# Embodied Information

In this project we examine the question of whether any algorithmical resp. non-algorithmical concept as well as any kind of scientific theory are ever be incomplete. We investigate this question with particular attention to the claims made that these concepts can contribute to the notions of observation, prediction, recollection and explanation. There’s some evidence about an interrelation between ideas within the philosophy of science – the Duhem-Quine thesis of underdetermination of observations and the observational/theoretical terms distinction failure – and the well known limitative theorems of Gödel and Tarski etc. Although these results original apply to deduction from axioms, we have a further evidence to assume that their extensions hold for general inference devices, i.e. recursive enumerable ones (like structural inductive) as well, as for other possible systems of logic with non-recursive sets of axioms resp. rules of inference and furthermore for any constraint satisfaction problem. This would imply that any idea or concept and any experience cannot be completely defined or contextualized. – So there’s a strong sense in which we remain under the shadow of chance and randomness.

The aim of this research work is to review, clarify, and critically analyze aspects of modern mathematical information theories. The emphasis is upon mathematical structures involved, rather than numerical computations. We will argue that theories and concepts of information and complexity can never be complete. For that reason particular attention will be paid to various provided measures of information and complexity and their dependence on algorithmical, resp. non-algorithmical concepts. We try to reveal the supposed conditions for incompleteness like computational irreducibility, arbitrariness, infinity, and self-awareness. Working hypothesis is that due to connections with disguised forms of the meta-mathematical theorems of Gödel and Tarski incompleteness is widely an epistemological limit which is manifest, e.g. in the non-existence of a procedure to determine valid empirical observations resp. the undefinability of valid observations, and we assume that this limit is not likely to be broken any time soon.

## Website

http://www.geisteswissenschaften.fu-berlin.de/v/embodiedinformation/

## Contact

Prof. Dr. Georg Trogemann

Professor for Experimental Computer Science

Phone: +49 – (0)221 – 20189 – 131

Fax: +49 – (0)221 – 20189 – 230

Mail: trogemann@khm.de

## Picture Credits

http://commons.wikimedia.org/wiki/Media:?-oracle.svg

Image depicting the character Mu in the ancient chinese oracle script. Mu or wú is a word which has been roughly translated as “no”, “none”, “null”, “without”, “no meaning”. In Japanese and Chinese mainly used as a prefix to imply the absence of something. It’s a word meaning neither yes nor no, i.e. this question has no unambiguous answer. Perhaps in the end this idea should be the only answer to the illusory question of lawful explanations of natural phenomena?

## References

(Keywords: Computation and Information, Entropie and Information, Physics and Information, Information Geometry, Information Theory, Algorithmic Information, Algorithmic Probability, Algorithmic Randomness, Complexity, Probability Theory, Probability and Measure, Statistical Inference, Induction)

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- Nihat Ay

“Information geometry on complexity and stochastic interaction” [preprint]

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- Carl T. Bergstrom, Michael Lachmann, “The fitness value of information”, q-bio.PE/0510007

- Patrick Billingsley, Probability and Measure, Wiley & Sons, N.Y.C. 1995

- Igor Bjelakovic, Tyll Krueger, Rainer Siegmund-Schultze, Arleta Szkola

“The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems,” math.DS/0207121

“Chained Typical Subspaces – a Quantum Version of Breiman’s Theorem,” quant-ph/0301177

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- Claudio Bonanno, “The Manneville map: topological, metric and algorithmic entropy,” math.DS/0107195

- Brillouin, Science and Information Theory, Dover Phoenix Editions: Dover Publications, 1962

- Paul Bohan Broderick, “On Communication and Computation”, Minds and Machines 14 (2004): 1–19 [“The most famous models of computation and communication, Turing Machines and (Shannon-style) information sources, are considered. The most significant difference lies in the types of state-transitions allowed in each sort of model. This difference does not correspond to the difference that would be expected after considering the ordinary usage of these terms.”]

- Kenneth P. Burnham, David R. Anderson, Model Selection and Inference: A Practical Information-Theoretic Approach

- Xavier Calmet, Jacques Calmet, “Dynamics of the Fisher Information Metric”, cond-mat/0410452 = Physical Review E 71 (2005): 056109

- C. J. Cellucci, A. M. Albano, P. E. Rapp, “Statistical validation of mutual information calculations: Comparison of alternative numerical algorithms”, Physical Review E 71 (2005): 066208

- Massimo Cencini, Alessandro Torcini, “A nonlinear marginal stability criterion for information propagation,” nlin.CD/0011044

- Gregory J. Chaitin

Algorithmic Information Theory [online]

Information, Randomness and Incompleteness [online]

Information-Theoretic Incompleteness [online]

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- Bernard Chazelle, The Discrepency Method: Randomness and Complexity, Cambridge UP, 2000

- J.-R. Chazottes, D. Gabrielli, “Large deviations for empirical entropies of Gibbsian sources”, math.PR/0406083 = Nonlinearity 18 (2005): 2545–2563 [large deviation principle in block entropies, and entropy rates estimated from those blocks – even as one lets the length of the blocks grow with the amount of data, provided the block-length doesn’t grow too quickly (only ln_2)]

- J.-R. Chazottes, E. Uglade, “Entropy estimation and fluctuations of Hitting and Recurrence Times for Gibbsian sources”, math.DS/0401093

- Tommy W. S. Chow, D. Huang, “Estimating Optimal Feature Subsets Using Efficient Estimation of High-Dimensional Mutual Information”, IEEE Transactions on Neural Networks 16 (2005): 213–224

- Bob Coecke, “Entropic Geometry from Logic,” quant-ph/0212065

- Thomas M. Cover, Joy A. Thomas, Elements of Information Theory, Wiley Series in Telecommunications and Signal Processing, 2006

- Gavin E. Crooks, “Measuring Thermodynamic Length”, Physical Review Letters 99 (2007): 100602 [“Thermodynamic length is a metric distance between equilibrium thermodynamic states. Among other interesting properties, this metric asymptotically bounds the dissipation induced by a finite time transformation of a thermodynamic system. It is also connected to the Jensen-Shannon divergence, Fisher information, and Rao’s entropy differential metric.”]

- Alfred Crosby, The Measure of Reality, Cambridge University Press, 1997

- Imre Csiszar, “The Method of Types”, IEEE Tranactions on Information Theory<.cite> 44 (1998): 2505–2523 [PDF]

- Imre Csiszar, Janos Korner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press 1997

- Imre Csiszar, Frantisek Matus, “Closures of exponential families”, Annals of Probability 33 (2005): 582–600 = math.PR/0503653

- Imre Csiszar, Paul Shields, Information Theory and Statistics: A Tutorial [Fulltext PDF]

- A. Daffertshofer, A. R. Plastino, “Landauer’s principle and the conservation of information”, Physics Letters A 342 (2005): 213–216

- Lukasz Debowski, “On vocabulary size of grammar-based codes”, cs.IT/0701047

- Gustavo Deco, Bernd Schurmann, Information Dynamics: Foundations and Applications, Springer, Berlin 2000

- Morris DeGroot, Mark J. Schervish, Probability and Statistics, Addison Wesley, 2001

- F. M. Dekking, C. Kraaikamp, H. P. Lopuhaä, L. E. Meester, A Modern Introduction to Probability and Statistics: Understanding How and Why [>>>]

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- Victor H. de la Pena, Tze Leung Lai, Qi-Man Shan, Self-Normalized Processes: Limit Theory and Statistical Applications [>>>]

- Amir Dembo, “Information Inequalities and Concentration of Measure”, The Annals of Probability 25 (1997): 927–939 [“We derive inequalities of the form \Delta(P,Q) =< H(P|R) + H(Q|R) which hold for every choice of probability measures P, Q, R, where H(P|R) denotes the relative entropy of P with respect to R and \Delta(P,Q) stands for a coupling type ‘distance’ between P and Q.”]

- Amir Dembo, I. Kontoyiannis, “Source Coding, Large Deviations, and Approximate Pattern Matching,” math.PR/0103007

- Steffen Dereich, “The quantization complexity of diffusion processes”, math.PR/0411597

- Joseph DeStefano, Erik Learned-Miller, “A Probabilistic Upper Bound on Differential Entropy”, cs.IT/0504091 [“A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution…”]

- Persi Diaconis, Svante Janson, “Graph limits and exchangeable random graphs”, arxiv:0712.2749

- David Doty, “Every sequence is compressible to a random one”, cs.IT/0511074 [“Kucera and Gacs independently showed that every infinite sequence is Turing reducible to a Martin-Löf random sequence. We extend this result to show that every infinite sequence S is Turing reducible to a Martin-Löf random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S.”]

- David Doty, Jared Nichols, “Pushdown Dimension”, cs.IT/0504047

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- Alvin W. Drake, Fundamentals of Applied Probability Theory, Mcgraw-Hill College 1967

- M. Drmota, W. Szpankowski, “Precise minimax redundancy and regret”, IEEE Transactions on Information Theory 50 (2004): 2686–2707

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- John Earman, John Norton, “Exorcist XIV: The wrath of Maxwell’s Demon”

“From Maxwell to Szilard”, Studies in the History and Philosophy of Modern Physics 29 (1998): 435–471

“From Szilard to Landauer and beyond”, Studies in the History and Philosophy of Modern Physics 30 (1999): 1–40

- Bruce R. Ebanks, Prasanna Sahoo, Wolfgang Sander, Characterization of Information Measures, Springer, 1998

- Karl-Erik Eriksson, Kristian Lindgren, Bengt Å. Månsson, Structure, Context, Complexity, Organization: Physical Aspects of Information and Value, Singapore : World Scientific, 1987

- Dave Feldman, Information Theory, Excess Entropy and Statistical Complexity

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- J. M. Finn, J. D. Goettee, Z. Toroczkai, M. Anghel, B. P. Wood, “Estimation of entropies and dimensions by nonlinear symbolic time series analysis”, Chaos 13 (2003): 444–456

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- Bert Fristedt, Lawrence Gray, A Modern Approach to Probability Theory, Birkhäuser, Boston, 1997

- Peter Gács, “Uniform test of algorithmic randomness over a general space”, Theoretical Computer Science 341 (2005): 91–137 [“The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Löf) and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin’s theory, and report further progress in the same framework….”]

- Travis Gagie, “Compressing Probability Distributions”, cs.IT/0506016 [Abstract (in full): “We show how to store good approximations of probability distributions in small space.”]

- Janos Galambos, Italo Simonelli, Bonferroni-type Inequalities with Applications, Springer, Berlin, 1996

- Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas, “Effective symbolic dynamics, random points, statistical behavior, complexity and entropy”, arxiv:0801.0209 [All, not almost all, Martin-Löf points are statistically typical.]

- Yun Gao, Ioannis Kontoyiannis, Elie Bienenstock, “From the entropy to the statistical structure of spike trains”, arxiv:0710.4117

- Pierre Gaspard, “Time-Reversed Dynamical Entropy and Irreversibility in Markovian Random Processes”, Journal of Statistical Physics 117 (2004): 599–615

- David Gelernter, Mirror Worlds, Oxford UP, 1993

- George M. Gemelos, Tsachy Weissman, “On the Entropy Rate of Pattern Processes”, cs.IT/0504046

- Neil Gershenfeld, The Physics of Information Technology, Cambridge UP, 2000

- Paolo Gibilisco, Tommaso Isola, “Uncertainty Principle and Quantum Fisher Information”, math-ph/0509046

- Paolo Gibilisco, Daniele Imparato, Tommaso Isola, “Uncertainty Principle and Quantum Fisher Information II” math-ph/0701062

- Josep Ginebra, “On the Measure of the Information in a Statistical Experiment”, Bayesian Analysis (2007): 167–212

- Clark Glymour, “Instrumental Probability”, Monist 84 (2001): 284–300 [PDF reprint]

- M. Godavarti, A. Hero, “Convergence of Differential Entropies”, IEEE Transactions on Information Theory 50 (2004): 171–176

- Stanford Goldman, Information Theory, Dover Publ., 1969 [some interesting time-series which has dropped out of most modern presentations]

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- H. Gopalkrishna Gadiyar, K. M. Sangeeta Maini, R. Padma, H. S. Sharatchandra, “Entropy and Hadamard matrices”, Journal of Physics A: Mathematical and General 36 (2003): L109–L112

- Alexander N. Gorban, Iliya V. Karlin, Hans Christian Ottinger, “The additive generalization of the Boltzmann entropy,” cond-mat/0209319 [a rediscovery of Renyi entropies…?]

- Peter Grassberger “Randomness, Information, and Complexity,” pp. 59–99 of Francisco Ramos-Gómez (ed.), Proceedings of the Fifth Mexican School on Statistical Physics (Singapore: World Scientific, 1989)

- Peter Grassberger, “Data Compression and Entropy Estimates by Non-sequential Recursive Pair Substitution,” physics/0207023

- R. M. Gray, Entropy and Information Theory [on-line.]

- Andreas Greven, Gerhard Keller, Gerald Warnecke (eds.), Entropy, Princeton UP, 2003

- Geoffrey R. Grimmett, David Stirzaker, Probability and Random Processes, Oxford UP, 2001

- Peter Grünwald, Paul Vitányi, “Shannon Information and Kolmogorov Complexity”, cs.IT/0410002

- Sudipto Guha, Andrew McGregor, Suresh Venkatasubramanian, “Streaming and Sublinear Approximation of Entropy and Information Distances”, 17th ACM-SIAM Symposium on Discrete Algorithms, 2006 [Link via Suresh]

- Ian Hacking

The Emergence of Probability, Cambridge UP, 2006

The Taming of Chance, Cambridge UP, 1990

- Michael J. W. Hall, “Universal Geometric Approach to Uncertainity, Entropy and Information,” physics/9903045

- Guangyue Han, Brian Marcus, “Analyticity of Entropy Rate in Families of Hidden Markov Chains”, math.PR/0507235

- Te Sun Han

“Hypothesis Testing with the General Source”, IEEE Transactions on Information Theory 46 (2000): 2415–2427 = math.PR/0004121 [“The asymptotically optimal hypothesis testing problem with the general sources as the null and alternative hypotheses is studied…. Our fundamental philosophy in doing so is first to convert all of the hypothesis testing problems completely to the pertinent computation problems in the large deviation-probability theory. … [This] enables us to establish quite compact general formulas of the optimal exponents of the second kind of error and correct testing probabbilities for the general sources including all nonstationary and/or nonergodic sources with arbitrary abstract alphabet (countable or uncountable). Such general formulas are presented from the information-spectrum point of view.”]

“Folklore in Source Coding: Information-Spectrum Approach”, IEEE Transactions on Information Theory 51 (2005): 747–753 [From the abstract: “we verify the validity of the folklore that the output from any source encoder working at the optimal coding rate with asymptotically vanishing probability of error looks like almost completely random.”]

“An information-spectrum approach to large deviation theorems”, cs.IT/0606104

- Te Sun Han, Kingo Kobayashi, Mathematics of Information and Coding [>>>]

- Masahito Hayashi, “Second order asymptotics in fixed-length source coding and intrinsic randomness”, cs.IT/0503089

- Nicolai T. A. Haydn, “The Central Limit Theorem for uniformly strong mixing measures”, arxiv:0903.1325

- Nicolai Haydn and Sandro Vaienti, “Fluctuations of the Metric Entropy for Mixing Measures”, Stochastics and Dynamics 4 (2004): 595–627

- Torbjorn Helvik, Kristian Lindgren, Mats G. Nordahl, “Continity of Information Transport in Surjective Cellular Automata”, Communications in Mathematical Physics 272 (2007): 53-74

- Alexander E. Holroyd, Terry Soo, “A Non-Measurable Set from Coin-Flips”, math.PR/0610705

- Hong-Da Chen, Chang-Heng Chang, Li-Ching Hsieh, Hoong-Chien Lee, “Divergence and Shannon Information in Genomes”, Physical Review Letters 94 (2005): 178103

- M. Hotta and I. Jochi, “Composability and Generalized Entropy,” cond-mat/9906377

- Marcus Hutter, “Distribution of Mutual Information,” cs.AI/0112019

Universal artificial intelligence: Sequential Decisions based on algorithmic probability. Springer, Berlin, 2004.

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- Shiro Ikeda, Toshiyuki Tanaka, Shun-ichi Amari, “Stochastic Reasoning, Free Energy, and Information Geometry”, Neural Computation 16 (2004): 1779–1810

- K. Iriyama, “Error Exponents for Hypothesis Testing of the General Source”, IEEE Transactions on Information Theory 51 (2005): 1517–1522

- K. Iwata, K. Ikeada, H. Sakai, “A Statistical Property of Multiagent Learning Based on Markov Decision Process”, IEEE Transactions on Neural Networks 17 (2006): 829–842 [The property is asymptotic equipartiton!]

- Aleks Jakulin, Ivan Bratko, “Quantifying and Visualizing Attribute Interactions”, cs.AI/0308002

- W. Janke, D.A. Johnston, R. Kenna, “Information Geometry and Phase Transitions”, cond-mat/0401092 = Physica A 336 (2004): 181–186

- Petr Jizba, Toshihico Arimitsu, “The world according to Renyi: Thermodynamics of multifractal systems,” cond-mat/0207707

- Oliver Johnson

“A conditional Entropy Power Inequality for dependent variables,” math.PR/0111021

“Entropy and a generalisation of `Poincare’s Observation’,” math.PR/0201273

- Oliver Johnson, Richard Samworth, “Central Limit Theorem and convergence to stable laws in Mallows distance”, math.PR/0406218

- Oliver Johnson, Andrew Barron, “Fisher Information inequalities and the Central Limit Theorem,” math.PR/0111020

- Mark Kac

Engimas of Chance, University of California Press, 1987

Probability and Related Topics in Physical Science, American Mathematical Society, 1957

Statistical Independence in Probability, Analysis and Number Theory, Mathematical Assn of Amer.,1959, Carus Mathematical Monographs, Nr.12

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- Olav Kallenberg, Probabilitic Symmetries and Invariance Principles [“This is the first comprehensive treatment of the three basic symmetries of probability theory – contractability, exchangeability, and rotatability – defined as invariance in distribution under contractions, permutations, and rotations.” >>>]

- Alexei Kaltchenko, “Algorithms for Estimating Information Distance with Applications to Bioinformatics and Linguistics”, cs.CC/0404039

- Robert E. Kass, Paul W. Vos, Geometrical Foundations of Asymptotic Inference, Wiley-Interscience, 1997

- Ido Kanter, Hanan Rosemarin, “Communication near the channel capacity with an absence of compression: Statistical Mechanical Approach,” cond-mat/0301005

- Hillol Kargupta, “Information Transmission in Genetic Algorithm and Shannon’s Second Theorem”, Proceedings of the 5th International Conference on Genetic Algorithms, 1993

- Matthew B. Kennel, Jonathon Shlens, Henry D. I. Abarbanel, E. J. Chichilnisky, “Estimating Entropy Rates with Bayesian Confidence Intervals”, Neural Computation 17 (2005): 1531–1576

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- Shiraj Khan, Sharba Bandyopadhyay, Auroop R. Ganguly, Sunil Saigal, David J. Erickson, Vladimir Protopopescu, George Ostrouchov, “Relative performance of mutual information estimation methods for quantifying the dependence among short and noisy data”, Physical Review E 76 (2007): 026209

- A. I. Khinchin, Mathematical Foundations of Information Theory, Dover Publ., 1957 [an axiomatic approach]

- Andrei Kolmogorov, Foundations of Probability Theory, Chelsea Publ.Co., N.Y.C., 1956

- Alexander Kraskov, Harald Stögbauer, Peter Grassberger, “Estimating Mutual Information”, cond-mat/0305641 = Physical Review E 69 (2004): 066138

- Rudolf Kulhavý Recursive Nonlinear Estimation: A Geometric Approach, (Lecture Notes in Control and Information Sciences. 216). Springer, London 1996

- Solomon Kullback, Information Theory and Statistics, Dover Publ., 1997

- Rolf Landauer, “The Physical Nature of Information,” Physics Letters A 217 (1996): 188–193

- Bernard H. Lavenda, “Information and coding discrimination of pseudo-additive entropies (PAE)”, cond-mat/0403591

- G. Lebanon, “Axiomatic Geometry of Conditional Models”, IEEE Transactions on Information Theory 51 (2005): 1283–1294

- Tue Lehn-schioler, Anant Hegde, Deniz Erdogmus, Jose C. Principe, “Vector quantization using information theoretic concepts”, Natural Computation 4 (2005): 39–51 [“it becomes clear that minimizing the free energy of the system is in fact equivalent to minimizing a divergence measure between the distribution of the data and the distribution of the processing elements, hence, the algorithm can be seen as a density matching method.”]

- Emmanuel Lesigne, Heads or Tails: An Introduction to Limit Theorems in Probability [>>>]

- Christophe Letellier, “Estimating the Shannon Entropy: Recurrence Plots versus Symbolic Dynamics”, Physical Review Letters 96 (2006): 254102

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- Lev B. Levitin, Tommaso Toffoli, “Thermodynamic Cost of Reversible Computing”, Physical Review Letters 99 (2007): 110502

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- F. Liang, A. Barron, “Exact Minimax Strategies for Predictive Density Estimation, Data Compression, and Model Selection”, IEEE Transactions on Information Theory 50 (2004): 2708–2726

- Douglas Lind, Brian Marcus, Symbolic Dynamics and Coding, Cambridge UP, 1995

- Christian Lindgren, “Information Theory for Complex Systems” (Online lecture notes, dated Jan. 2003)

- Seth Lloyd

“Use of Mutual Information to Decrease Entropy — Implications for the Second Law of Thermodynamics,” Physical Review A 39 (1989): 5378–5386

“Computational capacity of the universe,” quant-ph/0110141

- Michel Loève, Probability Theory, Princeton, D. Van Nostrand, 1963

- E. Lutwak, D. Yang, G. Zhang, “Cramer-Rao and Moment-Entropy Inequalities for Renyi Entropy and Generalized Fisher Information”, IEEE Transactions on Information Theory 51 (2005): 473–478

- Christian K. Machens, “Adaptive sampling by information maximization,” physics/0112070

- David J. C. MacKay

Information Theory, Inference and Learning Algorithms [>>>]

“Rate of Information Acquisition by a Species subjected to Natural Selection” [>>>]

- Donald Mackay, Information, Mechanism and Meaning [a notion of “meaning” out of information theory?]

- Andrew J. Majda, Rafail V. Abramov, Marcus J. Grote, Information Theory and Stochastic for Multiscale Nonlinear Systems [Sounds interesting, to judge from the >>>. PDF (draft?)]

- David Malone, Wayne J. Sullivan, “Guesswork and Entropy”, IEEE Transactions on Information Theory 50 (2004): 525–526

- Eddy Mayer-Wolf, Moshe Zakai, “Some relations between mutual information and estimation error on Wiener space”, math.PR/0610024

- James W. McAllister, “Effective Complexity as a Measure of Information Content”, Philosophy of Science 70 (2003): 302–307

- Robert J. McEliece, The Theory of Information and Coding, Kluwer Academic Publ., 2002

- N. Merhav, M. J. Weinberger, “On Universal Simulation of Information Sources Using Training Data”, IEEE Transactions on Information Theory 50 (2004): 5–20; +Addendum, IEEE Transactions on Information Theory 51 (2005): 3381–3383

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