Rota-Metropolis cubic logic and Ulam-Rényi games

  • F. Cicalese
  • D. Mundici
  • U. Vaccaro


In their paper [43] Rota and Metropolis considered the partially ordered set F n of all nonempty faces of the n-cube [0, 1] n for each n = 1, 2,…, equipped with the following operation: (⊔) taking the supremum AB of any two faces A and B of F n , together with the following two partially defined operations: (⊓) taking the set-theoretic intersection AB of any two intersecting faces A and B of F n , and (Δ) when a face A is contained in another face B, taking the antipode Δ (B, A) of A in B.


Winning Strategy Parity Check Matrix Noisy Channel Binary Expansion Comparison Question 
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  1. [1]
    Aigner, M. (1996): Searching with lies. J. Combin. Theory Ser. A 74, 43–56MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Aigner, M. (1988): Combinatorial search. Wiley-Teubner, Chichester-StuttgartMATHGoogle Scholar
  3. [3]
    Ambainis, A., Bloch, S.A., Schweizer, D.L. (1999): Delayed binary search, or playing twenty questions with a procrastinator. In: Proceedings of Tenth Annual ACM SIAM Symposium on Discrete Algorithms. Association for Computing Machinery, New York, pp.844–845Google Scholar
  4. [4]
    Aslam, J., Dhagat, A. (1991): Searching in the presence of linearly bounded errors. In: Proceedings of the Twenty-Third ACM Symposium on the Theory of Computing. Association for Computing Machinery, New York, pp. 486–493Google Scholar
  5. [5]
    Bar-Noy, A., Kipnis, S. (1994): Designing broadcast algorithms in the postal model for message-passing systems. Math. Systems Theory 27, 431–452MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Berlekamp, E.R. (1969): Block coding for the binary symmetric channel with noiseless, delayless feedback. In: Mann, H.B. (ed.) Error Correcting Codes, Wiley, New York, pp. 61–88Google Scholar
  7. [7]
    Borgstrom, R.S., Kosaraju, S. Rao (1993): Comparison-based search in the presence of errors. In: Proceedings of the Twenty-Fifth ACM Symposium on the Theory of Computing. Association for Computing Machinery, New York, pp. 130–136Google Scholar
  8. [8]
    Bose, R.C., Shrikhande, S.S., Parker, E.T. (1960): Further results on the construction of mutually orthogonal Latin squares and the falsity of a Euler’s conjecture. Canad. J. Math. 12, 189–203MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Brouwer, A.E., Shearer, J.B., Sloane, N.J.A., Smith, W.D. (1990): A new table of constant weight codes. IEEE Trans. Inform. Theory 36, 1334–1380MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Chen, W.Y.C., Oliveira, J.S. (1995): Implication algebras and the Metropolis-Rota axioms for cubic lattices. J. Algebra 171, 383–396MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Cicalese, F., Mundici, D. (1999): Optimal binary search with two unreliable tests and minimum adaptiveness. In: Nesetril, J. (ed.) Algorithms ESA 99. (Lecture Notes in Computer Science, vol. 1643) Springer, Berlin, pp. 257–266CrossRefGoogle Scholar
  12. [12] Cicalese, F., Mundici, D. (2000): Perfect two-fault tolerant search with minimum adaptiveness. Adv. Appl. Math. 25, 65–101MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Cicalese, F., Mundici, D. (2000): Optimal coding with one asymmetric error: below the sphere packing bound. In: Du, D.-Z. et. al. (eds.) Computing and Combinatorics. (Lecture Notes in Computer Science, vol. 1858). Springer, Berlin, pp. 159–169CrossRefGoogle Scholar
  14. [14]
    Cicalese, F., Vaccaro, U. (2000): Optimal strategies against a liar. Theoret. Comput. Sci. 230, 167–193MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Cicalese, F., Vaccaro, U. (2000): Coping with delays and time-outs in binary search procedures. PreprintGoogle Scholar
  16. [16]
    Cicalese, F., Mundici, D., Vaccaro, U. (2000): Least adaptive optimal search with unreliable tests. In: Halldorsson, M.M. (ed.) Algorithm Theory SWAT 2000. (Lecture Notes in Computer Science, vol. 1851). Springer, BerlinGoogle Scholar
  17. [17]
    Cignoli, R., D’Ottaviano, I.M.L., Mundici, D. (2000): Algebraic foundations of manyvalued reasoning. (Trends in Logic, vol. 7). Kluwer, DordrechtGoogle Scholar
  18. [18]
    Constantin, S.D., Rao, T.R.N. (1979): On the theory of binary asymmetric error correcting codes. Inform, and Control 40, 20–26MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Czyzowicz, J., Mundici, D., Pelc, A. (1989): Ulam’s searching game with lies. J. Combin. Theory Ser. A 52, 62–76MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Dhagat, A., Gács, P., Winkler, P. (1992): On playing “Twenty Questions” with a liar. In: Proceedings of the Third Annual ACM-SIAM Symposium on Discrete Algorithms. Association for Computing Machinery, New York, pp. 16–22Google Scholar
  21. [21] Du, D.-Z., Hwang, FK. (1993): Combinatorial group testing and its applications. World Scientific, SingaporeMATHGoogle Scholar
  22. [22]
    Golay, M.J.E. (1949): Notes on digital coding. Proc. IRE 37, 657Google Scholar
  23. [23]
    Hill, R. (1995): Searching with lies. In: Rowlinson, P. (ed.) Surveys in Combinatorics. Cambridge University Press, Cambridge, pp. 41–70CrossRefGoogle Scholar
  24. [24]
    Hill, R., Karim, J., Berlekamp, E.R. (1998): The solution of a problem of Ulam on searching with lies. In: Proceedings 1998 IEEE International Symposium on Information Theory. IEEE, Piscatawy, NJ, p. 244Google Scholar
  25. [25]
    Innes, D. (2000): Searching with a lie using only comparison questions. PreprintGoogle Scholar
  26. [26]
    Knill, E. (1995): Lower bounds for identifying subset members with subset queries. In: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms. Association for Computing Machinery, New York, pp. 369–377Google Scholar
  27. [27]
    MacWilliams, F.J., Sloane, N.J.A. (1977): The theory of error-correcting codes. North-Holland, AmsterdamGoogle Scholar
  28. [28]
    Malinowski, A. (1994): K-ary searching with a lie. Ars Combin. 37, 301–308MathSciNetMATHGoogle Scholar
  29. [29]
    McEliece, R.J., Rodemich, E.R., Rumsey, H.C., Welch, L.R. (1977): New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Trans. Inform. Theory 23, 157–166MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Mundici, D. (1989): The C*-algebras of three-valued logic. In: Ferro, R. et al. (eds.), Logic Colloquium’ 88. (Studies in Logic and the Foundations of Mathematics vol 127). North-Holland, Amsterdam, pp. 61–77Google Scholar
  31. [31]
    Mundici, D. (1992): The logic of Ulam’s game with lies. In: Bicchieri, C., Dalla Chiara, M. L. (eds.) Knowledge, Belief, and Strategic Interaction. (Cambridge Studies in Probability, Induction, and Decision Theory) Cambridge Universtity Press, Cambridge, pp. 275–284CrossRefGoogle Scholar
  32. [32]
    Mundici, D. (2001): Fault-tolerance and Rota-Metropolis cubic logic. In: Proceedings of the Second World Congress on Paraconsistency. Dekker, New YorkGoogle Scholar
  33. [33]
    Mundici, D., Trombetta, A. (1997): Optimal comparison strategies in Ulam’s searching game with two errors. Theoret. Comput. Sci. 182, 217–232MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Muthukrishnan, S. (1994): On optimal strategies for searching in presence of errors. In: Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms. Association for Computing Machinery, New York, pp. 680–689Google Scholar
  35. [35]
    Negro, A., Sereno, M. (1992): Ulam’s searching game with three lies. Adv. Appl. Math. 13, 404–428MathSciNetMATHCrossRefGoogle Scholar
  36. [36]
    Pedrotti, A. (1998): Reliable RAM computation in the presence of noise. PH.D. thesis, Scuola Normale Superiore, Pisa, ItalyGoogle Scholar
  37. [37]
    Pelc, A. (1987): Solution of Ulam’s problem on searching with a lie. J. Combin. Theory. Ser.A 44, 129–142MathSciNetMATHCrossRefGoogle Scholar
  38. [38]
    Pelc, A. (1989): Searching with known error probability. Theoret. Comput. Sci. 63, 185–202MathSciNetMATHCrossRefGoogle Scholar
  39. [39]
    Pelc, A. (1994): Searching with permanently faulty tests. Ars Combin. 38, 65–76Google Scholar
  40. [40]
    Pierce, J.R. (1978): Optical channels: practical limits with photon counting. IEEE Trans. Comm. 26, 1819–1821MathSciNetCrossRefGoogle Scholar
  41. [41]
    Rényi, A. (1976): Napló az informácioelméletrol. Gondolat, Budapest (Hungarian) [English transi.: A diary on information theory. Wiley, New York (1984)]Google Scholar
  42. [42]
    Rivest, R.L., Meyer, A.R., Kleitman, D.J., Winklmann, K., Spencer, J. (1980): Coping with errors in binary search procedures. J. Comput. System Sci. 20, 396–404MathSciNetMATHCrossRefGoogle Scholar
  43. [43]
    Rota, G.-C., Metropolis, N. (1978): Combinatorial structure of the faces of the n-cube. SIAM J. Appl. Math. 35, 689–694MathSciNetMATHCrossRefGoogle Scholar
  44. [44]
    Spencer, J. (1984): Guess a number-with lying. Math. Mag. 57, 105–108MathSciNetMATHCrossRefGoogle Scholar
  45. [45]
    Spencer, J. (1992): Ulam’s searching game with a fixed number of lies. Theoret. Comput. Sci. 95, 307–321MathSciNetMATHCrossRefGoogle Scholar
  46. [46]
    Spencer, J., Winkler, P. (1992): Three thresholds for a liar. Combin. Probab. Comput. 1, 81–93MathSciNetMATHCrossRefGoogle Scholar
  47. [47]
    Tietäväinen, A. (1973): On the nonexistence of perfect codes over finite fields. SIAM J. Appl. Math. 24, 88–96MathSciNetMATHCrossRefGoogle Scholar
  48. [48]
    Ulam, S.M. (1976): Adventures of a mathematician. Scribner’s, New YorkMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2001

Authors and Affiliations

  • F. Cicalese
  • D. Mundici
  • U. Vaccaro

There are no affiliations available

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