Chain rule for Kolmogorov complexity: Difference between revisions

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{{Short description|Lower bound for size of software program}}

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

{{inline|date=July 2014}}

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

:<math>

Line 6 ⟶ 8:

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 entropy|conditional]] and [[joint entropy]], and the fact from [[probability theory]] that the [[joint probability]] is the product of the [[marginal probability|marginal]] and [[conditional probability]]:

:<math>

P(X,Y) = P(X) P(Y|X) \,

</math>

:<math>

K(x) <\Rightarrow \log P(K(xX,y)Y) += \log(K(x,y) + KP(xX) + \log P(K(x,y)Y|X)

</math>

The equivalent statement for Kolmogorov complexity does not hold exactly; it is only true only up to a [[logarithm]]ic factorterm:

:<math>

K(Xx,Yy) = K(Xx) + K(Yy|Xx) + O(\log(K(Xx,Yy)))

</math>

(An exact version, {{math|1=''KP''(''Xx'',&nbsp; ''Yy'')&nbsp; =&nbsp; ''KP''(''Xx'')&nbsp; +&nbsp; ''KP''(''Yy''|{{!}}''Xx''*){{sup|&nbsplowast;}}) +&nbsp; ''O''(1)}},

holds for the prefix complexity ''KP'', where {{math|''X*x''{{sup|&lowast;}}}} is a shortest program for ''Xx''.)

It states that the shortest program to reproduceprinting ''X'' and ''Y'' is usingobtained by concatenating a programshortest toprogram reproduceprinting ''X'' andwith a program to reproduceprinting ''Y'' given ''X'', plus [[Big-O notation|at most]] a logarithmic factor. UsingThe thisresults statementimplies onethat can define [[Mutual information#Absolute mutual information|algorithmic mutual information]], an analogue of mutual information for Kolmogorov complexity]] is symmetric: {{tmath|1=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 {{math|1=''y''|{{!}}''x'' (log(''K''(''x'',&nbsp; ''y''))}} upper-bounds this length).

For the ≥ direction, it suffices to show that for all {{mvar|k,l}} such that {{tmath|1=k+l = K(x,y)}} we have that either

The ≥ direction is rather more difficult. The key to the proof is the

construction of the set <math>A = \{(u,z) : K(u,z) \le K(x,y)\}</math>; that is, the

construction of the set of all pairs <math>(u,z)</math> such that the shortest input (for

a [[universal Turing machine]]) that produces <math>(u,z)</math> (and some way to distinguish

<math>u</math> from <math>z</math>) is shorter than the shortest producing <math>(x,y)</math>. We will also

need <math>A_x = \{z: K(x,z) \le K(x,y)\}</math>, the subset of <math>A</math> where <math>x = u</math>.

[[recursively enumerable|Enumerating]] <math>A</math> is not hard (although not fast!). In parallel, simulate all

<math>2^{K(x,y)}</math> programs with length <math>\le K(x,y)</math>. Output each <math>u,z</math> as the

simulated program terminates; eventually, any particular such <math>u,z</math> will be

produced. We can restrict to ''A''_''x'' simply by ignoring any output where <math>u \ne x</math>. We will assume that the inequality does not hold and derive a contradiction by finding a short description for ''K''(''x'') in terms of its index in the enumeration of a subset of&nbsp;''A''.

:<math> K(yx|x) > K(xk,yl) -\le K(x)k + O(\log(K(x,y1))) \, </math>

In particular, we can describe ''y'' by giving the index in which ''y'' is output when enumerating ''A''_''x'' (which is less than or equal to |''A''_''x''|) along with ''x'' and the value of ''K''(''x'',&nbsp;''y''). Thus,

or

:<math> K(y|x) \le \log(|A_x|) + O(\log(K(x,y))) + O(1) \, </math>

:<math> K(y|x,k,l) \le \log(|A_x|) + O(\log(K(x,y)))l + O(1) \, </math>.

where the O(1) term is from the overhead of the enumerator, which does not depend on ''x'' or&nbsp;''y''.

Consider the list (''a''₁,''b''₁), (''a''₂,''b''₂), ..., (''a_e,b_e'') of all pairs {{tmath|1=(a,b)}} produced by programs of length exactly {{tmath|1=K(x,y)}} [hence {{tmath|K(a,b) \le K(x,y)}}]. Note that this list

Assume by way of contradiction

* contains the pair {{tmath|1=(x,y)}},

* can be [[recursively enumerable|enumerated]] given {{mvar|k}} and {{mvar|l}} (by running all programs of length {{tmath|1=K(x,y)}} in parallel),

* has at most 2<mathsup>2^{''K''(''x'',''y'')}</mathsup> elements, since(because there are onlyat thatmost many2^''n'' programs of length {{mvar|n}}).

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

:<math> K(y|x) > K(x,y) - K(x) + O(\log(K(x,y))) \, </math>

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

for some <math>x,y</math>.

Combining this with the above, we have

:<math>

\log(|A_x|) + O(\log(K(x,y))) + O(1) \ge K(y|x) > K(x,y) - K(x) + O(\log(K(x,y))) \,

</math>

So for any constant ''c'' and some&nbsp;''x'',&nbsp;''y'',

:<math> \log(|A_x|) \ge K(x,y) - K(x) + c \log(K(x,y)) \, </math>

Let

: <math>e = K(x,y) - K(x) + c \log K(x,y);\quad |A_x| \ge 2^e.</math>

Now, we can enumerate a set of possible values for <math>x</math> by finding all <math>u</math> such

that <math>|A_u| > 2^e</math> -- list only the <math>u</math> such that <math>y</math> is output after more than

<math>2^e</math> other values when enumerating <math>A_u = \{ z: K(u,z) \le K(x,y) \}</math>. <math>x</math> is

one such <math>u</math>. Let <math>U</math> be the set of all such <math>u</math>. We can enumerate <math>U</math> by

enumerating each <math>A_u</math> (in parallel, as usual) and outputting a <math>u</math> only when

<math>y</math> would be output for <math>A_u</math> and more than <math>2^e</math> other values for the <math>A_u</math>

would be output. Note that given <math>K(x,y)</math>, <math>e</math> and the index of <math>x</math> in <math>U</math> we

can enumerate <math>U</math> to recover <math>x</math>.

Note that <math>\{(u,z) : u \in U, z \in A_u\}</math> is a subset of <math>A</math>, and <math>A</math> has at

most <math>2^{K(x,y)}</math> elements, since there are only that many programs of length

<math>K(x,y)</math>. So we have:

:<math> |U| < \frac{|A|}{2^e} \le \frac{2^{K(x,y)}}{2^e} </math>

where the first inequality follows since each element of <math>U</math> corresponds to an

<math>A_u</math> with strictly more than <math>2^e</math> elements.

But this leads to a contradiction:

:<math>

K(x) < \log(K(x,y)) + \log(e) + \log(|U|) \le \log(K(x,y)) + \log(e) + K(x,y) - e

</math>

which when we substitute in <math>e</math> gives

:<math>

K(x) < \log(K(x,y)) + \log(K(x,y) + K(x) + \log(K(x,y))

+ K(x,y) - K(x,y) + K(x) - c \log K(x,y)

</math>

which for large enough <math>c</math> gives <math>K(x) < K(x)</math>.

==References==

* {{cite book

| last = Li

| first = Ming

| coauthorsauthor2 = Vit&aacute;nyiVitányi, Paul

| title = An introduction to Kolmogorov complexity and its applications

| ___location = New York

| publisher = [[Springer-Verlag]]

| date =February 1997

| month = February

| isbn = 0-387-94868-6 }}

* {{cite journal | last=Kolmogorov | first=A. | title=Logical basis for information theory and probability theory | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=14 | issue=5 | year=1968 | issn=0018-9448 | doi=10.1109/tit.1968.1054210 | pages=662–664| s2cid=11402549 }}

* {{cite journal | last1=Zvonkin | first1=A K | last2=Levin | first2=L A | 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 | publisher=IOP Publishing | volume=25 | issue=6 | date=1970-12-31 | issn=0036-0279 | doi=10.1070/rm1970v025n06abeh001269 | pages=83–124| bibcode=1970RuMaS..25...83Z | s2cid=250850390 }}

[[Category:Algorithmic information theory|*]]