The Active Element Machine



A new computing machine, called an active element machine (AEM), and the AEM programming language are presented. This computing model is motivated by the positive aspects of dendritic integration, inspired by biology, and traditional programming languages based on the register machine. Distinct from the traditional register machine, the fundamental computing elements . active elements . compute simultaneously. Distinct from traditional programming languages, all active element commands have an explicit reference to time. These attributes make the AEM an inherently parallel machine, enable the AEM to change its architecture (program) as it is executing its program. Using a random bit source from the environment and the Meta command, we show how to generate an AEM that represents an arbitrary real number in [0,1]. Exploiting the randomness from the environment, this example is extended to an AEM that can recognize an arbitrary binary language L ⊆ {0,1}*. Finally, we demonstrate an AEM that finds the Ramsey number r(3,3), illustrating how parallel AEM algorithms and time in the commands help compute an NP-hard problem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abelson, H., Sussman, G.J.: Structure and Interpretation of Computer Programs. MIT Press, Cambridge (1996)MATHGoogle Scholar
  2. 2.
    Agnew, G.B.: Random sources for cryptographic systems. In: Price, W.L., Chaum, D. (eds.) EUROCRYPT 1987. LNCS, vol. 304, pp. 77–81. Springer, Heidelberg (1988)Google Scholar
  3. 3.
    Ajtai, M., Komlós, J., Szemerédi, E.: A note on Ramsey numbers. Journal Combinatorial Theory 29(3), 354–360 (1980)MATHCrossRefGoogle Scholar
  4. 4.
    Ajtai, M., Komlós, J., Szemerédi, E.: A dense infinite Sidon sequence. European J. Combin. 2(1), 1–11 (1981)MathSciNetMATHGoogle Scholar
  5. 5.
    Ajtai, M., Komlós, J., Szemerédi, E.: Sorting in c logn parallel steps. Combinatorica 3(1), 1–19 (1983)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Alon, N.: Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory. Combinatorica 6(3), 207–219 (1986)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Alon, N.: Explicit Ramsey graphs and orthonormal labelings. Electron. J. Combin. Research Paper 12 1, 8 (1994) (electronic)Google Scholar
  8. 8.
    Alon, N.: The Shannon capacity of a union. Combinatorica 18(3), 301–310 (1998)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Axelrod, R., Hamilton, W.D.: The Evolution of Cooperation. Science New Series 211(4489), 1390–1396 (1981)MathSciNetMATHGoogle Scholar
  10. 10.
    Burr, S.A.: Determining Generalized Ramsey Numbers is NP-hard. Ars Combinatoria 17, 21–25 (1984)MathSciNetMATHGoogle Scholar
  11. 11.
    Calude, C.S., Dinneen, M.J., Dumitrescu, M., Svozil, K.: Experimental Evidence of Quantum Randomness Incomputability. Phys. Rev. A 82, 022102, 1–8 (2010)CrossRefGoogle Scholar
  12. 12.
    Cook, S.A.: The complexity of theorem proving procedures. In: Proceedings, Third Annual ACM Symposium on the Theory of Computing, pp. 151–158. ACM, New York (1971)CrossRefGoogle Scholar
  13. 13.
    Davis, M.: Computability and Unsolvability. Dover Publications, New York (1982)MATHGoogle Scholar
  14. 14.
    Eberhart, R.C., Kennedy, J.: A new optimizer using particle swarm theory. In: Proceedings of the Sixth International Symposium on Micro Machine and Human Science, Nagoya, Japan, pp. 39–43 (1995)Google Scholar
  15. 15.
    Erdös, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, 464–470 (1935)Google Scholar
  16. 16.
    Erdös, P., Rado, R.: Combinatorial theorems on classifications of subsets of a given set. Proc. London Math. 3(2), 417–439 (1952)CrossRefGoogle Scholar
  17. 17.
    Fiske, M.S.: Machine Learning. US 7,249,116 B2 (2003),, or
  18. 18.
    Fiske, M.S.: Effector Machine Computation. US 7,398,260 B2 (2004), Provisional 60/456,715 (2003), or
  19. 19.
    Fiske, M.S.: Active Element Machine Computation. US Application 20070288668 (2007), or
  20. 20.
    Furstenberg, H.: Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981)MATHGoogle Scholar
  21. 21.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman (1979)Google Scholar
  22. 22.
    Graham, R.L., Rodl, V.: Numbers in Ramsey Theory. Surveys in Combinatorics. LMS Lecture Note Series 123. Cambridge University Press (1987)Google Scholar
  23. 23.
    Graham, R.L., Rothschild, B.L.: A Survey of Finite Ramsey Theorems. In: Proc. 2nd Louisiana State Univ. Conference on Combinatorics, Graph Theory and Computation, pp. 21–40 (1971)Google Scholar
  24. 24.
    Hales, A.W., Jewett, R.I.: Regularity and positional games. Trans. Amer. Math. Soc. 106, 222–229 (1963)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Hertz, J., Krogh, A., Palmer, R.G.: Introduction To The Theory of Neural Computation. Addison-Wesley Publishing Company, California (1991)Google Scholar
  26. 26.
    Holland, J.H.: Outline for a logical theory of adaptive systems. JACM 3, 297–314 (1962)CrossRefGoogle Scholar
  27. 27.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. MIT Press, Cambridge (1992)Google Scholar
  28. 28.
    Quantique, I.D., Quantis, S.: Random Number Generation Using Quantum Optics (electronic), Geneva, Switzerland (2001-2011),
  29. 29.
    Koza, J.R.: Genetic Programming: On the Programming of Computer by Means of Natural Selection. MIT Press, Cambridge (1992)Google Scholar
  30. 30.
    Lee, E.A.: Computing Needs Time. Technical Report No. UCB/EECS-2009-30 (electronic). Electrical Engineering and Computer Sciences, University of California at Berkeley (2009),
  31. 31.
    McCulloch, W.S.: Walter Pitts. A logical calculus immanent in nervous activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    McKay, B.D., Radziszowski, S.P.: Subgraph counting identities and Ramsey numbers. Journal Combinatorial Theory 69(2), 193–209 (1997)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Minsky, M.: Computation: Finite and Infinite Machines, 1st edn. Prentice-Hall, Inc., Englewood Cliffs (1967)MATHGoogle Scholar
  34. 34.
    Paris, J., Harrington, L.: A mathematical incompleteness in Peano arithmetic. In: Barwise, J. (ed.) Handbook for Mathematical Logic. North-Holland (1977)Google Scholar
  35. 35.
    Rall, W.: The Theoretical Foundation of Dendritic Function. In: Segev, I., Rinzel, J., Shepherd, G. (eds.) Selected Papers of Wilfrid Rall with Commentaries. MIT Press, Cambridge (1995)Google Scholar
  36. 36.
    Ramsey, F.P.: On a problem of formal logic. Proc. London Math. Soc. Series 2(30), 264–286 (1930)CrossRefGoogle Scholar
  37. 37.
    Roberts, F.S.: Applications of Ramsey theory. Discrete Appl. Math. 9(3), 251–261 (1984)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Robinson, A.: Non-standard Analysis. Princeton University Press, Princeton (1996), Revised EditionMATHGoogle Scholar
  39. 39.
    Rothschild, B.L.: A generalization of Ramsey’s theorem and a conjecture of Erdös. Doctoral Dissertation. Yale University, New Haven, Connecticut (1967)Google Scholar
  40. 40.
    Schur, I.: Uber die Kongruenz x m + y m ≡ z m ( mod p). Deutsche Math. 25, 114–117 (1916)MATHGoogle Scholar
  41. 41.
    Shannon, C.E.: The zero error capacity of a noisy channel. Transactions on Information Theory Institute of Radio Engineers IT-2, 8–19 (1956)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Spencer, J.H.: Ten Lectures on the Probabilistic Method, p. 4. SIAM (1994)Google Scholar
  43. 43.
    Stefanov, A., Gisin, N., Guinnard, O., Guinnard, L., Zbinden, H.: Optical quantum random number generator. Journal of Modern Optics 10, 1362–3044 (2000)Google Scholar
  44. 44.
    Sturgis, H.E., Shepherdson, J.C.: Computability of Recursive Functions. J. Assoc. Comput. Mach. 10, 217–255 (1963)MathSciNetMATHGoogle Scholar
  45. 45.
    Turing, A.M.: On computable numbers, with an application to the Entscheidungsproblem. Proc. London Math. Soc. Series 2(42) (Parts 3 and 4), 230–265 (1936); A correction, ibid. 43, 544–546 (1937)Google Scholar
  46. 46.
    Yao, A.C.C.: Should tables be sorted? J. Assoc. Comput. Mach. 28(3), 615–628 (1981)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Aemea InstituteSan FranciscoUSA

Personalised recommendations