“Chain rule for Kolmogorov complexity”的意思、由来-开放百科全书

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Chain rule for Kolmogorov complexity

释义

Proof
References

The chain rule{{cn|date=July 2014}} for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states:

That is, the combined randomness of two sequences X and Y is the sum of the randomness of X plus whatever randomness is left in Y once we know X.

This follows immediately from the definitions of conditional and joint entropy, and the fact from probability theory that the joint probability is the product of the marginal and conditional probability:

The equivalent statement for Kolmogorov complexity does not hold exactly; it is true only up to a logarithmic term:

(An exact version, KP(x, y) = KP(x) + KP(y|x*) + O(1),

holds for the prefix complexity KP, where x* is a shortest program for x.)

It states that the shortest program printing X and Y is obtained by concatenating a shortest program printing X with a program printing Y given X, plus at most a logarithmic factor. The results implies that algorithmic mutual information, an analogue of mutual information for Kolmogorov complexity is symmetric: I(x:y) = I(y:x) + O(log K(x,y)) for all x,y.

Proof

The ≤ direction is obvious: we can write a program to produce x and y by concatenating a program to produce x, a program to produce y given

access to x, and (whence the log term) the length of one of the programs, so

that we know where to separate the two programs for x and y|x (log(K(x, y)) upper-bounds this length).

For the ≥ direction, it suffices to show that for all k,l such that k+l = K(x,y) we have that either

Consider the list (a₁,b₁), (a₂,b₂), ..., (a_e,b_e) of all pairs (a,b) produced by programs of length exactly K(x,y) [hence K(a,b) ≤ K(x,y)]. Note that this list

contains the pair (x,y),
can be enumerated given k and l (by running all programs of length K(x,y) in parallel),
has at most 2^K(x,y) elements (because there are at most 2ⁿ programs of length n).

First, suppose that x appears less than 2^l times as first element. We can specify y given x,k,l by enumerating (a₁,b₁), (a₂,b₂), ... and then selecting (x,y) in the sub-list of pairs (x,b). By assumption, the index of (x,y) in this sub-list is less than 2^l and hence, there is a program for y given x,k,l of length l + O(1).

Now, suppose that x appears at least 2^l times as first element. This can happen for at most 2^K(x,y)-l = 2^k different strings. These strings can be enumerated given k,l and hence x can be specified by its index in this enumeration. The corresponding program for x has size k + O(1). Theorem proved.

References

{{cite book

| last = Li
| first = Ming
| author2 = Vitányi, Paul
| title = An introduction to Kolmogorov complexity and its applications
| location = New York
| publisher = Springer-Verlag
|date=February 1997
| isbn = 0-387-94868-6 }}

{{cite news

| last = Kolmogorov
| first = A.N.
| title = Logical basis for information theory and probability theory
| journal = IEEE Transactions on Information Theory
| number = 5
| volume = 14
| pages = 662–664
| year = 1968 }}

{{cite news

| first = A.
| last = Zvonkin
|author2=L. Levin
| title = The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms.
| journal = Russian Mathematical Surveys
| volume = 25
| number = 6
| pages = 83–124
| year = 1970}}

5 : Algorithmic information theory|Information theory|Computability theory|Theory of computation|Articles containing proofs

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