Additivity of on-line decision complexity is violated by a linear term in the length of a binary string

On-line decision complexity has been introduced and investigated in [?, ?]. It also naturally appears in the definition of ideal influence tests [?, ?]. A natural question is whether algorithmic mutual information of two time series x, y, can be decomposed into an information flow going from x to y, a flow going from y to x, and an information flow instantaneously present in both strings. It turns out [?] that this question is related to the question of defining a decomposition of K(x, y) with l(x) = l(y) as the sum of the complexity of predicting x i+1 given x 1 ...x i and y 1 ...y i , i n, and the complexity of predicting y i+1 given x 1 ...x i+1 and y 1 ...y i . It will be shown that using on-line decision complexity for this complexity, this sum exceeds K(x, y) by a linear constant in l(x). A modification of this definition of on-line decision complexity will be shown to have an approximate decomposition [?, ?].

Non-additivity of decision complexity was also shown in [?], in the context of randomness defined by supermartingales. Using natural definitions for randomness a paradox is shown: if the even bits of z given the past uneven bits of z are random, and also the uneven bits of z given the past even bits of z are random, than it is possible that z is not random. The proof of this result implies that additivity of on-line decision complexity is violated by a logarithmic term.

For excellent introductions to Kolmogorov complexity we refer to [?, ?].

Let ω, ω <ω , 2 N and 2 <ω denote the set of the Natural numbers, the set of finite sequences of Natural numbers, the binary strings of length N, and the binary strings of finite length. Other definitions are analogue. Let ǫ denote the empty sequence. Remark that there is a natural bijection between ω and 2 <ω , defined by:

[ω] is the set of nested sequences of Natural numbers, with finite depth. Mathematically, it is the closure of ω under the mapping f (S) = S <ω . Remark that there is a computable bijection between ω and [ω], therefore most complexity and computability results in ω also hold in [ω].

An interpreter Φ is a partial computable function from 2 <ω × [ω] → [ω]. An interpreter is prefix-free if for any x, the set D x of all p where Φ(p|x) is defined, is prefix-free. Let Φ be some fixed optimal universal prefix-free interpreter.

For any x ∈ 2 <ω , l(x) denotes the length of x. For any x ∈ ω <ω , l(x) corresponds to the length of some prefix-free encoding of x on a binary tape:

For x, y ∈ [ω], the Kolmogorov complexity K(x|y), is defined as:

The Kolmogorov complexity of elements in 2 <ω is defined by using the computable bijection mentioned in the beginning of this section.

The on-line decision complexity is defined by:

This definition differs slightly with the definition of [?], with respect that A ∈ ω n is chosen, in stead of A ∈ 2 n . Also a shorter notation [?] will be used:

3 Main result and proof tactic Proposition 3.1.

In [?] and repeated in [?, ?], it is proven that for any n there is an x ∈ 2 n such that:

Let y be the binary expansion of K(x). From this and equation ( 4) it can be shown that

By inserting zeros at the right places in x, y, it can be shown that there exists infinity many x, y with l(x) = l(y):

This shows proposition 3.1 for a logarithmic term in l(x). It seems natural to think that such a result can be improved to a linear term, by concatenating such strings. This is what eventually will happen in the proof, at equation ( 12). However, to be able to add up these differences, conditional complexities must add up in some way to on-line decision complexity, in what extend this is possible is still an open problem. Happily, Lemma 4.4 can circumvent this, if some extra information is available. This information is stored in sequences u and v and is added to x and y. Adding this information requires, some more bounds to make the proof work: (10), (11). The proof below provides all technical details.

First some definitions and lemmas are given. f (x) + g(x) is short for f (x) g(x) + O(1), and (1) Lemmas 4.1, 4.2, and 4.3 provide observations, known within the community, and stated here explicitly for later reference.

Lemma 4.1.

Proof. For U, V ∈ ω, prefix-free complexity satisfies additivity [?]:

Since there is a computable bijection between ω and [ω], this result also applies to

Inductive application of both equations above on A i proves the lemma.

Lemma 4.2. For a, b ∈ ω and c ∈ ω <ω :

Proof. The proof below, shows the unconditioned version of the lemma, since the proof of the conditioned version is the same. In [?] and exercise 3.3.7 in [?] it is stated that for every w ∈ ω, and n K(w):

and

Therefore, for c constant, there are an O(1) number of programs that compute a, b and have length K(a, b) + c. Let S be the set of these programs. Remark that the elements of S can be enumerated given a, b, K(a, b) and therefore, for any p ∈ S, using (4), we have:

By equation ( 2), we have:

Combining with equation (

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