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Chain rule for Kolmogorov complexity

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The chain rule[citation needed] for Kolmogorov complexity is an analogue of the chain rule for information entropy, which states:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle H(X,Y) = H(X) + H(Y|X) }

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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P(X,Y) = P(X) P(Y|X) }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle \Rightarrow \log P(X,Y) = \log P(X) + \log P(Y|X) }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle K(x,y) = K(x) + K(y|x) + O(\log(K(x,y))) }

(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: ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle I(x:y) = I(y:x) + O(\log K(x,y))} ⁠ for all x,y.

Proof

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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 ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle k+l = K(x,y)} ⁠ we have that either

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle K(x|k,l) \le k + O(1)}

or

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle K(y|x,k,l) \le l + O(1)} .

Consider the list (a1,b1), (a2,b2), ..., (ae,be) of all pairs ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (a,b)} ⁠ produced by programs of length exactly ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle K(x,y)} ⁠ [hence ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle K(a,b) \le K(x,y)} ⁠]. Note that this list

  • contains the pair ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (x,y)} ⁠,
  • can be enumerated given k and l (by running all programs of length ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle K(x,y)} ⁠ in parallel),
  • has at most 2K(x,y) elements (because there are at most 2n programs of length n).

First, suppose that x appears less than 2l times as first element. We can specify y given x,k,l by enumerating (a1,b1), (a2,b2), ... and then selecting ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (x,y)} ⁠ in the sub-list of pairs ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (x,b)} ⁠. By assumption, the index of ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (x,y)} ⁠ in this sub-list is less than 2l and hence, there is a program for y given x,k,l of length ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle l + O(1)} ⁠. Now, suppose that x appears at least 2l times as first element. This can happen for at most 2K(x,y)−l = 2k 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 ⁠Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle k + O(1)} ⁠. Theorem proved.

References

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  • Li, Ming; Vitányi, Paul (February 1997). An introduction to Kolmogorov complexity and its applications. New York: Springer-Verlag. ISBN 0-387-94868-6.
  • Kolmogorov, A. (1968). "Logical basis for information theory and probability theory". IEEE Transactions on Information Theory. 14 (5). Institute of Electrical and Electronics Engineers (IEEE): 662–664. doi:10.1109/tit.1968.1054210. ISSN 0018-9448. S2CID 11402549.
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