# Information Distance in Multiples Paul M. B. Vitányi

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2 2452 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 57, NO 4, APRIL 2011 information about all the others [all elements of ], the right-h side (RHS) is interpreted as the program length of the most specialized object that is similar to all the others The paper [27] develops the stated results applications It does not develop the theory in any detail That is the purpose of the present paper B Results Information distance for multiples, that is, finite lists, appears both practically theoretically promising In nearly all cases below the results imply the corresponding ones for the pairwise information distance defined as follows The information distance in [3] between strings is In the current paper These two definitions coincide for since up to an additive constant term We investigate the maximal overlap of information (Theorem 31) which for specializes to Theorem 34 in [3], Corollary 32 shows (I1) Corollary 33 shows that the LHS of (I2) can be taken to correspond to a single program embodying the most comprehensive object that contains the most information about all the others as stated but not argued or proved in [27]; metricity (Theorem 41) universality (Theorem 52) which for (for metricity) (for universality) specialize to Theorem 42 in [3]; additivity (Theorem 61); minimum overlap of information (Theorem 71) which for specializes to [29, Theorem 837] the nonmetricity of normalized information distance for lists of more than two elements certain proposals of the normalizing factor (Section VIII) In contrast, for lists of two elements we can normalize the information distance as in Lemma V4 Theorem V7 of [26] The definitions are of necessity new as are the proof ideas Remarkably, the new notation proofs for the general case are simpler than the mentioned existing proofs for the particular case of pairwise information distance II PRELIMINARIES Kolmogorov Complexity: This is the information in a single object [21] The notion has been the subject of a plethora of papers Informally, the Kolmogorov complexity of a finite binary string is the length of the shortest string from which the original can be losslessly reconstructed by an effective general-purpose computer such as a particular universal Turing machine Hence it constitutes a lower bound on how far a lossless compression program can compress For technical reasons we choose Turing machines with a separate read-only input tape, that is scanned from left to right without backing up, a separate work tape on which the computation takes place a separate output tape Upon halting, the initial segment of the input that has been scanned is called the input program the contents of the output tape is called the output By construction, the set of halting programs is prefix free We call the reference universal prefix Turing machine This leads to the definition of prefix Kolmogorov complexity which we shall designate simply as Kolmogorov complexity Formally, the conditional Kolmogorov complexity is the length of the shortest input such that the reference universal prefix Turing machine on input with auxiliary information outputs The unconditional Kolmogorov complexity is defined by where is the empty string (of length 0) In these definitions both can consist of a nonempty finite lists of finite binary strings For more details theorems that are used in the present work see Appendix I Lists: A list is a multiple of finite binary strings in length-increasing lexicographic order If is a list, then some or all of its elements may be equal Thus, a list is not a set but an ordered bag of elements With some abuse of the common set-membership notation we write for every to mean that is an element of list The conditional prefix Kolmogorov complexity of a list given an element is the length of a shortest program for the reference universal Turing machine that with input outputs the list The prefix Kolmogorov complexity of a list is defined by One can also put lists in the conditional such as or We will use the straightforward laws up to an additive constant term, for equals the list with the element deleted Information Distance: To obtain the pairwise information distance in [3] we take in (I1) Then(I1) is equivalent to III MAXIMAL OVERLAP We use the notation terminology of Section I-A Define, We prove a maximal overlap theorem: the information needed to go from any to any in can be divided in two parts: a single string of length a string of length (possibly depending on ), everything up to an additive logarithmic term Theorem 31: A single program of length bits concatenated with a string of bits, possibly depending on, suffice to find from for every To find an arbitrary element from it suffices to concatenate at most another bits, possibly depending on Proof: Enumerate the finite binary strings lexicographic length-increasing as Let be a graph defined as follows Let be the set of finite binary strings the set of vectors of strings in defined by such that Given the set can be enumerated Define Define by length-increasing lexicographic enumerating put with if for some, where is chosen as follows It is the th string of length where is the number of times we have used So the first times we choose an edge we use, the next we use so on In this way, so that By adding to we take care that the degree of is at most not at most as it could be without the prefix The degree of a node is trivially

6 2456 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 57, NO 4, APRIL 2011 ditive constant term with the logarithm of also with the logarithm of This result is called the Coding Theorem since it shows that the shortest upper semicomputable code is a Shannon-Fano code of the greatest lower semicomputable probability mass function REFERENCES [1] C Ané M Serson, Missing the forest for the trees: Phylogenetic compression its implications for inferring complex evolutionary histories, Systemat Biol, vol 54, no 1, pp , 2005 [2] D Benedetto, E Caglioti, V Loreto, Language trees zipping, Phys Rev Lett, vol 88, no 4, p , 2002 [3] C H Bennett, P Gács, M Li, P M B Vitányi, W Zurek, Information distance, IEEE Trans Inf Theory, vol 44, pp , 1998 [4] M Cebrián, M Alfonseca, A Ortega, The normalized compression distance is resistant to noise, IEEE Trans Inf Theory, vol 53, pp , 2007 [5] X Chen, B Francia, M Li, B Mckinnon, A Seker, Shared information program plagiarism detection, IEEE Trans Inf Theory, 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York: Idea Group Inc, 2008, ch IX, pp [42] X Zhang, Y Hao, X Zhu, M Li, Information distance from a question to an answer, in Proc 13th ACM SIGKDD Int Conf Knowledge Discovery Data Mining, 2007, pp [43] J Zhou, S Wang, C Cao, A Google-based statistical acquisition model of Chinese lexical concepts, in Proc 2nd Conf Knowl Sci, Eng Manage, Lect Notes Comp Sci, 2007, vol 4798, pp [44] A K Zvonkin L A Levin, The complexity of finite objects the development of the concepts of information romness by means of the theory of algorithms, Russian Math Surveys, vol 25, no 6, pp , 1970 Paul M B Vitányi received the PhD degree from the Free University of Amsterdam in 1978 He is a CWI Fellow with the National Research Institute for Mathematics Computer Science, Netherls, CWI, Professor of Computer Science with the University of Amsterdam He has worked on cellular automata, computational complexity, distributed parallel computing, machine learning prediction, physics of computation, Kolmogorov complexity, information theory, quantum computing, publishing about 200 research papers some books Together with M Li, they pioneered applications of Kolmogorov complexity coauthored An Introduction to Kolmogorov Complexity its Applications (New York: Springer-Verlag, 1993), (3rd ed 2008) parts of which have been translated into Chinese, Russian, Japanese Web page: ~paulv/ Dr Vitányi serves on the editorial boards of Distributed Computing ( ), Information Processing Letters, Theory of Computing Systems, Parallel Processing Letters, International Journal of Foundations of Computer Science, Entropy, Information, Journal of Computer Systems Sciences (guest editor), elsewhere He received a Knighthood (Ridder in de Orde van de Nederlse Leeuw) in 2007

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