A Survey on Continuous Time Computations

  • Olivier Bournez
  • Manuel L. Campagnolo

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature.


Hybrid System Continuous Time Turing Machine Initial Value Problem Computable Function 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aaronson, S. (2005). NP-complete problems and physical reality. ACM SIGACT News, 36(1):30-52.Google Scholar
  2. 2.
    Abdi, H. (1994). A neural network primer. Journal of Biological Systems, 2:247-281.Google Scholar
  3. 3.
    Adleman, L. M. (1994). Molecular computation of solutions to combinatorial problems. Science, 266:1021-1024.Google Scholar
  4. 4.
    Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T. A., Ho, P. H., Nicollin, X., Oliv-ero, A., Sifakis, J., and Yovine, S. (1995). The algorithmic analysis of hybrid systems. Theoretical Computer Science, 138(1):3-34.MATHMathSciNetGoogle Scholar
  5. 5.
    Alur, R. and Dill, D. L. (1990). Automata for modeling real-time systems. In Pater-son, M., editor, Automata, Languages and Programming, 17th International Colloquium, ICALP90, Warwick University, England, July 16-20, 1990, Proceedings, volume 443 of Lecture Notes in Computer Science, pages 322-335. Springer.Google Scholar
  6. 6.
    Alur, R. and Dill, D. L. (1994). A theory of timed automata. Theoretical Computer Science, 126(2):183-235.MATHMathSciNetGoogle Scholar
  7. 7.
    Alur, R. and Madhusudan, P. (2004). Decision problems for timed automata: A survey. In Bernardo, M. and Corradini, F., editors, Formal Methods for the Design of Real-Time Systems, International School on Formal Methods for the Design of Computer, Com-munication and Software Systems, SFM-RT 2004, Bertinoro, Italy, September 13-18, 2004, Revised Lectures, volume 3185 of Lecture Notes in Computer Science, pages 1-24. Springer.Google Scholar
  8. 8.
    Arnold, V. I. (1989). Mathematical methods of classical mechanics, volume 60 of Grad-uate Texts in Mathematics. Springer, second edition.Google Scholar
  9. 9.
    Artobolevskii, I. (1964). Mechanisms for the Generation of Plane Curves. Macmillan, New York. Translated by R.D. Wills and W. Johnson.Google Scholar
  10. 10.
    Asarin (2004). Challenges in timed languages: From applied theory to basic theory. Bulletin of the European Association for Theoretical Computer Science, 83:106-120.MATHMathSciNetGoogle Scholar
  11. 11.
    Asarin, E. (1995). Chaos and undecidabilty (draft). Avalaible in http://www. liafa.jussieu.fr/$\tilde{\}$asarin/.
  12. 12.
    Asarin, E. (1998). Equations on timed languages. In Henzinger, T. A. and Sastry, S., editors, Hybrid Systems: Computation and Control, First International Workshop, HSCC’98, Berkeley, CA, April 13-15, 1998, Proceedings, volume 1386 of Lecture Notes in Computer Science, pages 1-12. Springer.Google Scholar
  13. 13.
    Asarin, E. (2006). Noise and decidability. Continuous Dynamics and Computabil-ity Colloquium. Video and sound available trough “Diffusion des savoirs de l’Ecole Normale Supérieure,” on http://www.diffusion.ens.fr/en/index.php? res=conf\&idconf=1226.
  14. 14.
    Asarin, E. and Bouajjani, A. (2001). Perturbed Turing machines and hybrid systems. In Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science, pages 269-278, Los Alamitos, CA. IEEE Computer Society Press.Google Scholar
  15. 15.
    Asarin, E., Caspi, P., and Maler, O. (1997). A Kleene theorem for timed automata. In Proceedings, 12th Annual IEEE Symposium on Logic in Computer Science, pages 160-171, Warsaw, Poland. IEEE Computer Society Press.Google Scholar
  16. 16.
    Asarin, E., Caspi, P., and Maler, O. (2002). Timed regular expressions. Journal of the ACM, 49(2):172-206.MathSciNetGoogle Scholar
  17. 17.
    Asarin, E. and Collins, P. (2005). Noisy Turing machines. In Caires, L., Italiano, G. F., Monteiro, L., Palamidessi, C., and Yung, M., editors, Automata, Languages and Pro-gramming, 32nd International Colloquium, ICALP 2005, Lisbon, Portugal, July 11-15, 2005, Proceedings, volume 3580 of Lecture Notes in Computer Science, pages 1031-1042. Springer.Google Scholar
  18. 18.
    Asarin, E. and Dima, C. (2002). Balanced timed regular expressions. Electronic Notes in Theoretical Computer Science, 68(5).Google Scholar
  19. 19.
    Asarin, E. and Maler, O. (1998). Achilles and the tortoise climbing up the arithmetical hierarchy. Journal of Computer and System Sciences, 57(3):389-398.MATHMathSciNetGoogle Scholar
  20. 20.
    Asarin, E., Maler, O., and Pnueli, A. (1995). Reachability analysis of dynamical systems having piecewise-constant derivatives. Theoretical Computer Science, 138(1):35-65.MATHMathSciNetGoogle Scholar
  21. 21.
    Asarin, E. and Schneider, G. (2002). Widening the boundary between decidable and undecidable hybrid systems. In Brim, L., Jancar, P., Kretínský, M., and Kucera, A., editors, CONCUR 2002 - Concurrency Theory, 13th International Conference, Brno, Czech Republic, August 20-23, 2002, Proceedings, volume 2421 of Lecture Notes in Computer Science, pages 193-208. Springer.Google Scholar
  22. 22.
    Asarin, E., Schneider, G., and Yovine, S. (2001). On the decidability of the reachability problem for planar differential inclusions. In Benedetto, M. D. D. and Sangiovanni-Vincentelli, A. L., editors, Hybrid Systems: Computation and Control, 4th International Workshop, HSCC 2001, Rome, Italy, March 28-30, 2001, Proceedings, volume 2034 of Lecture Notes in Computer Science, pages 89-104. Springer.Google Scholar
  23. 23.
    Beauquier, D. (1998). Pumping lemmas for timed automata. In Nivat, M., editor, Foun-dations of Software Science and Computation Structure, First International Conference, FoSSaCS’98, Held as Part of the European Joint Conferences on the Theory and Prac-tice of Software, ETAPS’98, Lisbon, Portugal, March 28 - April 4, 1998, Proceedings, volume 1378 of Lecture Notes in Computer Science, pages 81-94. Springer.Google Scholar
  24. 24.
    Ben-Hur, A., Feinberg, J., Fishman, S., and Siegelmann, H. T. (2003). Probabilistic anal-ysis of a differential equation for linear programming. Journal of Complexity, 19(4):474-510.MathSciNetGoogle Scholar
  25. 25.
    Ben-Hur, A., Feinberg, J., Fishman, S., and Siegelmann, H. T. (2004a). Random ma-trix theory for the analysis of the performance of an analog computer: a scaling theory. Physics Letters A, 323(3-4):204-209.MATHMathSciNetGoogle Scholar
  26. 26.
    Ben-Hur, A., Roitershtein, A., and Siegelmann, H. T. (2004b). On probabilistic analog automata. Theoretical Computer Science, 320(2-3):449-464.MATHMathSciNetGoogle Scholar
  27. 27.
    Ben-Hur, A., Siegelmann, H. T., and Fishman, S. (2002). A theory of complexity for continuous time systems. Journal of Complexity, 18(1):51-86.MATHMathSciNetGoogle Scholar
  28. 28.
    Blondel, V. D. and Tsitsiklis, J. N. (1999). Complexity of stability and controllability of elementary hybrid systems. Automatica, 35(3):479-489.MATHMathSciNetGoogle Scholar
  29. 29.
    Blondel, V. D. and Tsitsiklis, J. N. (2000). A survey of computational complexity results in systems and control. Automatica, 36(9):1249-1274.MATHMathSciNetGoogle Scholar
  30. 30.
    Blum, L., Cucker, F., Shub, M., and Smale, S. (1998). Complexity and Real Computation. Springer.Google Scholar
  31. 31.
    Blum, L., Shub, M., and Smale, S. (1989). On a theory of computation and complexity over the real numbers; NP completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society, 21(1):1-46.MATHMathSciNetGoogle Scholar
  32. 32.
    Bournez, O. (1999a). Achilles and the Tortoise climbing up the hyper-arithmetical hier-archy. Theoretical Computer Science, 210(1):21-71.MATHMathSciNetGoogle Scholar
  33. 33.
    Bournez, O. (1999b). Complexité Algorithmique des Systèmes Dynamiques Continus et Hybrides. PhD thesis, Ecole Normale Supérieure de Lyon.Google Scholar
  34. 34.
    Bournez, O. (2006). How much can analog and hybrid systems be proved (super-)Turing. Applied Mathematics and Computation, 178(1):58-71.MATHMathSciNetGoogle Scholar
  35. 35.
    Bournez, O., Campagnolo, M. L., Graça, D. S., and Hainry, E. (2007). Polynomial differential equations compute all real computable functions on computable compact intervals. Journal of Complexity. To appear.Google Scholar
  36. 36.
    Bournez, O. and Hainry, E. (2005). Elementarily computable functions over the real numbers and R-sub-recursive functions. Theoretical Computer Science, 348(2-3): 130-147.MATHMathSciNetGoogle Scholar
  37. 37.
    Bournez, O. and Hainry, E. (2006). Recursive analysis characterized as a class of real recursive functions. Fundamenta Informaticae, 74(4):409-433.MATHMathSciNetGoogle Scholar
  38. 38.
    Bouyer, P., Dufourd, C., Fleury, E., and Petit, A. (2000a). Are timed automata updatable? In Emerson, E. A. and Sistla, A. P., editors, Computer Aided Verification, 12th Interna-tional Conference, CAV 2000, Chicago, IL, July 15-19, 2000, Proceedings, volume 1855 of Lecture Notes in Computer Science, pages 464-479. Springer.Google Scholar
  39. 39.
    Bouyer, P., Dufourd, C., Fleury, E., and Petit, A. (2000b). Expressiveness of updatable timed automata. In Nielsen, M. and Rovan, B., editors, Mathematical Foundations of Computer Science 2000, 25th International Symposium, MFCS 2000, Bratislava, Slo-vakia, August 28 - September 1, 2000, Proceedings, volume 1893 of Lecture Notes in Computer Science, pages 232-242. Springer.Google Scholar
  40. 40.
    Bouyer, P. and Petit, A. (1999). Decomposition and composition of timed automata. In Wiedermann, J., van Emde Boas, P., and Nielsen, M., editors, Automata, Languages and Programming, 26th International Colloquium, ICALP’99, Prague, Czech Republic, July 11-15, 1999, Proceedings, volume 1644 of Lecture Notes in Computer Science, pages 210-219. Springer.Google Scholar
  41. 41.
    Bouyer, P. and Petit, A. (2002). A Kleene/Büchi-like theorem for clock languages. Jour-nal of Automata, Languages and Combinatorics, 7(2):167-186.MATHMathSciNetGoogle Scholar
  42. 42.
    Bowles, M. D. (1996). U.S. technological enthusiasm and British technological skepticism in the age of the analog brain. IEEE Annals of the History of Computing, 18(4): 5-15.Google Scholar
  43. 43.
    Branicky, M. S. (1995a). Studies in Hybrid Systems: Modeling, Analysis, and Control. PhD thesis, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA.Google Scholar
  44. 44.
    Branicky, M. S. (1995b). Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoretical Computer Science, 138(1):67-100.MATHMathSciNetGoogle Scholar
  45. 45.
    Brihaye, T. (2006). A note on the undecidability of the reachability problem for o-minimal dynamical systems. Math. Log. Q, 52(2):165-170.MATHMathSciNetGoogle Scholar
  46. 46.
    Brihaye, Th. and Michaux, Ch. (2005). On the expressiveness and decidability of o-minimal hybrid systems. Journal of Complexity, 21(4):447-478.MATHMathSciNetGoogle Scholar
  47. 47.
    Brockett, R. W. (1989). Smooth dynamical systems which realize arithmetical and logical operations. In Nijmeijer, H. and Schumacher, J. M., editors, Three Decades of Mathemat-ical Systems Theory, volume 135 of Lecture Notes in Computer Science, pages 19-30. Springer.Google Scholar
  48. 48.
    Brockett, R. W. (1991). Dynamical systems that sort lists, diagonalize matrices, and solve linear programming problems. Linear Algebra and its Applications, 146:79-91.MATHMathSciNetGoogle Scholar
  49. 49.
    Brockett, R. W. (1994). Dynamical systems and their associated automata. In U. Helmke, R. M. and Saurer, J., editors, Systems and Networks: Mathematical Theory and Applications, volume 77, pages 49-69. Akademi-Verlag, Berlin.Google Scholar
  50. 50.
    Bush, V. (1931). The differential analyser. Journal of the Franklin Institute, 212(4): 447-488.MATHGoogle Scholar
  51. 51.
    Calude, C. S. and Pavlov, B. (2002). Coins, Quantum measurements, and Turing’s bar-rier. Quantum Information Processing, 1(1-2):107-127.MathSciNetGoogle Scholar
  52. 52.
    Campagnolo, M., Moore, C., and Costa, J. F. (2000). Iteration, inequalities, and differ-entiability in analog computers. Journal of Complexity, 16(4):642-660.MATHMathSciNetGoogle Scholar
  53. 53.
    Campagnolo, M., Moore, C., and Costa, J. F. (2002). An analog characterization of the Grzegorczyk hierarchy. Journal of Complexity, 18(4):977-1000.MATHMathSciNetGoogle Scholar
  54. 54.
    Campagnolo, M. L. (2001). Computational complexity of real valued recursive functions and analog circuits. PhD thesis, IST, Universidade Técnica de Lisboa.Google Scholar
  55. 55.
    Campagnolo, M. L. (2002). The complexity of real recursive functions. In Calude, C., Dinneen, M., and Peper, F., editors, Unconventional Models of Computation, UMC’02, Volume 2509 in Lecture Notes in Computer Science, pages 1-14. Springer.Google Scholar
  56. 56.
    Campagnolo, M. L. (2004). Continuous time computation with restricted integration capabilities. Theoretical Computer Science, 317(4):147-165.MATHMathSciNetGoogle Scholar
  57. 57.
    Campagnolo, M. L. and Ojakian, K. (2007). The elementary computable functions over the real numbers: applying two new techniques. Archive for Mathematical Logic. To appear.Google Scholar
  58. 58.
    Casey, M. (1996). The dynamics of discrete-time computation, with application to re-current neural networks and finite state machine extraction. Neural Computation, 8: 1135-1178.Google Scholar
  59. 59.
    Casey, M. (1998). Correction to proof that recurrent neural networks can robustly recog-nize only regular languages. Neural Computation, 10:1067-1069.Google Scholar
  60. 60.
    Ceraens, K. and Viksna, J. (1996). Deciding reachability for planar multi-polynomial systems. In Hybrid Systems III, volume 1066 of Lecture Notes in Computer Science, page 389. Springer-Verlag.Google Scholar
  61. 61.
    Church, A. (1936). An unsolvable problem of elementary number theory. American Journal of Mathematics,, 58:345-363. Reprinted in [73].Google Scholar
  62. 62.
    Clote, P. (1998). Computational models and function algebras. In Griffor, E. R., editor, Handbook of Computability Theory, pages 589-681. North-Holland, Amsterdam.Google Scholar
  63. 63.
    Coddington, E. A. and Levinson, N. (1972). Theory of Ordinary Differentiel Equations. McGraw-Hill.Google Scholar
  64. 64.
    Collins, P. (2005). Continuity and computability on reachable sets. Theoretical Computer Science, 341:162-195.MATHMathSciNetGoogle Scholar
  65. 65.
    Collins, P. and Lygeros, J. (2005). Computability of finite-time reachable sets for hybrid systems. In Proceedings of the 44th IEEE Conference on Decision and Control and the European Control Conference, pages 4688-4693. IEEE Computer Society Press.Google Scholar
  66. 66.
    Collins, P. and van Schuppen, J. H. (2004). Observability of piecewise-affine hybrid sys-tems. In Alur, R. and Pappas, G. J., editors, Hybrid Systems: Computation and Control, 7th International Workshop, HSCC 2004, Philadelphia, PA, March 25-27, 2004, Pro-ceedings, volume 2993 of Lecture Notes in Computer Science, pages 265-279. Springer.Google Scholar
  67. 67.
    Copeland, B. J. (1998). Even Turing machines can compute uncomputable functions. In Calude, C., Casti, J., and Dinneen, M., editors, Unconventional Models of Computations. Springer.Google Scholar
  68. 68.
    Copeland, B. J. (2002). Accelerating Turing machines. Minds and Machines, 12: 281-301.MATHGoogle Scholar
  69. 69.
    Costa, J. F. and Mycka, J. (2006). The conjecture P = NP given by some analytic condition. In Bekmann, A., Berger, U., Löwe, B., and Tucker, J., editors, Logical Approaches to Computational Barriers, Second conference on Computability in Europe, CiE 2006, pages 47-57, Swansea, UK. Report CSR 7-26, Report Series, University of Wales Swansea Press, 2006.Google Scholar
  70. 70.
    Coward, D. (2006). Doug Coward’s Analog Computer Museum. http://dcoward best.vwh.net/analog/.
  71. 71.
    Davies, E. B. (2001). Building infinite machines. The British Journal for the Philosophy of Science, 52:671-682.MATHGoogle Scholar
  72. 72.
    Dee, D. and Ghil, M. (1984). Boolean difference equations, I: Formulation and dynamic behavior. SIAM Journal on Applied Mathematics, 44(1):111-126.MATHMathSciNetGoogle Scholar
  73. 73.
    Davis M. (ed.) (1965) The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions, Raven, NY.Google Scholar
  74. 74.
    Delvenne, J.-C., Kurka, P., and Blondel, V. D. (2004). Computational universality in symbolic dynamical systems. In Margenstern, M., editor, MCU: International Confer-ence on Machines, Computations, and Universality, volume 3354 of Lecture Notes in Computer Science, pages 104-115. Springer.Google Scholar
  75. 75.
    Deutsch, D. (1985). Quantum theory, the Church-Turing principle and the universal quantum computer. Proceedings of the Royal Society (London), Series A, 400:97-117.MATHMathSciNetGoogle Scholar
  76. 76.
    Durand-Lose, J. (2005). Abstract geometrical computation: Turing-computing ability and undecidability. In Cooper, S. B., Löwe, B., and Torenvliet, L., editors, New Com-putational Paradigms, First Conference on Computability in Europe, CiE 2005, Amster-dam, The Netherlands, June 8-12, 2005, Proceedings, volume 3526 of Lecture Notes in Computer Science, pages 106-116. Springer.Google Scholar
  77. 77.
    Earman, J. and Norton, J. D. (1993). Forever is a day: Supertasks in Pitowksy and Malament-Hogarth spacetimes. Philosophy of Science, 60(1):22-42.MathSciNetGoogle Scholar
  78. 78.
    Etesi, G. and Németi, I. (2002). Non-Turing computations via Malament-Hogarth space-times. International Journal Theoretical Physics, 41:341-370.MATHGoogle Scholar
  79. 79.
    Faybusovich, L. (1991a). Dynamical systems which solve optimization problems with linear constraints. IMA Journal of Mathematical Control and Information, 8:135-149.MATHMathSciNetGoogle Scholar
  80. 80.
    Faybusovich, L. (1991b). Hamiltonian structure of dynamical systems which solve linear programming problems. Physics, D53:217-232.MathSciNetGoogle Scholar
  81. 81.
    Filippov, A. (1988). Differential equations with discontinuous right-hand sides. Kluwer Academic Publishers.Google Scholar
  82. 82.
    Finkel, O. (2006). On the shuffle of regular timed languages. Bulletin of the European Association for Theoretical Computer Science, 88:182-184. Technical Contributions.Google Scholar
  83. 83.
    Foy, J. (2004). A dynamical system which must be stable whose stability cannot be proved. Theoretical Computer Science, 328(3):355-361.MATHMathSciNetGoogle Scholar
  84. 84.
    Francisco, A. P. L. (2002). Finite automata over continuous time. Diploma Thesis. Universidade Técnica de Lisboa, Instituto Superior Técnico.Google Scholar
  85. 85.
    Fränzle, M. (1999). Analysis of hybrid systems: An ounce of realism can save an infin-ity of states. In Flum, J. and Rodríguez-Artalejo, M., editors, Computer Science Logic (CSL’99), volume 1683 of Lecture Notes in Computer Science, pages 126-140. Springer Verlag.Google Scholar
  86. 86.
    Gori, M. and Meer, K. (2002). A step towards a complexity theory for analog systems. Mathematical Logic Quarterly, 48(Suppl. 1):45-58.MATHMathSciNetGoogle Scholar
  87. 87.
    Graça, D. (2002). The general purpose analog computer and recursive functions over the reals. Master’s thesis, IST, Universidade Técnica de Lisboa.Google Scholar
  88. 88.
    Graça, D. S. (2004). Some recent developments on Shannon’s general purpose analog computer. Mathematical Logic Quarterly, 50(4-5):473-485.MATHGoogle Scholar
  89. 89.
    Graça, D. S. and Costa, J. F. (2003). Analog computers and recursive functions over the reals. Journal of Complexity, 19(5):644-664.MATHMathSciNetGoogle Scholar
  90. 90.
    Graça, D., Campagnolo, M., and Buescu, J. (2005). Robust simulations of Turing ma-chines with analytic maps and flows. In Cooper, B., Loewe, B., and Torenvliet, L., editors, Proceedings of CiE’05, New Computational Paradigms, volume 3526 of Lecture Notes in Computer Science, pages 169-179. Springer.Google Scholar
  91. 91.
    Graça, D. S., Campagnolo, M. L., and Buescu, J. (2007). Computability with polynomial differential equations. Advances in Applied Mathematics. To appear.Google Scholar
  92. 92.
    Graça, D. S., Zhong, N., and Buescu, J. (2006). Computability, noncomputability and undecidability of maximal intervals of IVPs. Transactions of the American Mathematical Society. To appear.Google Scholar
  93. 93.
    Grigorieff, S. and Margenstern, M. (2004). Register cellular automata in the hyperbolic plane. Fundamenta Informaticae, 1(61):19-27.MathSciNetGoogle Scholar
  94. 94.
    Gruska, J. (1997). Foundations of Computing. International Thomson Publishing.Google Scholar
  95. 95.
    Gupta, V., A., T., and Jagadeesan, R. (1997). Robust timed automata. In Maler, O., editor, Hybrid and Real-Time Systems, International Workshop. HART’97, Grenoble, France, March 26-28, 1997, Proceedings, volume 1201 of Lecture Notes in Computer Science, pages 331-345. Springer.Google Scholar
  96. 96.
    Head, T. (1987). Formal language theory and DNA: An analysis of the generative capac-ity of specific recombinant behaviors. Bulletin of Mathematical Biology, 49:737-759.MATHMathSciNetGoogle Scholar
  97. 97.
    Helmke, U. and Moore, J. (1994). Optimization and Dynamical Systems. Communica-tions and Control Engineering Series. Springer Verlag, London.Google Scholar
  98. 98.
    Henzinger, T. A., Kopke, P. W., Puri, A., and Varaiya, P. (1998). What’s decidable about hybrid automata? Journal of Computer and System Sciences, 57(1):94-124.MATHMathSciNetGoogle Scholar
  99. 99.
    Henzinger, T. A. and Raskin, J.-F. (2000). Robust undecidability of timed and hybrid systems. In Lynch, N. A. and Krogh, B. H., editors, Hybrid Systems: Computation and Control, Third International Workshop, HSCC 2000, Pittsburgh, PA, March 23-25, 2000, Proceedings, volume 1790 of Lecture Notes in Computer Science, pages 145-159. Springer.Google Scholar
  100. 100.
    Hirsch, M. W., Smale, S., and Devaney, R. (2003). Differential Equations, Dynamical Systems, and an Introduction to Chaos. Elsevier Academic Press.Google Scholar
  101. 101.
    Hogarth, M. (1994). Non-Turing computers and non-Turing computability. In Proceedings of the Philosophy of Science Association (PSA’94), volume 1, pages 126-138.Google Scholar
  102. 102.
    Hogarth, M. (1996). Predictability, Computability and Spacetime. PhD thesis, Sidney Sussex College, Cambridge.Google Scholar
  103. 103.
    Hogarth, M. (2006). Non-Turing computers are the new non-Eucliedean geometries. In Future Trends in Hypercomputation. Sheffield, 11-13 September 2006. Available for download on www.hypercomputation.net.
  104. 104.
    Hogarth, M. L. (1992). Does general relativity allow an observer to view an eternity in a finite time? Foundations of Physics Letters, 5:173-181.MathSciNetGoogle Scholar
  105. 105.
    Hopfield, J. J. (1984). Neural networks with graded responses have collective computational properties like those of two-state neurons. Proceedings of the National Academy of Sciences of the United States of America, 81:3088-3092.Google Scholar
  106. 106.
    Hopfield, J. J. and Tank, D. W. (1985). ‘Neural’ computation of decisions in optimization problems. Biological Cybernetics, 52:141-152.MATHMathSciNetGoogle Scholar
  107. 107.
    Hoyrup, M. (2006). Dynamical systems: stability and simulability. Technical report, Département d’Informatique, ENS Paris.Google Scholar
  108. 108.
    Kempe, A. (1876). On a general method of describing plane curves of the n-th degree by linkwork. Proceedings of the London Mathematical Society, 7:213-216.Google Scholar
  109. 109.
    Kieu, T. D. (2004). Hypercomputation with quantum adiabatic processes. Theoretical Computer Science, 317(1-3):93-104.MATHMathSciNetGoogle Scholar
  110. 110.
    Kleene, S. C. (1936). General recursive functions of natural numbers. Mathematical Annals, 112:727-742. Reprinted in [73].Google Scholar
  111. 111.
    Ko, K.-I. (1983). On the computational complexity of ordinary differential equations. Information and Control, 58(1-3):157-194.MATHMathSciNetGoogle Scholar
  112. 112.
    Ko, K.-I. (1991). Complexity Theory of Real Functions. Progress in Theoretical Computer Science. Birkhäuser, Boston.MATHGoogle Scholar
  113. 113.
    Koiran, P. (2001). The topological entropy of iterated piecewise affine maps is uncomputable. Discrete Mathematics & Theoretical Computer Science, 4(2):351-356.MATHGoogle Scholar
  114. 114.
    Koiran, P., Cosnard, M., and Garzon, M. (1994). Computability with low-dimensional dynamical systems. Theoretical Computer Science, 132(1-2):113-128.MATHMathSciNetGoogle Scholar
  115. 115.
    Koiran, P. and Moore, C. (1999). Closed-form analytic maps in one and two dimensions can simulate universal Turing machines. Theoretical Computer Science, 210(1):217-223.MATHMathSciNetGoogle Scholar
  116. 116.
    Korovina, M. V. and Vorobjov, N. (2004). Pfaffian hybrid systems. In Marcinkowski, J. and Tarlecki, A., editors, Computer Science Logic, 18th International Workshop, CSL 2004, 13th Annual Conference of the EACSL, Karpacz, Poland, September 20-24, 2004, Proceedings, volume 3210 of Lecture Notes in Computer Science, pages 430-441. Springer.Google Scholar
  117. 117.
    Korovina, M. V. and Vorobjov, N. (2006). Upper and lower bounds on sizes of finite bisimulations of Pfaffian hybrid systems. In Beckmann, A., Berger, U., Löwe, B., and Tucker, J. V., editors, Logical Approaches to Computational Barriers, Second Conference on Computability in Europe, CiE 2006, Swansea, UK, June 30-July 5, 2006, Proceedings, volume 3988 of Lecture Notes in Computer Science, pages 267-276. Springer.Google Scholar
  118. 118.
    Kurganskyy, O. and Potapov, I. (2005). Computation in one-dimensional piecewise maps and planar pseudo-billiard systems. In Calude, C., Dinneen, M. J., Paun, G., Pérez-Jiménez, M. J., and Rozenberg, G., editors, Unconventional Computation, 4th Interna-tional Conference, UC 2005, Sevilla, Spain, October 3-7, 2005, Proceedings, volume 3699 of Lecture Notes in Computer Science, pages 169-175. Springer.Google Scholar
  119. 119.
    Lafferriere, G. and Pappas, G. J. (2000). O-minimal hybrid systems. Mathematics of Control, Signals, and Systems, 13:1-21.MATHMathSciNetGoogle Scholar
  120. 120.
    Legenstein, R. and Maass, W. (2007). What makes a dynamical system computationally powerful? In Haykin, S., Principe, J. C., Sejnowski, T., and McWhirter, J., editors, New Directions in Statistical Signal Processing: From Systems to Brain, pages 127-154. MIT Press, Cambridge, MA.Google Scholar
  121. 121.
    Lipshitz, L. and Rubel, L. A. (1987). A differentially algebraic replacement theorem, and analog computability. Proceedings of the American Mathematical Society, 99(2): 367-372.MATHMathSciNetGoogle Scholar
  122. 122.
    Lipton, R. J. (1995). DNA solution of hard computational problems. Science, 268: 542-545.Google Scholar
  123. 123.
    Loff, B. (2007). A functional characterisation of the analytical hierarchy. In Computability in Europe 2007: Computation and Logic in the Real World.Google Scholar
  124. 124.
    Loff, B., Costa, J. F., and Mycka, J. (2007a). Computability on reals, infinite limits and differential equations. Applied Mathematics and Computation. To appear.Google Scholar
  125. 125.
    Loff, B., Costa, J. F., and Mycka, J. (2007b). The new promise of analog computation. In Computability in Europe 2007: Computation and Logic in the Real World.Google Scholar
  126. 126.
    Maass, W. (1996a). Lower bounds for the computational power of networks of spiking neurons. Neural Computation, 8(1):1-40.MATHMathSciNetGoogle Scholar
  127. 127.
    Maass, W. (1996b). On the computational power of noisy spiking neurons. In Touretzky, D., Mozer, M. C., and Hasselmo, M. E., editors, Advances in Neural Information Processing Systems, volume 8, pages 211-217. MIT Press, Cambridge, MA.Google Scholar
  128. 128.
    Maass, W. (1997a). A model for fast analog computations with noisy spiking neurons. In Bower, J., editor, Computational Neuroscience: Trends in research, pages 123-127.Google Scholar
  129. 129.
    Maass, W. (1997b). Networks of spiking neurons: the third generation of neural network models. Neural Networks, 10:1659-1671.Google Scholar
  130. 130.
    Maass, W. (1999). Computing with spiking neurons. In Maass, W. and Bishop, C. M., editors, Pulsed Neural Networks, pages 55-85. MIT Press, Cambridge, MA.Google Scholar
  131. 131.
    Maass, W. (2002). Computing with spikes. Special Issue on Foundations of Information Processing of TELEMATIK, 8(1):32-36.Google Scholar
  132. 132.
    Maass, W. (2003). Computation with spiking neurons. In Arbib, M. A., editor, The Handbook of Brain Theory and Neural Networks, pages 1080-1083. MIT Press, Cambridge, MA. 2nd edition.Google Scholar
  133. 133.
    Maass, W. and Bishop, C. (1998). Pulsed Neural Networks. MIT Press, Cambridge, MA.MATHGoogle Scholar
  134. 134.
    Maass, W., Joshi, P., and Sontag, E. D. (2007). Computational aspects of feedback in neural circuits. Public Library of Science Computational Biology, 3(1):1-20. e165.MathSciNetGoogle Scholar
  135. 135.
    Maass, W. and Natschläger, T. (2000). A model for fast analog computation based on unreliable synapses. Neural Computation, 12(7):1679-1704. Google Scholar
  136. 136.
    Maass, W. and Orponen, P. (1998). On the effect of analog noise in discrete-time analog computations. Neural Computation, 10(5):1071-1095.Google Scholar
  137. 137.
    Maass, W. and Ruf, B. (1999). On computation with pulses. Information and Computation, 148(2):202-218.MATHMathSciNetGoogle Scholar
  138. 138.
    Maass, W. and Sontag, E. (1999). Analog neural nets with gaussian or other common noise distributions cannot recognize arbitrary regular languages. Neural Computation, 11(3):771-782.Google Scholar
  139. 139.
    MacLennan, B. J. (2001). Can differential equations compute? citeseer.ist. psu.edu/maclennan01can.html.Google Scholar
  140. 140.
    Mills, J. (1995). Programmable VLSI extended analog computer for cyclotron beam control. Technical Report 441, Indiana University Computer Science.Google Scholar
  141. 141.
    Mills, J. W., Himebaugh, B., Allred, A., Bulwinkle, D., Deckard, N., Gopalakrishnan, N., Miller, J., Miller, T., Nagai, K., Nakamura, J., Ololoweye, B., Vlas, R., Whitener, P., Ye, M., , and Zhang, C. (2005). Extended analog computers: A unifying paradigm for VLSI, plastic and colloidal computing systems. In Workshop on Unique Chips and Systems (UCAS-1). Held in conjunction with IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS05), Austin, Texas.Google Scholar
  142. 142.
    Müller, N. and Moiske, B. (1993). Solving initial value problems in polynomial time. In Proc. 22 JAIIO - PANEL ’93, Part 2, pages 283-293.Google Scholar
  143. 143.
    Moore, C. (1990). Unpredictability and undecidability in dynamical systems. Physical Review Letters, 64(20):2354-2357.MATHMathSciNetGoogle Scholar
  144. 144.
    Moore, C. (1991). Generalized shifts: unpredictability and undecidability in dynamical systems. Nonlinearity, 4(3):199-230.MATHMathSciNetGoogle Scholar
  145. 145.
    Moore, C. (1996). Recursion theory on the reals and continuous-time computation. Theoretical Computer Science, 162(1):23-44.MATHMathSciNetGoogle Scholar
  146. 146.
    Moore, C. (1998a). Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201(1-2):99-136.MATHMathSciNetGoogle Scholar
  147. 147.
    Moore, C. (1998b). Finite-dimensional analog computers: Flows, maps, and recurrent neural networks. In Calude, C. S., Casti, J. L., and Dinneen, M. J., editors, Unconventional Models of Computation (UMC’98). Springer.Google Scholar
  148. 148.
    Murray, J. D. (2002). Mathematical Biology. I: An Introduction. Springer, third edition.Google Scholar
  149. 149.
    Mycka, J. and Costa, J. F. (2004). Real recursive functions and their hierarchy. Journal of Complexity, 20(6):835-857.MATHMathSciNetGoogle Scholar
  150. 150.
    Mycka, J. and Costa, J. F. (2005). What lies beyond the mountains? Computational systems beyond the Turing limit. European Association for Theoretical Computer Science Bulletin, 85:181-189.Google Scholar
  151. 151.
    Mycka, J. and Costa, J. F. (2006). The P = NP conjecture in the context of real and complex analysis. Journal of Complexity, 22(2):287-303.MATHMathSciNetGoogle Scholar
  152. 152.
    Mycka, J. and Costa, J. F. (2007). A new conceptual framework for analog computation. Theoretical Computer Science, 374:277-290.MATHMathSciNetGoogle Scholar
  153. 153.
    Natschläger, T. and Maass, W. (2002). Spiking neurons and the induction of finite state machines. Theoretical Computer Science: Special Issue on Natural Computing, 287(1):251-265.MATHGoogle Scholar
  154. 154.
    Németi, I. and Andréka, H. (2006). New physics and hypercomputation. In Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., and Stuller, J., editors, SOFSEM 2006: Theory and Practice of Computer Science, 32nd Conference on Current Trends in Theory and Prac-tice of Computer Science, Merín, Czech Republic, January 21-27, 2006, Proceedings, volume 3831 of Lecture Notes in Computer Science, page 63. Springer.Google Scholar
  155. 155.
    Németi, I. and Dávid, G. (2006). Relativistic computers and the Turing barrier. Applied Mathematics and Computation, 178:118-142.MATHMathSciNetGoogle Scholar
  156. 156.
    Nicollin, X., Olivero, A., Sifakis, J., and Yovine, S. (1993). An approach to the description and analysis of hybrid systems. In Grossman, R. L., Nerode, A., Ravn, A. P., and Rischel, H., editors, Hybrid Systems, volume 736 of Lecture Notes in Computer Science, pages 149-178. Springer.Google Scholar
  157. 157.
    Omohundro, S. (1984). Modelling cellular automata with partial differential equations. Physica D, 10D(1-2):128-134.MathSciNetGoogle Scholar
  158. 158.
    Orponen, P. (1994). Computational complexity of neural networks: a survey. Nordic Journal of Computing, 1(1):94-110.MathSciNetGoogle Scholar
  159. 159.
    Orponen, P. (1996). The computational power of discrete Hopfield nets with hidden units. Neural Computation, 8(2):403-415.MathSciNetGoogle Scholar
  160. 160.
    Orponen, P. (1997). A survey of continuous-time computation theory. In Du, D.-Z. and Ko, K.-I., editors, Advances in Algorithms, Languages, and Complexity, pages 209-224. Kluwer Academic Publishers.Google Scholar
  161. 161.
    Orponen, P. and Šíma, J. (2000). A continuous-time Hopfield net simulation of discrete neural networks. In Proceedings of the 2nd International ICSC Symposium on Neural Computations (NC’2000), pages 36-42, Berlin, Germany. ICSC Academic Press, Wetaskiwin (Canada)Google Scholar
  162. 162.
    Papadimitriou, C. (2001). Algorithms, games, and the Internet. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing: Hersonissos, Crete, Greece, July 6-8, 2001, pages 749-753, New York, NY. ACM Press.Google Scholar
  163. 163.
    P ăun, G. (2002). Membrane Computing. An Introduction. Springer-Verlag, Berlin.Google Scholar
  164. 164.
    Post, E. (1946). A variant of a recursively unsolvable problem. Bulletin of the American Math. Soc., 52:264-268.MATHMathSciNetGoogle Scholar
  165. 165.
    Pour-El, M. and Zhong, N. (1997). The wave equation with computable initial data whose unique solution is nowhere computable. Mathematical Logic Quarterly, 43(4):499-509.MATHMathSciNetGoogle Scholar
  166. 166.
    Pour-El, M. B. (1974). Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers). Transactions of the American Mathematical Society, 199:1-28.MATHMathSciNetGoogle Scholar
  167. 167.
    Pour-El, M. B. and Richards, J. I. (1979). A computable ordinary differential equation which possesses no computable solution. Annals of Mathematical Logic, 17:61-90.MATHMathSciNetGoogle Scholar
  168. 168.
    Pour-El, M. B. and Richards, J. I. (1981). The wave equation with computable initial data such that its unique solution is not computable. Advances in Mathematics, 39: 215-239.MATHMathSciNetGoogle Scholar
  169. 169.
    Pour-El, M. B. and Richards, J. I. (1989). Computability in Analysis and Physics.Springer.Google Scholar
  170. 170.
    Puri, A. (1998). Dynamical properties of timed automata. In Ravn, A. P. and Rischel, H., editors, Formal Techniques in Real-Time and Fault-Tolerant Systems, 5th International Symposium, FTRTFT’98, Lyngby, Denmark, September 14-18, 1998, Proceedings, vol-ume 1486 of Lecture Notes in Computer Science, pages 210-227. Springer.Google Scholar
  171. 171.
    Puri, A. and Varaiya, P. (1994). Decidability of hybrid systems with rectangular differ-ential inclusion. In Dill, D. L., editor, Computer Aided Verification, 6th International Conference, CAV ’94, Stanford, CA. June 21-23, 1994, Proceedings, volume 818 of Lec-ture Notes in Computer Science, pages 95-104. Springer.Google Scholar
  172. 172.
    Rabin, M. O. (1963). Probabilistic automata. Information and Control, 6(3):230-245.Google Scholar
  173. 173.
    Rabinovich, A. (2003). Automata over continuous time. Theoretical Computer Science, 300(1-3):331-363.MATHMathSciNetGoogle Scholar
  174. 174.
    Rabinovich, A. M. and Trakhtenbrot, B. A. (1997). From finite automata toward hybrid systems (extended abstract). In Chlebus, B. S. and Czaja, L., editors, Fundamentals of Computation Theory, 11th International Symposium, FCT ’97, Kraków, Poland, September 1-3, 1997, Proceedings, volume 1279 of Lecture Notes in Computer Science, pages 411-422. Springer.Google Scholar
  175. 175.
    Rubel, L. A. (1989). A survey of transcendentally transcendental functions. American Mathematical Monthly, 96(9):777-788.MATHMathSciNetGoogle Scholar
  176. 176.
    Rubel, L. A. (1993). The extended analog computer. Advances in Applied Mathematics, 14:39-50.MATHMathSciNetGoogle Scholar
  177. 177.
    Ruohonen, K. (1993). Undecidability of event detection for ODEs. Journal of Information Processing and Cybernetics, 29:101-113.MATHGoogle Scholar
  178. 178.
    Ruohonen, K. (1994). Event detection for ODEs and nonrecursive hierarchies. In Karhumäki, J. and Maurer, H., editors, Proceedings of the Colloquium in Honor of Arto Salomaa. Results and Trends in Theoretical Computer Science (Graz, Austria, June 10-11,1994), volume 812 of Lecture Notes in Computer Science, pages 358-371. Springer, Berlin.Google Scholar
  179. 179.
    Ruohonen, K. (1996). An effective Cauchy-Peano existence theorem for unique solutions. International Journal of Foundations of Computer Science, 7(2):151-160.MATHGoogle Scholar
  180. 180.
    Ruohonen, K. (1997a). Decidability and complexity of event detection problems for ODEs. Complexity, 2(6):41-53.MathSciNetGoogle Scholar
  181. 181.
    Ruohonen, K. (1997b). Undecidable event detection problems for ODEs of dimension one and two. Theoretical Informatics and Applications, 31(1):67-79.MATHMathSciNetGoogle Scholar
  182. 182.
    Ruohonen, K. (2004). Chomskian hierarchies of families of sets of piecewise continuous functions. Theory of Computing Systems, 37(5):609-638.MATHMathSciNetGoogle Scholar
  183. 183.
    Shannon, C. E. (1941). Mathematical theory of the differential analyser. Journal of Mathematics and Physics MIT, 20:337-354.MATHMathSciNetGoogle Scholar
  184. 184.
    Shor, P. W. (1994). Algorithms for quantum computation: Discrete logarithms and factoring. In Goldwasser, S., editor, Proceedings of the 35th Annual Symposium on Founda-tions of Computer Science, pages 124-134, Los Alamitos, CA. IEEE Computer Society Press.Google Scholar
  185. 185.
    Siegelmann, H. T. and Fishman, S. (1998). Analog computation with dynamical systems. Physica D, 120:214-235.MATHGoogle Scholar
  186. 186.
    Siegelmann, H. T. and Sontag, E. D. (1994). Analog computation via neural networks. Theoretical Computer Science, 131(2):331-360.MATHMathSciNetGoogle Scholar
  187. 187.
    Siegelmann, H. T. and Sontag, E. D. (1995). On the computational power of neural nets. Journal of Computer and System Sciences, 50(1):132-150.MATHMathSciNetGoogle Scholar
  188. 188.
    Šíma and Orponen (2003a). Exponential transients in continuous-time Liapunov systems. Theoretical Computer Science, 306(1-3):353-372.MATHMathSciNetGoogle Scholar
  189. 189.
    Šíma, J. and Orponen, P. (2003b). Continuous-time symmetric Hopfield nets are computationally universal. Neural Computation, 15(3):693-733.Google Scholar
  190. 190.
    Šíma, J. and Orponen, P. (2003c). General-purpose computation with neural networks: A survey of complexity theoretic results. Neural Computation, 15(12):2727-2778.MATHGoogle Scholar
  191. 191.
    Smith, W. D. (1998). Plane mechanisms and the downhill principle. http:// citeseer.ist.psu.edu/475350.html.
  192. 192.
    Smith, W. D. (2006). Church’s thesis meets the N-body problem. Applied Mathematics and Computation, 178(1):154-183.MATHMathSciNetGoogle Scholar
  193. 193.
    Stoll, H. M. and Lee, L. S. (1988). A continuous-time optical neural network. In IEEE Second International Conference on Neural Networks (2nd ICNN’88), volume II, pages 373-384, San Diego, CA. IEEE Society Press.Google Scholar
  194. 194.
    Svoboda, A. (1948). Computing Mechanisms and Linkages. McGraw Hill. Reprinted by Dover Publications in 1965.Google Scholar
  195. 195.
    Thomson, W. (1876). On an instrument for calculating the integral of the product of two given functions. In Proceedings of the Royal Society of London, volume 24, pages 266-276.Google Scholar
  196. 196.
    Trakhtenbrot, B. (1995). Origins and metamorphoses of the trinity: Logic, nets, automata. In Kozen, D., editor, Proceedings of the 10th Annual IEEE Symposium on Logic in Computer Science San Diego, CA, June 26-29, 1995, pages 506-507. IEEE Computer Society, Press.Google Scholar
  197. 197.
    Trakhtenbrot, B. A. (1999). Automata and their interaction: Definitional suggestions. In Ciobanu, G. and Paun, G., editors, Fundamentals of Computation Theory, 12th Interna-tional Symposium, FCT ’99, Iasi, Romania, August 30 - September 3, 1999, Proceedings, volume 1684 of Lecture Notes in Computer Science, pages 54-89. Springer.Google Scholar
  198. 198.
    Tucker, J. V. and Zucker, J. I. (2007). Computability of analog networks. Theoretical Computer Science, 371(1-2):115-146.MATHMathSciNetGoogle Scholar
  199. 199.
    Turing, A. (1936). On computable numbers, with an application to the EntscheidungsproblemֶProceedings of the London Mathematical Society, 42(2):230-265. Reprinted in [73].MATHGoogle Scholar
  200. 200.
    Vergis, A., Steiglitz, K., and Dickinson, B. (1986). The complexity of analog computation. Mathematics and Computers in Simulation, 28(2):91-113.MATHGoogle Scholar
  201. 201.
    Weihrauch, K. (2000). Computable Analysis. Springer.Google Scholar
  202. 202.
    Weihrauch, K. and Zhong, N. (2002). Is wave propagation computable or can wave computers beat the Turing machine? Proceedings of the London Mathematical Society, 85 (3):312-332.MATHMathSciNetGoogle Scholar
  203. 203.
    Welch, P. D. (2006). The extent of computation in Malament-Hogarth spacetimes. http://www.citebase.org/abstract?id=oai:arXiv.org:gr-qc/0609035.
  204. 204.
    Williams, M. R. (1996). About this issue. IEEE Annals of the History of Computing, 18(4).Google Scholar
  205. 205.
    Woods, D. and Naughton, T. J. (2005). An optical model of computation. Theoretical Computer Science, 334(1-3):227-258.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Olivier Bournez
    • 1
  • Manuel L. Campagnolo
    • 2
  1. 1.INRIA LorraineFrance
  2. 2.DM/ISA, Technical University of LisbonPortugal

Personalised recommendations