Chain rule for Kolmogorov complexity: Difference between revisions

Revision as of 12:38, 24 July 2014

Latest revision as of 04:41, 2 December 2024

(14 intermediate revisions by 13 users not shown)

Line 1:

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

{{inline}}

{{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 7 ⟶ 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>

\Rightarrow \log P(X,Y) = \log P(X) + \log P(Y|X)

</math>

Line 19 ⟶ 23:

</math>

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

holds for the prefix complexity ''KP'', where {{math|''x*''{{sup|&lowast;}}}} 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 [[Big-O notation|at most]] a logarithmic factor. The results implies that [[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==

Line 28 ⟶ 32:

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

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

or

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

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

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

* 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^{''K''(''x'',''y'')}'' elements (because there are at most 2^''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)''}}.

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.

==References==

Line 46 ⟶ 54:

| last = Li

| first = Ming

| coauthorsauthor2 = Vit&aacute;nyiVitányi, Paul

| title = An introduction to Kolmogorov complexity and its applications

| ___location = New York

Line 53 ⟶ 61:

| 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 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 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 }}

* {{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}}

[[Category:Algorithmic information theory|*]]