DOCSLIB.ORG

Combinatorial Reasoning in Information Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Combinatorial Reasoning in Information Theory Combinatorial Reasoning in Information Theory Noga Alon, Tel Aviv U. ISIT 2009 1 Combinatorial Reasoning is crucial in Information Theory Google lists 245,000 sites with the words “Information Theory ” and “ Combinatorics ” 2 The Shannon Capacity of Graphs The (and) product G x H of two graphs G=(V,E) and H=(V’,E’) is the graph on V x V’, where (v,v’) = (u,u’) are adjacent iff (u=v or uv є E) and (u’=v’ or u’v’ є E’) The n-th power Gn of G is the product of n copies of G. 3 Shannon Capacity Let α ( G n ) denote the independence number of G n. The Shannon capacity of G is n 1/n n 1/n c(G) = lim [α(G )] ( = supn[α(G )] ) n →∞ 4 Motivation output input A channel has an input set X, an output set Y, and a fan-out set S Y for each x X. x ⊂ ∈ The graph of the channel is G=(X,E), where xx’ є E iff x,x’ can be confused , that is, iff S S = x ∩ x′ ∅ 5 α(G) = the maximum number of distinct messages the channel can communicate in a single use (with no errors) α(Gn)= the maximum number of distinct messages the channel can communicate in n uses. c(G) = the maximum number of messages per use the channel can communicate (with long messages) 6 There are several upper bounds for the Shannon Capacity: Combinatorial [ Shannon(56)] Geometric [ Lovász(79), Schrijver (80)] Algebraic [Haemers(79), A (98)] 7 Theorem (A-98): For every k there are graphs G and H so that c(G), c(H) ≤ k and yet c(G + H) kΩ(log k/ log log k) ≥ where G+H is the disjoint union of G and H. This answers a problem of Shannon. abcd… …a b א ב c אבגד… 8 Multiple channels and privileged users (A+Lubetzky 07 ): For any fixed t and any family of subsets of 1, 2, . , t , there are graphs G ,G ,...,GF so that { } 1 2 t c(i I Gi) is “large” if I contains a member of , and i∈s “small” otherwise. F For example, the capacity can be large iff I is of size at least k. 9 Not much is known on the Shannon capacity of graphs: What is the maximum possible capacity of a disjoint union of two graphs, each having capacity at most k ? Is the maximum possible capacity of a graph G with independence number 2 bounded by an absolute constant ? Is the problem of deciding if the Shannon capacity of a given input graph is at least a given real x decidable ? What is the expected value of c(G(n,½)) ? 10 Broadcasting with side information A sender holds a word x=x 1x2…x n of n blocks x i, each consisting of t bits, which he wants to broadcast to m receivers . Each receiver Ri is interested in x f(i) and has a prior side information consisting of some other blocks x j . βt=minimum number of bits that have to be transmitted to allow each R i to recover x f(i) β β β = lim t ( = inf t ). t →∞ t t 11 Motivation [Birk and Kol (98)] [Bar-Yossef,Birk,Jayram and Kol (06)]: Applications such as Video on Demand 12 The representing directed hypergraph: Vertices =[n]={1,2, …, n} (=indices of blocks) Edges: for each receiver R j there is a directed edge (f(j),N(j)), where N(j) is the set of indices of all blocks known to R i. Thus βt=βt(H), βs+t (H) ≤βs(H)+ βt(H), β(H)=inf βt(H)/t In words: β is the average asymptotic number of encoding bits needed per bit in each input block. 13 Example: H=([5],{ (1,2),(2,1),(3,{4,5}),(4,{3,5}),(5,{3,4}) } 5 1 2 4 3 βt(H) ≥ 2t as receivers R 1 , R 3 have to get x 1 , x 3 βt(H) ≤ 2t : it is enough to transmit x x , x x x . 1 ⊕ 2 3 ⊕ 4 ⊕ 5 14 * A related parameter: βt = β1(t H) , where t H is the disjoint union of t copies of H. * In words: βt (H) is the minimum number of bits required if the network topology is replicated t independent times. β∗(H) = limt βt∗(H)/t = inftβt∗(H)/t. →∞ Facts: α(H) β(H) β∗(H) β (H) ≤ ≤ ≤ 1 α(H) = max S : S V, v S e = (v, J) E, J S = . {| | ⊂ ∀ ∈ ∃ ∈ ∩ ∅} β (H) 1 ≤ min r : V = V V e = (v, J) E i, v V ,V v J { 1 ∪ · · · ∪ r ∀ ∈ ∃ ∈ i i − ⊂ } 15 Computing β*(H) Given H=([n],E) and t=1, two input strings x, y 0, 1 n ∈ { } are confusable if there exists a receiver (=directed edge) (i,J) in E such that x =y for all j in J, and yet x = y . j j i i Let λ (= λn) denote the maximum cardinality of a set of input strings that are pairwise non-confusable. Theorem 1 [A, Lubetzky, Stav(08)]: β∗(H) = limt β1(tH)/t = n log2 λ. →∞ − 16 Corollary [ALS]: For H = C = (Z ,E) where E = (i, i 1, i + 1 ): i Z , 5 5 { { − } ∈ 5} β (H) = 3, β∗(H) = 5 log 5 = 2.68.. 1 − 2 This implies that in Network Coding (introduced by Ahlswede,Cai, Li and Yeung ), nonlinear capacity may exceed linear capacity [Earlier examples given by Dougherty, Freiling and Zeger (05), and by Chan and Grant (07). ] 17 Theorem 2 [A, Weinstein (08)]: For every constant C there is an (explicit) hypergraph H for which * β (H)<3 and β1(H) > C. 1001..0 0001..1 0011..0 1100..0 (r) (r) (b) (b) (g) (g) (l) (l) R1 ,...,Rm R1 ,...,Rm R1 ,...,Rm R1 ,...,Rm 18 Theorem 3 [ A,Hassidim,Lubetzky,Stav,Weinsten (08)]: For every constant C there is an (explicit) hypergraph H for which β(H)=2 and yet β*(H)>C. 19 Codes for disjoint unions of hypergraphs Definition: The confusion graph C(H) of a directed hypergraph H=([n],E) describing a broadcast network is the undirected graph on {0,1}n, where x,y are adjacent iff for some e=(i,J) in E, x = y and yet x = y for all j J. i i j j ∈ Here each block is of length 1, the vertices are all possible input words, and two are adjacent iff they are confusable. Observation 1: The number of words in the optimal code for H is χ(C(H). That is, β (H) = log χ(C(H)) . 1 ⌈ 2 ⌉ 20 Definition: The OR graph product of G 1 and G 2 is the graph on V(G 1) x V(G 2), where (u,v) and (u’,v’) are adjacent if either uu E(G ) or vv E(G ) (or both). ′ ∈ 1 ′ ∈ 2 k LetG∨ denote the k-fold OR product of G. Observation 2: For any pair H 1 and H 2 of directed hypergraphs, the confusion graph of their disjoint union is the OR product of C(H 1) and C(H 2). Theorem [McEliece+Posner(71), Berge+Simonovits(74 )]: t 1/t For every graph G limt [χ(G∨ )] = χf (G) →∞ where χf (G) is the fractional chromatic number of G. 21 For a directed hypergraph H=([n],E), G=C(H) is a Cayley Graph and thus χf (G) = V (G) /α(G). n | | Hence χf (C(H)) = 2 /λ(H), where λ(H) is the maximum cardinality of a set of pairwise non-confusable words. Therefore n β∗(H) = log[2 /λ(H)] = n log λ(H) − proving Theorem 1. 22 Confusion graphs with χ much bigger than χf : An auxiliary graph: put n=2k, and let G be the Cayley Graph on (Z_2)k in which i,j are adjacent if the Hamming distance i j between i and j is at least k √k . | ⊕ | − 100 √k Fact 1:χ(G) > 100 and χf (G) < 2.05. Proof: The Kneser Graph k √k K(k, 2 200 ) is a subgraph of G, and hence by Lovász [78] the− chromatic number k √k √k is at least k 2( 2 200 ) + 2 > 100 . The fractional −chromatic− number is small as that’s a Cayley Graph and there is an independent set of size k 2k i<k/2 √k/200 i > 2.05 . 23 − Let H=([n],E) be the directed hypergraph on [n], where N=2k, and for each i,j satisfying √k i j k 100 (i,V-{i,j}),(j,V-{i,j}) are directed edges.| ⊕ | ≥ − Let C=C(H) be the corresponding confusion graph . This is the Cayley Graph on (Z_2)n whose generators are all vectors e and all e e with i j k √k . i i ⊕ j | ⊕ | ≥ − 100 24 √log n Fact 2: χ(C) χ(G) . ≥ ≥ 100 Proof: The induced subgraph of C on the vectors e i is G. Fact 3: χ (C) 2χ (G) < 4.1. f ≤ f Proof (sketch): Let I be a maximum independent set in k G. Then for all j Z2 and all ǫ 0, 1 , the set n ∈ ∈ { } Ij = u Z2 : u ǫ(mod 2), j i iui I is an independent{ ∈ set| of| ≡C, and its expected⊕ size∈ for} n random j, ǫ is 2 . 2χf (G) Thus, there is a directed hypergraph H on [n] so that √k β (H) log( ) = Ω(log log n), β∗(H) log 4.05 < 3.
Load more

Recommended publications
  • Information Theory Techniques for Multimedia Data Classification and Retrieval
    INFORMATION THEORY TECHNIQUES FOR MULTIMEDIA DATA CLASSIFICATION AND RETRIEVAL Marius Vila Duran Dipòsit legal: Gi. 1379-2015 http://hdl.handle.net/10803/302664 http://creativecommons.org/licenses/by-nc-sa/4.0/deed.ca Aquesta obra està subjecta a una llicència Creative Commons Reconeixement- NoComercial-CompartirIgual Esta obra está bajo una licencia Creative Commons Reconocimiento-NoComercial- CompartirIgual This work is licensed under a Creative Commons Attribution-NonCommercial- ShareAlike licence DOCTORAL THESIS Information theory techniques for multimedia data classification and retrieval Marius VILA DURAN 2015 DOCTORAL THESIS Information theory techniques for multimedia data classification and retrieval Author: Marius VILA DURAN 2015 Doctoral Programme in Technology Advisors: Dr. Miquel FEIXAS FEIXAS Dr. Mateu SBERT CASASAYAS This manuscript has been presented to opt for the doctoral degree from the University of Girona List of publications Publications that support the contents of this thesis: "Tsallis Mutual Information for Document Classification", Marius Vila, Anton • Bardera, Miquel Feixas, Mateu Sbert. Entropy, vol. 13, no. 9, pages 1694-1707, 2011. "Tsallis entropy-based information measure for shot boundary detection and • keyframe selection", Marius Vila, Anton Bardera, Qing Xu, Miquel Feixas, Mateu Sbert. Signal, Image and Video Processing, vol. 7, no. 3, pages 507-520, 2013. "Analysis of image informativeness measures", Marius Vila, Anton Bardera, • Miquel Feixas, Philippe Bekaert, Mateu Sbert. IEEE International Conference on Image Processing pages 1086-1090, October 2014. "Image-based Similarity Measures for Invoice Classification", Marius Vila, Anton • Bardera, Miquel Feixas, Mateu Sbert. Submitted. List of figures 2.1 Plot of binary entropy..............................7 2.2 Venn diagram of Shannon’s information measures............ 10 3.1 Computation of the normalized compression distance using an image compressor....................................
    [Show full text]
  • 2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (Chair), Bobby Kleinberg September 14Th, 2020
    2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (chair), Bobby Kleinberg September 14th, 2020 SIGACT Mission Statement: The primary mission of ACM SIGACT (Association for Computing Machinery Special Interest Group on Algorithms and Computation Theory) is to foster and promote the discovery and dissemination of high quality research in the domain of theoretical computer science. The field of theoretical computer science is the rigorous study of all computational phenomena - natural, artificial or man-made. This includes the diverse areas of algorithms, data structures, complexity theory, distributed computation, parallel computation, VLSI, machine learning, computational biology, computational geometry, information theory, cryptography, quantum computation, computational number theory and algebra, program semantics and verification, automata theory, and the study of randomness. Work in this field is often distinguished by its emphasis on mathematical technique and rigor. 1. Awards ▪ 2020 Gödel Prize: This was awarded to Robin A. Moser and Gábor Tardos for their paper “A constructive proof of the general Lovász Local Lemma”, Journal of the ACM, Vol 57 (2), 2010. The Lovász Local Lemma (LLL) is a fundamental tool of the probabilistic method. It enables one to show the existence of certain objects even though they occur with exponentially small probability. The original proof was not algorithmic, and subsequent algorithmic versions had significant losses in parameters. This paper provides a simple, powerful algorithmic paradigm that converts almost all known applications of the LLL into randomized algorithms matching the bounds of the existence proof. The paper further gives a derandomized algorithm, a parallel algorithm, and an extension to the “lopsided” LLL.
    [Show full text]
  • Information Theory for Intelligent People
    Information Theory for Intelligent People Simon DeDeo∗ September 9, 2018 Contents 1 Twenty Questions 1 2 Sidebar: Information on Ice 4 3 Encoding and Memory 4 4 Coarse-graining 5 5 Alternatives to Entropy? 7 6 Coding Failure, Cognitive Surprise, and Kullback-Leibler Divergence 7 7 Einstein and Cromwell's Rule 10 8 Mutual Information 10 9 Jensen-Shannon Distance 11 10 A Note on Measuring Information 12 11 Minds and Information 13 1 Twenty Questions The story of information theory begins with the children's game usually known as \twenty ques- tions". The first player (the \adult") in this two-player game thinks of something, and by a series of yes-no questions, the other player (the \child") attempts to guess what it is. \Is it bigger than a breadbox?" No. \Does it have fur?" Yes. \Is it a mammal?" No. And so forth. If you play this game for a while, you learn that some questions work better than others. Children usually learn that it's a good idea to eliminate general categories first before becoming ∗Article modified from text for the Santa Fe Institute Complex Systems Summer School 2012, and updated for a meeting of the Indiana University Center for 18th Century Studies in 2015, a Colloquium at Tufts University Department of Cognitive Science on Play and String Theory in 2016, and a meeting of the SFI ACtioN Network in 2017. Please send corrections, comments and feedback to [email protected]; http://santafe.edu/~simon. 1 more specific, for example. If you ask on the first round \is it a carburetor?" you are likely wasting time|unless you're playing the game on Car Talk.
    [Show full text]
  • An Information-Theoretic Approach to Normal Forms for Relational and XML Data
    An Information-Theoretic Approach to Normal Forms for Relational and XML Data Marcelo Arenas Leonid Libkin University of Toronto University of Toronto [email protected] [email protected] ABSTRACT Several papers [13, 28, 18] attempted a more formal eval- uation of normal forms, by relating it to the elimination Normalization as a way of producing good database de- of update anomalies. Another criterion is the existence signs is a well-understood topic. However, the same prob- of algorithms that produce good designs: for example, we lem of distinguishing well-designed databases from poorly know that every database scheme can be losslessly decom- designed ones arises in other data models, in particular, posed into one in BCNF, but some constraints may be lost XML. While in the relational world the criteria for be- along the way. ing well-designed are usually very intuitive and clear to state, they become more obscure when one moves to more The previous work was specific for the relational model. complex data models. As new data formats such as XML are becoming critically important, classical database theory problems have to be Our goal is to provide a set of tools for testing when a revisited in the new context [26, 24]. However, there is condition on a database design, specified by a normal as yet no consensus on how to address the problem of form, corresponds to a good design. We use techniques well-designed data in the XML setting [10, 3]. of information theory, and define a measure of informa- tion content of elements in a database with respect to a set It is problematic to evaluate XML normal forms based of constraints.
    [Show full text]
  • 1 Applications of Information Theory to Biology Information Theory Offers A
    Pacific Symposium on Biocomputing 5:597-598 (2000) Applications of Information Theory to Biology T. Gregory Dewey Keck Graduate Institute of Applied Life Sciences 535 Watson Drive, Claremont CA 91711, USA Hanspeter Herzel Innovationskolleg Theoretische Biologie, HU Berlin, Invalidenstrasse 43, D-10115, Berlin, Germany Information theory offers a number of fertile applications to biology. These applications range from statistical inference to foundational issues. There are a number of statistical analysis tools that can be considered information theoretical techniques. Such techniques have been the topic of PSB session tracks in 1998 and 1999. The two main tools are algorithmic complexity (including both MML and MDL) and maximum entropy. Applications of MML include sequence searching and alignment, phylogenetic trees, structural classes in proteins, protein potentials, calcium and neural spike dynamics and DNA structure. Maximum entropy methods have appeared in nucleotide sequence analysis (Cosmi et al., 1990), protein dynamics (Steinbach, 1996), peptide structure (Zal et al., 1996) and drug absorption (Charter, 1991). Foundational aspects of information theory are particularly relevant in several areas of molecular biology. Well-studied applications are the recognition of DNA binding sites (Schneider 1986), multiple alignment (Altschul 1991) or gene-finding using a linguistic approach (Dong & Searls 1994). Calcium oscillations and genetic networks have also been studied with information-theoretic tools. In all these cases, the information content of the system or phenomena is of intrinsic interest in its own right. In the present symposium, we see three applications of information theory that involve some aspect of sequence analysis. In all three cases, fundamental information-theoretical properties of the problem of interest are used to develop analyses of practical importance.
    [Show full text]
  • Efficient Space-Time Sampling with Pixel-Wise Coded Exposure For
    IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 1 Efficient Space-Time Sampling with Pixel-wise Coded Exposure for High Speed Imaging Dengyu Liu, Jinwei Gu, Yasunobu Hitomi, Mohit Gupta, Tomoo Mitsunaga, Shree K. Nayar Abstract—Cameras face a fundamental tradeoff between spatial and temporal resolution. Digital still cameras can capture images with high spatial resolution, but most high-speed video cameras have relatively low spatial resolution. It is hard to overcome this tradeoff without incurring a significant increase in hardware costs. In this paper, we propose techniques for sampling, representing and reconstructing the space-time volume in order to overcome this tradeoff. Our approach has two important distinctions compared to previous works: (1) we achieve sparse representation of videos by learning an over-complete dictionary on video patches, and (2) we adhere to practical hardware constraints on sampling schemes imposed by architectures of current image sensors, which means that our sampling function can be implemented on CMOS image sensors with modified control units in the future. We evaluate components of our approach - sampling function and sparse representation by comparing them to several existing approaches. We also implement a prototype imaging system with pixel-wise coded exposure control using a Liquid Crystal on Silicon (LCoS) device. System characteristics such as field of view, Modulation Transfer Function (MTF) are evaluated for our imaging system. Both simulations and experiments on a wide range of
    [Show full text]
  • Combinatorial Foundations of Information Theory and the Calculus of Probabilities
    Home Search Collections Journals About Contact us My IOPscience Combinatorial foundations of information theory and the calculus of probabilities This article has been downloaded from IOPscience. Please scroll down to see the full text article. 1983 Russ. Math. Surv. 38 29 (http://iopscience.iop.org/0036-0279/38/4/R04) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 129.236.21.103 The article was downloaded on 13/01/2011 at 16:58 Please note that terms and conditions apply. Uspekhi Mat. Nauk 38:4 (1983), 27-36 Russian Math. Surveys 38:4 (1983), 29-40 Combinatorial foundations of information theory and the calculus of probabilities*1 ] A.N. Kolmogorov CONTENTS §1. The growing role of finite mathematics 29 §2. Information theory 31 §3. The definition of "complexity" 32 §4. Regularity and randomness 34 §5. The stability of frequencies 34 §6. Infinite random sequences 35 §7. Relative complexity and quantity of information 37 §8. Barzdin's theorem 38 §9. Conclusion 39 References 39 §1. The growing role of finite mathematics I wish to begin with some arguments that go beyond the framework of the basic subject of my talk. The formalization of mathematics according to Hilbert is nothing but the theory of operations on schemes of a special form consisting of finitely many signs arranged in some order or another and linked by some connections or other. For instance, according to Bourbaki's conception, the entire set theory investigates exclusively expressions composed of the signs D, τ, V, Ί, =, 6, => and of "letters" connected by "links" Π as, for instance, in the expression ΙΓ 1 ι—71 which is the "empty set".
    [Show full text]
  • Information Theory and Communication - Tibor Nemetz
    PROBABILITY AND STATISTICS – Vol. II - Information Theory and Communication - Tibor Nemetz INFORMATION THEORY AND COMMUNICATION Tibor Nemetz Rényi Mathematical Institute, Hungarian Academy of Sciences, Budapest, Hungary Keywords: Shannon theory, alphabet, capacity, (transmission) channel, channel coding, cryptology, data processing, data compression, digital signature, discrete memoryless source, entropy, error-correcting codes, error-detecting codes, fidelity criterion, hashing, Internet, measures of information, memoryless channel, modulation, multi-user communication, multi-port systems, networks, noise, public key cryptography, quantization, rate-distortion theory, redundancy, reliable communication, (information) source, source coding, (information) transmission, white Gaussian noise. Contents 1. Introduction 1.1. The Origins 1.2. Shannon Theory 1.3. Future in Present 2. Information source 3. Source coding 3.1. Uniquely Decodable Codes 3.2. Entropy 3.3. Source Coding with Small Decoding Error Probability 3.4. Universal Codes 3.5. Facsimile Coding 3.6. Electronic Correspondence 4. Measures of information 5. Transmission channel 5.1. Classification of Channels 5.2. The Noisy Channel Coding Problem 5.3. Error Detecting and Correcting Codes 6. The practice of classical telecommunication 6.1. Analog-to-Digital Conversion 6.2. Quantization 6.3 ModulationUNESCO – EOLSS 6.4. Multiplexing 6.5. Multiple Access 7. Mobile communicationSAMPLE CHAPTERS 8. Cryptology 8.1. Classical Cryptography 8.1.1 Simple Substitution 8.1.2. Transposition 8.1.3. Polyalphabetic Methods 8.1.4. One Time Pad 8.1.5. DES: Data Encryption Standard. AES: Advanced Encryption Standard 8.1.6. Vocoders 8.2. Public Key Cryptography ©Encyclopedia of Life Support Systems (EOLSS) PROBABILITY AND STATISTICS – Vol. II - Information Theory and Communication - Tibor Nemetz 8.2.1.
    [Show full text]
  • Wisdom - the Blurry Top of Human Cognition in the DIKW-Model? Anett Hoppe1 Rudolf Seising2 Andreas Nürnberger2 Constanze Wenzel3
    EUSFLAT-LFA 2011 July 2011 Aix-les-Bains, France Wisdom - the blurry top of human cognition in the DIKW-model? Anett Hoppe1 Rudolf Seising2 Andreas Nürnberger2 Constanze Wenzel3 1European Centre for Soft Computing, Email: fi[email protected] 2Otto-von-Guericke-University Magdeburg, Email: [email protected] 3Otto-von-Guericke-University Magdeburg, Email: [email protected] Abstract draws from a real integrational view throughout scientific communities. Therefore, in Section 3, we Wisdom is an ancient concept, that experiences a review definitions used in different domains in order renaissance since the last century throughout sev- to outline their common points and discrepancies. eral scientific communities. In each of them it is In Section 4, we finally try to apply the found interpreted in rather different ways - from a key definitions to the DIKW-model. We argue that ability to succesful aging to a human peak perfo- none of the definitions we found in several different mance transcending mere knowledge. As a result, domains suggest a positioning of wisdom in a chain miscellaneous definitions and models exist. There with data, information and knowledge except the is for instance the DIKW-hierarchy that tempts to one, Ackhoff proposed himself to justify his model. integrate the concept of wisdom into information Section 5 offers a closing reflection on how our science, still without providing a proper definition. conclusions might affect the view of computer The work at hand tries to sum up current ap- science, artificial and computational intelligence/ proaches (out of computer science as well as others) soft computing on wisdom and proposes directions with a focus on their usefulness for the positioning of further research.
    [Show full text]
  • Kolmogorov Complexity in Perspective. Part II: Classification, Information Processing and Duality Marie Ferbus-Zanda
    Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality Marie Ferbus-Zanda To cite this version: Marie Ferbus-Zanda. Kolmogorov Complexity in perspective. Part II: Classification, Information Processing and Duality. Synthese, Springer Verlag (Germany), 2010, pp.00. hal-00525505 HAL Id: hal-00525505 https://hal.archives-ouvertes.fr/hal-00525505 Submitted on 13 Oct 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Kolmogorov Complexity in perspective Part II: Classification, Information Processing and Duality∗ Marie Ferbus-Zanda LIAFA, CNRS & Universit´eParis Diderot - Paris 7 Case 7014 75205 Paris Cedex 13 France [email protected] Abstract We survey diverse approaches to the notion of information: from Shan- non entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts published in a same volume. Part II is dedicated to the relation between logic and information sys- tem, within the scope of Kolmogorov algorithmic information theory. We present a recent application of Kolmogorov complexity: classification us- ing compression, an idea with provocative implementation by authors such as Bennett, Vit´anyi and Cilibrasi.
    [Show full text]
  • An Affective Computational Model for Machine Consciousness
    An affective computational model for machine consciousness Rohitash Chandra Artificial Intelligence and Cybernetics Research Group, Software Foundation, Nausori, Fiji Abstract In the past, several models of consciousness have become popular and have led to the development of models for machine con- sciousness with varying degrees of success and challenges for simulation and implementations. Moreover, affective computing attributes that involve emotions, behavior and personality have not been the focus of models of consciousness as they lacked moti- vation for deployment in software applications and robots. The affective attributes are important factors for the future of machine consciousness with the rise of technologies that can assist humans. Personality and affection hence can give an additional flavor for the computational model of consciousness in humanoid robotics. Recent advances in areas of machine learning with a focus on deep learning can further help in developing aspects of machine consciousness in areas that can better replicate human sensory per- ceptions such as speech recognition and vision. With such advancements, one encounters further challenges in developing models that can synchronize different aspects of affective computing. In this paper, we review some existing models of consciousnesses and present an affective computational model that would enable the human touch and feel for robotic systems. Keywords: Machine consciousness, cognitive systems, affective computing, consciousness, machine learning 1. Introduction that is not merely for survival. They feature social attributes such as empathy which is similar to humans [12, 13]. High The definition of consciousness has been a major challenge level of curiosity and creativity are major attributes of con- for simulating or modelling human consciousness [1,2].
    [Show full text]
  • Combinatorial Reasoning in Information Theory
    Combinatorial Reasoning in Information Theory Noga Alon∗ Abstract Combinatorial techniques play a crucial role in the investigation of problems in Informa- tion Theory. We describe a few representative examples, focusing on the tools applied, and mentioning several open problems. 1 Introduction Combinatorial ideas play a prominent role in the study of problems in Information theory. Indeed, the whole theory can be developed using a combinatorial approach, as done, for example, in [12]. In this brief survey we discuss several examples in which tools from Combinatorics and Graph Theory are applied in the investigation of problems in Information Theory. The combinatorial approach seems especially powerful for tackling problems in zero-error information theory which deals with scenarios in which no positive probability of error is tolerated. Problems of this type are discussed in a significant number of papers starting with [23], and are also the focus of the present short paper. This is not meant to be a comprehensive treatment of the subject, but hopefully provides an interesting description of several intriguing information theoretic results obtained by combinatorial reasoning. 2 The Shannon Capacity of graphs For an undirected graph G = (V; E), let G^n denote the graph whose vertex set is V n in which two distinct vertices (u1; u2; : : : ; un) and (v1; v2; : : : ; vn) are adjacent iff for all i between 1 and n ^n 1=n either ui = vi or uivi 2 E. The Shannon capacity c(G) of G is the limit limn!1(α(G )) , where α(G^n) is the maximum size of an independent set of vertices in G^n.
    [Show full text]