DALEY, ROBERT P.

Daley, Robert P.

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A note on a result of Kamae

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Daley, R.P.
Osaka J. Math.
12 (1975), 283-284
A NOTE ON A RESULT OF KAMAE
ROBERT P. DALEY
(Received August 12, 1974)
In [3] Kamae proved that for each constant c, there exists a finite string y
such that
K(y)-K(y\x)>c
for all but finitely many finite strings x, where K( ) and K( \ •) are the uncon-
ditional and conditional minimal-program complexity measures respectively of
Kolmogorov [4], By considering infinite sequences we are able to obtain a
slightly stronger statement of this result.
Let X00 denote the set of all infinite binary strings. For x^X00 let xn denote
the initial segment of x of length n, i.e., x"=x(l) x(ri). To simplify matters
we will associate with each finite binary string y the integer n whose binary
representation is l y. By this means we will consider complexity expressions of
the form K(xn), K(xn\m) and K(m) for x^X°° and integers n and m. Then
K(x1t I n) is Kolmogorov's restricted conditional complexity. By a recursively
enumerable sequence we mean the characteristic sequence of a recursively
CO CO ^^
enumerable set. By "3n" and "V/z" we mean respectively "There exist in-
finitely many integers n" and "For all but finitely many integers n."
Theorem. There exists a recursively enumerable sequence x, such that
\fc3n\fm. K(xn\n)~K(xn\m)>c.
Proof. We need the following two lemmas.
Lemma. For every recursively enumerable sequence x}
BCiVnVm. K(xn\m)<K(n)+c1.
Proof. Let h be a total recursive function which enumerates the Γs of
x, i.e., #(*')= 1^ 3> h(j)=i. Define the total recursive function/(τz, m) as follows:
Step 1: Enumerate via h the first m Γs of the sequence x, i.e. compute h(l),-••,
h(nί).
Step 2: Output the following finite stringy of length n— y(i)=l^i appears on
* This work was supported by NSF Grant GJ-43223.
284 R.P. DALEY
the list generated in Step 1, i.e. 3j<m. h(j)=i.
Using/ and the asymptotically optimal algorithm for K( \ •) the result follows.
Lemma. There exists a recursively enumerable sequence x such that Vn.
K(x2"\2n)>n-l.
Proof. Let A by the asymptotically optimal algorithm for K( | ) and
define the sequence x as follows: For each n and each m such that 1 <m<2n~l,
x(2n~l+m)=l^mth digit of A(m, 2n) is 0. Clearly x is recursively enumerable
and x2n^pA(y, 2n) for all programs j; of length<#— 1.
Combining these two lemmas we have that there exists a recursively enu-
merable sequence x such that
f. K(x2"\2n)-K(xzn m)>n-\-c
Λ
-K(2n) .
While it is true that K(2n) for most integers of the form 2n is about n, there are
infinitely many such integers which have descriptions of arbitrarily (in an
CO
effective sense) short length relative to n, i.e. Vc"3n. n— K(2n)>c. (see [1])
Combining these two inequalities yields the result. Q.E.D.
Recently, Chaitin (see [2]) has shown that a sequence x is recursive
<=^3^Vw. jK(#w) <£"(#)-(-£. Combining this result with the first lemma above
we have immediately
Corollary. For every non-recursive recursively enumerable sequence x,
VcBnVm. K(xn)-K(x"\m)>c.
However, the above theorem is the best result obtainable for any recursively
enumerable sequence in view of the following result which is proved in [1].
Theorem. For every recursively enumerable sequence x
3c3n. K(xn\n)<c.
THE UNIVERSITY OF CHICAGO
References
[1] R. Daley: Non-complex sequences-characterizations and examples, Proc. 15th
Annual IEEE Symposium on Switching and Automata Theory, 1974.
[2] G. Chaitin: information theoretic characterizations of recursive infinite sequences, to
appear.
[3] T. Kamae: On Kolmogorov's complexity and information, Osaka J. Math. 10
(1973), 305-307.
[4] A. Kolmogorov: Three approaches for defining the concept of information quantity,
Problemy Peredaci informacii 1 (1965), 3-11.


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