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Chaitin. Algorithmic information theory

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229

We now replace the above (n − 2)-term recurrence for Sn by a twoterm recurrence.3

The rst step is to eliminate the annoying middle term by completing the square. We replace the original generating function F by a new generating function whose coe cients are the same for all terms of degree 2 or higher:

G(x) F (x) + 12 (− x − 1) :

With this modi ed generating function, we have

G(x)2

=

F x 2

x

1]

F x + 1

[ x

1]2

 

( )

+ [

( ) 2

4

 

 

= − x − x2 + 41 [− x − 1]

P (x);

 

 

where we introduce the notation P for the second degree polynomial on the right-hand side of this equation. I.e.,

G(x)2 = P (x):

Di erentiating with respect to x, we obtain

2G(x)G0(x) = P 0(x):

Multiplying both sides by G(x),

2G(x)2G0(x) = P 0(x)G(x);

and thus

2P (x)G0(x) = P 0(x)G(x);

from which we now derive a recurrence for calculating Sn from Sn−1 and Sn−2, instead of needing all previous values.

We have

G(x)2 = P (x);

that is,

G(x)2 = − x − x2 + 14 [− x − 1]2 :

3I am grateful to my colleague Victor Miller for suggesting the method we use to do this.

230

APPENDIX B. S-EXPRESSIONS OF SIZE N

Expanding the square,

 

 

 

 

 

 

 

 

 

P (x) = − x − x2 +

1

h 2x2 + 2 x + 1i :

 

 

 

4

Collecting terms,

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

1

 

 

P (x) =

 

2

1 x2

 

x +

 

:

 

4

2

4

Di erentiating,

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P 0(x) =

 

2

2 x −

 

:

 

2

2

We have seen that

2P (x) X(n + 1)Sn+1xn = P 0(x) X Snxn;

where it is understood that the low order terms of the sums have been \modi ed." Substituting in P (x) and P 0(x), and multiplying through by 2, we obtain

h i h i

( 2 4)x2 2 x + 1 X(n + 1)Sn+1xn = ( 2 4)x − X Snxn:

I.e.,

P[( 2 4)(n − 1)Sn−1 2 nSn + (n + 1)Sn+1] xn

=P [( 2 4)Sn−1 − Sn] xn:

We have thus obtained the following remarkable recurrence for n 3:

h i

nSn = ( 2 4)(n − 3) Sn−2 + [2 (n − 1) ] Sn−1: (B.2)

If exact rather than asymptotic values of Sn are desired, this is an excellent technique for calculating them.

We now derive Sn ( + 2)2Sn−1 from this recurrence. For n 4 we have, since we know that Sn−1 is greater than or equal to Sn−2,

h i h i

Sn ( 2 + 4) + (2 + ) Sn−1 ( + 2)2 Sn−1:

In the special case that = 0, one of the terms of recurrence (B.2) drops out, and we have

n − 3

Sn = 4 n Sn−2:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

231

From this it can be shown by induction that

 

 

 

 

 

 

 

 

 

 

 

 

S2n

= 2 2n 1

 

n ! =

2 2n 1 n!n!

;

 

 

 

 

 

 

 

 

 

 

1

 

 

1

2n

1

 

 

1 (2n)!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

which with Stirling's formula [see Feller (1970)]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n! p

 

 

 

 

nn+ 21 e−n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

yields the asymptotic estimate we used before. For

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p

 

 

(2n)2n+ 21 e2n

 

1 22n

 

1 1

 

 

(2n)!

 

1

 

 

2

 

 

S2n =

 

 

 

 

 

 

 

 

 

 

 

 

=

 

 

p

 

:

 

2

2n − 1

n!n!

 

4n

hp

 

nn+ 21 e−ni2

4n

 

 

 

 

 

 

 

 

 

n

 

 

 

 

 

2

 

For large n recurrence (B.2) is essentially

 

 

 

 

 

 

 

( 2 4)Sn−2 2 Sn−1 + Sn = 0

 

(n very large):

(B.3)

Recurrences such as (B.3) are well known. See, for example, the discussion of \Recurring series," and \Solution of di erence equations," Exercises 15{16, Chapter VIII, pp. 392{393, Hardy (1952). The limiting ratio Sn=Sn−1 ! must satisfy the following equation:

( 2 4) 2 x + x2 = 0:

This quadratic equation factors nicely:

(x − ( + 2)) (x − ( 2)) = 0:

Thus the two roots are:

1 = 2;2 = + 2:

The larger root 2 agrees with our previous asymptotic estimate for

Sn=Sn−1.

232

APPENDIX B. S-EXPRESSIONS OF SIZE N

Appendix C

Back Cover

G.J. Chaitin, the inventor of algorithmic information theory, presents in this book the strongest possible version of G¨odel's incompleteness theorem, using an information theoretic approach based on the size of computer programs.

An exponential diophantine equation is explicitly constructed with the property that certain assertions are independent mathematical facts, that is, irreducible mathematical information that cannot be compressed into any nite set of axioms.

This is the rst book on this subject and will be of interest to computer scientists, mathematicians, physicists and philosophers interested in the nature of randomness and in the limitations of the axiomatic method.

\Gregory Chaitin. . . has proved the ultimate in undecidability theorems. . . , that the logical structure of arithmetic can be random. . . The assumption that the formal structure of arithmetic is precise and regular turns out to have been a time-bomb, and Chaitin has just pushed the detonator." Ian Stewart in Nature

\No one, but no one, is exploring to greater depths the amazing insights and theorems that flow from G¨odel's work on undecidability than Gregory Chaitin. His exciting discoveries and speculations invade such areas as logic, induction, simplicity, the

233

234

APPENDIX C. BACK COVER

philosophy of mathematics and science, randomness, proof theory, chaos, information theory, computer complexity, diophantine analysis, and even the origin and evolution of life. If you haven't yet encountered his brilliant, clear, creative, wide-ranging mind, this is the book to read and absorb." Martin Gardner