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Coding with Feedback and Searching with Lies

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Entropy, Search, Complexity

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 16))

Abstract

This paper gives a broad overview of the area of searching with errors and the related field of error-correcting coding. In the vast literature regarding this problem, many papers simultaneously deal with various sorts of restrictions on the searching protocol. We partition this survey into sections, choosing the most appropriate section for each topic.

Supported in part by INTAS-00-738.

A survey of results in coding with feedback and searching with lies until 2000.

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References

  1. R. Ahlswede, A constructive proof of the coding theorem for discrete memoryless channels with feedback, in: Transactions of the Sixth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, Tech. Univ., Prague (1973), pp. 39–50.

    Google Scholar 

  2. R. Ahlswede, Channels with arbitrarily varying channel probability functions in the presence of noiseless feedback, Z. Wahrsch. th. u. verw. Geb., 25 (1973), 239–252.

    Article  MATH  MathSciNet  Google Scholar 

  3. R. Ahlswede, Concepts of performance parameters for channels, General Theory of Information Transfer and Combinatorics, Lecture Notes in Computer Science, Vol. 4123 (2006).

    Google Scholar 

  4. R. Ahlswede, Elimination of correlation in random codes for AVC, Z. Wahrsch. th. u. verw. Geb., 44 (1978), 159–175.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Ahlswede and G. Dueck, Identification in the presence of feedback — a discovery of new capacity formulas, IEEE Trans. Inform. Theory, 35, no. 1 (1989), 30–36.

    Article  MATH  MathSciNet  Google Scholar 

  6. R. Ahlswede and G. Dueck, Identification via channels, IEEE Trans. Inform. Theory, Vol. 35 (1989), 15–29.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Ahlswede, N. Cai and C. Deppe, An isoperimetric theorem for sequences generated by feedback and feedback codes for unequal protection of messages, Problems in Information Transmission, Vol. 37, No. 4 (2001), 332–338.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. Ahlswede, General Theory of information transfer, Preprint 97-118, SFB 343, Universität Bielefeld, 1997.

    Google Scholar 

  9. R. Ahlswede and I. Wegener, Suchprobleme, Teubner (1979), English translation: Wiley (1987), Russian translation: MIR (1982).

    Google Scholar 

  10. R. Ahlswede and N. Cai, The AVC with noiseless feedback and maximal error probability: a capacity formula with a trichotomy, in: Numbers, Information and Complexity (Bielefeld, 1998), Kluwer Acad. Publ. (Boston, MA, 2000), pp. 151–176.

    Google Scholar 

  11. M. Aigner, Combinatorial Search, Wiley-Teubner (1988).

    Google Scholar 

  12. M. Aigner, Ulams Millionenspiel, Math. Semesterber., vol. 42 (1995), 71–80.

    Article  MATH  MathSciNet  Google Scholar 

  13. M. Aigner, Searching with lies, J. Combin. Theory Ser. A, vol. 74 (1996), 43–56.

    Article  MATH  MathSciNet  Google Scholar 

  14. J. A. Aslam and A. Dhagat, Searching in the presence of linearly bounded errors, Proceedings of the 23rd Annual ACM Symposium of Theory of Computing (1991).

    Google Scholar 

  15. V. B. Balakirsky, A direct approach to searching with lies, Preprint 98-070, University of Bielefeld (1998).

    Google Scholar 

  16. R. Beigel, Unbounded searching algorithms, SIAM J. Comput., Vol. 19, no. 3 (1990), 522–537.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. L. Bentley and C. C.-C. Yao, An almost optimal algorithm for unbounded searching, Inform. Process. Lett., 5 (1976), 82–87.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. R. Berlekamp, Block coding for the binary symmetric channel with noiseless, delayless feedback, in: H. B. Mann, Error Correcting Codes, Wiley (1968), pp. 61–85.

    Google Scholar 

  19. E. R. Berlekamp, R. Hill and J. Karim, The solution of a problem of Ulam on searching with lies, Proc. IEEE Int. Symp. on Inform. Theory (MIT, Cambridge, MA USA, 1998), p. 244.

    Google Scholar 

  20. A. de Bonis, L. Gargano and U. Vaccaro, Group testing with unreliable tests, Inf. Sci., vol. 96, no. 1–2 (1997), 1–12.

    MATH  Google Scholar 

  21. M. V. Burnashev, Data transmission over a discrete channel with feedback, Problems Inform. Transmission, vol. 12 (1976), 250–265, translated from Problemy Peredachi Informatsii, vol. 12, no. 4 (1976), 10–30 (in Russian).

    Google Scholar 

  22. F. Cicalese, Reliable computation with unreliable information, PhD thesis, Univ. of Salerno, Dept. of Computer Science (2000).

    Google Scholar 

  23. F. Cicalese and U. Vaccaro, An improved heuristic for the Ulam-Rényi game, Inf. Proc. Letters, Vol. 73 (2000), 119–124.

    Article  MATH  MathSciNet  Google Scholar 

  24. F. Cicalese and D. Mundici, Learning and the Art of Fault-Tolerant Guesswork, Handbook Chapter, In: Perspectives on Adaptivity and Learning. Stamatescu, I. et al., Eds., Springer-Verlag (2003), pp. 115–140.

    Google Scholar 

  25. F. Cicalese, D. Mundici and U. Vaccaro, Least adaptive optimal search with unreliable tests, Theor. comput. Sci., vol. 270, no. 1–2 (2002), 877–893.

    Article  MATH  MathSciNet  Google Scholar 

  26. F. Cicalese and D. Mundici, Perfect two-fault tolerant search with minimum adaptiveness, Adv. Appl. Math. 25, no. 1 (2000), 65–101.

    Article  MATH  MathSciNet  Google Scholar 

  27. F. Cicalese and D. Mundici, Optimal binary search with two unreliable tests and minimum adaptiveness, Lect. Notes Comput. Sci., 1643 (1999), 257–266.

    Google Scholar 

  28. F. Cicalese and D. Mundici, Optimal coding with one asymmetric error: below the sphere packing bound, Lecture Notes in Comput. Sci., 1858 (2000), 159–169.

    Google Scholar 

  29. F. Cicalese and U. Vaccaro, Optimal strategies against a liar, Theoretical Computer Science, Vol. 230 (2000), 167–193.

    Article  MATH  MathSciNet  Google Scholar 

  30. F. Cicalese, D. Mundici and U. Vaccaro, Rota-Metropolis cubic logic and Ulam-Rényi games, in: Algebraic combinatorics and computer science: a tribute to Gian-Carlo Rota, Springer-Italia (Milan, 2001), pp. 197–244.

    Google Scholar 

  31. F. Cicalese and D. Mundici, Optimal Searching Strategies for two Asymmetric Lies, preprint (2002).

    Google Scholar 

  32. J. Czyzowicz, D. Mundici and A. Pelc, Solution of Ulam’s problem on binary search with two lies, J. Combin. Theory Ser. A, vol. 49 (1988), 384–388.

    Article  MATH  MathSciNet  Google Scholar 

  33. J. Czyzowicz, D. Mundici and A. Pelc, Ulam’s searching game with lies, J. Combin. Theory Ser. A, vol. 52 (1989), 62–76.

    Article  MATH  MathSciNet  Google Scholar 

  34. J. Czyzowicz, K. B. Lakshmanan and A. Pelc, Searching with a forbidden lie pattern in responses, Inform. Process. Lett., vol. 37 (1991), 127–132.

    Article  MATH  MathSciNet  Google Scholar 

  35. C. Deppe, Strategies for the Rényi-Ulam Game with fixed number of lies, Theoretical Computer Science, 314 (2004), 45–55.

    Article  MATH  MathSciNet  Google Scholar 

  36. C. Deppe, Solution of Ulam’s searching game with three lies or an optimal adaptive strategy for binary three-error-correcting codes, Discrete Math., 224, no. 1–3 (2000), 79–98.

    Article  MATH  MathSciNet  Google Scholar 

  37. D. DesJardins, Precise coding with noiseless Feedback, PhD thesis, University of California at Berkeley, Department of Mathematics (2001).

    Google Scholar 

  38. A. Dhagat, P. Gacs and P. Winkler, On playing twenty questions with a liar, BUCS Tech Report #91-006, Boston University (1992).

    Google Scholar 

  39. D.-Z. Du and F. K. Hwang, Combinatorial Group Testing, World Scientific (Singapore, 1993).

    Google Scholar 

  40. W. Guzicki, Ulam’s searching game with two lies, J. Combin. Theory Ser. A, vol. 54 (1990), 1–19.

    Article  MATH  MathSciNet  Google Scholar 

  41. L. H. Harper, Optimal numberings and isoperimetric problems on graphs, J. Combinatorial Theory, 1 (1966), 385–393.

    MATH  MathSciNet  Google Scholar 

  42. R. Hill, Searching with lies, Surveys in Combinatorics, Lecture Note Series, 218 (1995), 41–70.

    Google Scholar 

  43. R. Hill and J. P. Karim, Searching with lies: the Ulam problem, Discrete Math., vol. 106/107 (1992), 273–283.

    Article  MathSciNet  Google Scholar 

  44. M. Horstein, Sequential transmission using noiseless feedback, IEEE Trans. Inform. Theory, vol. 9, no. 3 (1963), 136–142.

    Article  MATH  Google Scholar 

  45. J. P. Karim, Searching with lies: the Ulam problem, PhD thesis, University of Salford, Department of Mathematics and Computer Science.

    Google Scholar 

  46. G. Katona, A theorem of finite sets, in: Theory of graphs (Proc. Colloq., Tihany, 1966), Academic Press (New York, 1968), pp. 187–207.

    Google Scholar 

  47. D. J. Kleitman, A. R. Meyer, R. L. Rivest, J. Spencer and K. Winklmann, Coping with errors in binary search procedures, J. Comput. System Sci., vol. 20 (1980), 396–404.

    Article  MATH  MathSciNet  Google Scholar 

  48. D. E. Knuth, Supernatural numbers, The Mathematical Gardner, ed. D. A. Klarner, Wadsworth International (1981), pp. 310–325.

    Google Scholar 

  49. N. L. Komarova, P. Niyogi and M. A. Nowak, Evolution of universal grammar, Scienc,e 291 (2001), 114–118.

    MathSciNet  Google Scholar 

  50. N. L. Komarova and I. Riven, Mathematics of Learning, Arxiv, Preprintserver, math.PR/0105235, (http://arxiv.org), 2001.

    Google Scholar 

  51. N. L. Komarova and M. A. Nowak, Natural selection of the critical period for grammar acquisition, Proc. Royal Soc. B., 268, No. 1472 (2001), 1189–1196.

    Article  Google Scholar 

  52. N. L. Komarova, P. Niyogi and M. A. Nowak, The evolutionary dynamics of grammar acquisition, J. Theor. Biology 209 (2001), 43–59.

    Article  Google Scholar 

  53. K. B. Lakshmanan and B. Ravikumar, Coping with known patterns of lies in a search game, Theoret. Comput. Sci., vol. 33 (1984), 85–94.

    Article  MATH  MathSciNet  Google Scholar 

  54. E. L. Lawler and S. Sarkissian, An algorithm for Ulam’s game and its application to error correcting codes, Inform. Process. Lett., vol. 56 (1995), 89–93.

    Article  MATH  MathSciNet  Google Scholar 

  55. K. Leweling, Codierung eines diskreten gedächtnislosen Kanals mit Rückkopplung, Diploma Thesis, Fakultät für Mathematik, UniversitÄat Bielefeld (1975).

    Google Scholar 

  56. F. J. MacWilliams and N. J. A. Sloane, The theory of error-correcting codes, II. North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co. (Amsterdam-New York-Oxford, 1977).

    Google Scholar 

  57. A. Malinowski, K-ary searching with a lie, ARS Combin., vol. 37 (1994), 301–308.

    MATH  MathSciNet  Google Scholar 

  58. D. Mundici, The complexity of adaptive error-correcting codes, Springer Lecture Notes in Computer Science, 533 (1991), 300–307.

    Google Scholar 

  59. D. Mundici and A. Trombetta, Optimal comparison strategies in Ulam’s searching game with two lies, Theoretical Computer Science, vol. 182 (1997), 217–232.

    Article  MATH  MathSciNet  Google Scholar 

  60. A. Negro and M. Sereno, Ulam’s searching game with three lies, Adv. in Appl. Math., 13 (1992), 404–428.

    Article  MATH  MathSciNet  Google Scholar 

  61. A. Negro and M. Sereno, Solution of Ulam’s problem on binary search with three lies, J. Combin. Theory Ser. A, vol. 59 (1992), 149–154.

    Article  MATH  MathSciNet  Google Scholar 

  62. I. Niven, Coding theory applied to a problem of Ulam, Math. Mag., vol. 61 (1988), 275–281.

    Article  MATH  MathSciNet  Google Scholar 

  63. A. Pelc, Coding with bounded error fraction, Ars Combinatoria, vol. 24 (1987), 17–22.

    MATH  MathSciNet  Google Scholar 

  64. A. Pelc, Detecting errors in searching games, J. Combin. Theory Ser. A, vol. 51 (1989), 43–54.

    Article  MATH  MathSciNet  Google Scholar 

  65. A. Pelc, Detecting a counterfeit coin with unreliable weighings, Ars Combinatoria, vol. 27 (1989), 181–192.

    MATH  MathSciNet  Google Scholar 

  66. A. Pelc, Solution of Ulam’s problem on searching with a lie, J. Combin. Theory Ser. A, vol. 44 (1987), 129–140.

    Article  MATH  MathSciNet  Google Scholar 

  67. A. Pelc, Searching with known error probability, Theoret. Comput. Sci., 63 (1989), 185–202.

    Article  MATH  MathSciNet  Google Scholar 

  68. A. Pelc, Searching games with errors — fifty years of coping with liars, Theoret. Comput. Sci., 270 (2002), 71–109.

    Article  MATH  MathSciNet  Google Scholar 

  69. A. Rényi, A Diary on Information Theory, Wiley (1987), translation of Napló az információelméletről (1976).

    Google Scholar 

  70. A. Rényi, On a problem of information theory, MTA Mat. Kut. Int. Közl., 6B (1961), 505–516.

    Google Scholar 

  71. G. Rong and L. Yizhong, Generalized Solution of Ulam’s Problem, Chinese Journal of Contemporary Mathematics, vol. 13 (1992), 323–331.

    MathSciNet  Google Scholar 

  72. J. P. M. Schalkwijk, A class of simple and optimal strategies for block coding on the binary symmetric channel with noiseless feedback, IEEE Trans. Inform. Theory, vol. 17 (1971), no. 3.

    Google Scholar 

  73. C. E. Shannon, The zero-error capacity of a noisy channel, IRE Trans. Inform. Th., 3 (1956), 3–15.

    Article  Google Scholar 

  74. C. E. Shannon, Two-way communication channels, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I (1961), pp. 611–644.

    MathSciNet  Google Scholar 

  75. J. Spencer, Guess a number — with lying, Math. Mag., vol. 57, no. 2 (1984), 105–108.

    Article  MATH  MathSciNet  Google Scholar 

  76. J. Spencer, Ulam’s searching game with a fixed number of lies, Theoret. Comput. Sci., vol. 95 (1992), 307–321.

    Article  MATH  MathSciNet  Google Scholar 

  77. J. Spencer and P. Winkler, Three thresholds for a liar, Combin. Probab. Comput., vol. 1 (1992), 81–93.

    Article  MATH  MathSciNet  Google Scholar 

  78. S. Ulam, Adventures of a Mathematician, Scribner (NY, 1976).

    MATH  Google Scholar 

  79. T. Verhoeff, An update table of minimum distance bounds for binary codes, IEEE Trans. Inform. Theory, vol. 33 (1987), 665–680.

    Article  MATH  MathSciNet  Google Scholar 

  80. T. Veugen, Multiple-repetition coding for channels with feedback, PhD thesis, Eindhoven University of Technology (1992).

    Google Scholar 

  81. K. Sh. Zigangirov, Number of correctable errors for transmission over a binary symmetrical channel with feedback, Problems Inform. Transmission, vol. 12 (1976), 85–97, translated from Problemi Peredachi Informatsii, vol. 12, no. 3 (1976), 3–19 (in Russian).

    Google Scholar 

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© 2007 János Bolyai Mathematical Society and Springer-Verlag

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Deppe, C. (2007). Coding with Feedback and Searching with Lies. In: Csiszár, I., Katona, G.O.H., Tardos, G., Wiener, G. (eds) Entropy, Search, Complexity. Bolyai Society Mathematical Studies, vol 16. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32777-6_2

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