=> i

(imp.14)

L14

=> i

(imp.15)

L15

=> i

(imp.16)

L16

=> i

(eq.7)

i = L1 + L2 + L3 + L4 + L5 + L6 + L7 + L8 + L9 +

L10

+ L11 + L12 + L13 + L14 + L15 + L16

Equations for starting & halting:

(imp.17)

1 => L1

(eq.8)

q ** time = q * L10 + q * L16

2.5. AN EXAMPLE OF ARITHMETIZATION

53

In other words,

(eq.9)

ic = 1075605632 * L1

+

1083994240

* L2

+

1079799936

*

L3

+ 1088188544

*

L4

+ 1077702784 *

L5 + 1086091392

*

L6

+

1081897088

*

L7

+

1090285696

*

L8

+

1073901696

*

L9

+

278839193728 *

L10 + 275349532800

*

L11 + 277497016448

*

L12 +

276423274624

*

L13

+

278570758272

*

L14

+

275886403712

*

L15

+

278033887360

*

L16

(imp.41)

q * next.ic =>

ic

(imp.42)

ic => q * next.ic + q - 1

(imp.43)

set.C.L2 => 1079799936 * i

(imp.44)

set.C.L2 => q.minus.1 * L2

(imp.45)

1079799936 * i => set.C.L2 + q.minus.1 * i -

q.minus.1 * L2

(imp.46)

set.A.L11

=> 184 * i

(imp.47)

set.A.L11

=> q.minus.1 * L11

(imp.48)

184 * i => set.A.L11 + q.minus.1 * i - q.minus.1 *

L11

L12: SET A B

(imp.49)

set.A.L12 => B

(imp.50)

set.A.L12 => q.minus.1 * L12

(imp.51)

B => set.A.L12 + q.minus.1 * i - q.minus.1 * L12

L13: RIGHT C

(imp.52)

set.C.L13 => shift.C

(imp.53)

set.C.L13 => q.minus.1 * L13

(imp.54)

shift.C => set.C.L13 + q.minus.1 * i - q.minus.1 *

L13

L14: LEFT A C'b'

(imp.55)

set.A.L14 => 256 * A + 120 * i

(imp.56)

set.A.L14 => q.minus.1 * L14

(imp.57)

256 * A + 120 * i => set.A.L14 + q.minus.1 * i -

q.minus.1 * L14

(imp.58)

set.A.L15 => 256 * A + char.B

56

CHAPTER 2. REGISTER MACHINES

(imp.74)

B => dont.set.B + q.minus.1 * set.B

(imp.75)

256

* shift.B => B

(imp.76)

256

* shift.B => q.minus.1 * i - 255 * i

(imp.77)

B => 256 * shift.B + 255 * i

(eq.15)

B = 256 * shift.B + char.B

Register C

...................................................

(imp.78)

C => q.minus.1 * i

(eq.16)

C + output.C * q ** time = input.C + q * set.C.L2

+ q * set.C.L13 + q * dont.set.C

(eq.17)

set.C = L2 + L13

(imp.79)

dont.set.C => C

(imp.80)

dont.set.C => q.minus.1 * i - q.minus.1 * set.C

(imp.81)

C => dont.set.C + q.minus.1 * set.C

(imp.82)

256

* shift.C => C

(imp.83)

256

* shift.C => q.minus.1 * i - 255 * i

(imp.84)

C => 256 * shift.C + 255 * i

(imp.85)

ge.A.C'a' => i

(imp.86)

256

* ge.A.C'a' => 256 * i + char.A - 184 * i

(imp.87)

256

* i + char.A - 184 * i => 256 * ge.A.C'a' +

255

* i

(imp.88)

ge.C'a'.A => i

(imp.89)

256

* ge.C'a'.A => 256 * i + 184 * i - char.A

(imp.90)

256

* i + 184 * i - char.A => 256 * ge.C'a'.A +

255

* i

(imp.91)

eq.A.C'a' => i

2.5. AN EXAMPLE OF ARITHMETIZATION

57

(imp.92)

2 * eq.A.C'a' => ge.A.C'a' + ge.C'a'.A

(imp.93)

ge.A.C'a' + ge.C'a'.A => 2 * eq.A.C'a' + i

Compare A B

..................................................

(imp.94)

ge.A.B => i

(imp.95)

256

* ge.A.B => 256 * i + char.A - char.B

(imp.96)

256

* i + char.A - char.B => 256 * ge.A.B + 255 *

i

(imp.97)

ge.B.A => i

(imp.98)

256

* ge.B.A => 256 * i + char.B - char.A

(imp.99)

256

* i + char.B - char.A => 256 * ge.B.A + 255 *

i

(imp.100)

eq.A.B => i

(imp.101)

2 * eq.A.B => ge.A.B + ge.B.A

(imp.102)

ge.A.B + ge.B.A => 2 * eq.A.B + i

Compare A C'b' ...............................................

Note: C'b' is 120

(imp.103)

ge.A.C'b' => i

(imp.104)

256

* ge.A.C'b' => 256 * i + char.A - 120 * i

(imp.105)

256

* i + char.A - 120 * i => 256 * ge.A.C'b' +

255

* i

(imp.106)

ge.C'b'.A => i

(imp.107)

256

* ge.C'b'.A => 256 * i + 120 * i - char.A

(imp.108)

256

* i + 120 * i - char.A => 256 * ge.C'b'.A +

255

* i

(imp.109)

eq.A.C'b' => i

(imp.110)

2 * eq.A.C'b' => ge.A.C'b' + ge.C'b'.A

(imp.111)

ge.A.C'b' + ge.C'b'.A => 2 * eq.A.C'b' + i