A Survey on Continuous Time Computations

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.

[1]  Ning Zhong,et al.  Computability, noncomputability and undecidability of maximal intervals of IVPs , 2009 .

[2]  Anders P. Ravn,et al.  Formal Techniques in Real-Time and Fault-Tolerant Systems , 1998, Lecture Notes in Computer Science.

[3]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

[4]  Rajeev Alur,et al.  Decision Problems for Timed Automata: A Survey , 2004, SFM.

[5]  Lee A. Rubel,et al.  The Extended Analog Computer , 1993 .

[6]  B. Jack Copeland,et al.  Accelerating Turing Machines , 2002, Minds and Machines.

[7]  J. Lygeros,et al.  Computability of finite-time reachable sets for hybrid systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[8]  Brian Cantwell Smith,et al.  The Foundations of Computing , 1996 .

[9]  Sergio Yovine,et al.  On the Decidability of the Reachability Problem for Planar Differential Inclusions , 2001, HSCC.

[10]  Boris A. Trakhtenbrot,et al.  Automata and Their Interaction: Definitional Suggestions , 1999, FCT.

[11]  Allen Miller What Lies Beyond , 1968 .

[12]  Olivier Bournez,et al.  Polynomial differential equations compute all real computable functions on computable compact intervals , 2007, J. Complex..

[13]  Patricia Bouyer,et al.  A Kleene/Büchi-like Theorem for Clock Languages , 2001, J. Autom. Lang. Comb..

[14]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[15]  Antonín Svoboda,et al.  Computing Mechanisms and Linkages , 1965 .

[16]  Ecnica De Lisboa,et al.  The General Purpose Analog Computer and Recursive Functions Over the Reals , 2002 .

[17]  Christos H. Papadimitriou,et al.  Algorithms, Games, and the Internet , 2001, ICALP.

[18]  A. Kempe On a General Method of describing Plane Curves of the nth degree by Linkwork , 1875 .

[19]  Manuel Lameiras Campagnolo,et al.  The Complexity of Real Recursive Functions , 2002, UMC.

[20]  M. Hogarth PREDICTABILITY, COMPUTABILITY, AND SPACETIME , 2002 .

[21]  Emil L. Post A variant of a recursively unsolvable problem , 1946 .

[22]  Michael S. Branicky,et al.  Universal Computation and Other Capabilities of Hybrid and Continuous Dynamical Systems , 1995, Theor. Comput. Sci..

[23]  L. Faybusovich Dynamical Systems which Solve Optimization Problems with Linear Constraints , 1991 .

[24]  John L. Casti,et al.  Unconventional Models of Computation , 2002, Lecture Notes in Computer Science.

[25]  Eugene Asarin,et al.  Balanced timed regular expressions , 2002, Electron. Notes Theor. Comput. Sci..

[26]  J. V. Tucker,et al.  Computability of analog networks , 2007, Theor. Comput. Sci..

[27]  Ecnica De Lisboa,et al.  Computational complexity of real valued recursive functions and analog circuits , 2001 .

[28]  Pieter Collins,et al.  Continuity and computability of reachable sets , 2005, Theor. Comput. Sci..

[29]  Keijo Ruohonen An Effective Cauchy-Peano Existence Theorem for Unique Solutions , 1996, Int. J. Found. Comput. Sci..

[30]  Manuel Lameiras Campagnolo,et al.  The elementary computable functions over the real numbers: applying two new techniques , 2008, Arch. Math. Log..

[31]  Manuel Lameiras Campagnolo Continuous-time computation with restricted integration capabilities , 2004, Theor. Comput. Sci..

[32]  Wolfgang Maass,et al.  A Model for Fast Analog Computation Based on Unreliable Synapses , 2000, Neural Computation.

[33]  Wolfgang Maass,et al.  Computing with spiking neurons , 1999 .

[34]  Roger W. Brockett,et al.  Smooth dynamical systems which realize arithmetical and logical operations , 1989 .

[35]  Wolfgang Maass,et al.  Lower Bounds for the Computational Power of Networks of Spiking Neurons , 1996, Neural Computation.

[36]  V. Arnold Mathematical Methods of Classical Mechanics , 1974 .

[37]  John Foy,et al.  A dynamical system which must be stable whose stability cannot be proved , 2004, Theor. Comput. Sci..

[38]  M. Hogarth Non-Turing Computers and Non-Turing Computability , 1994, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.

[39]  R. Brockett,et al.  Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems , 1988, Proceedings of the 27th IEEE Conference on Decision and Control.

[40]  Olivier Bournez How much can analog and hybrid systems be proved (super-)Turing , 2006, Appl. Math. Comput..

[41]  Peter Clote,et al.  Computational Models and Function Algebras , 1994, LCC.

[42]  Pravin Varaiya,et al.  Decidability of Hybrid Systems with Rectangular Differential Inclusion , 1994, CAV.

[43]  Azaria Paz,et al.  Probabilistic automata , 2003 .

[44]  Patricia Bouyer,et al.  Decomposition and Composition of Timed Automata , 1999, ICALP.

[45]  B. Jack Copeland,et al.  EVEN TURING MACHINES CAN COMPUTE UNCOMPUTABLE FUNCTIONS , 1998 .

[46]  Pekka Orponen,et al.  Continuous-Time Symmetric Hopfield Nets Are Computationally Universal , 2003, Neural Computation.

[47]  Eugene Asarin Equations on Timed Languages , 1998, HSCC.

[48]  Mike Casey,et al.  The Dynamics of Discrete-Time Computation, with Application to Recurrent Neural Networks and Finite State Machine Extraction , 1996, Neural Computation.

[49]  John N. Tsitsiklis,et al.  A survey of computational complexity results in systems and control , 2000, Autom..

[50]  Peter W. Shor,et al.  Algorithms for quantum computation: discrete logarithms and factoring , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[51]  Wofgang Maas,et al.  Networks of spiking neurons: the third generation of neural network models , 1997 .

[52]  Klaus Weihrauch,et al.  Is wave propagation computable or can wave computers beat the turing machine? , 2002 .

[53]  Ker-I Ko,et al.  Complexity Theory of Real Functions , 1991, Progress in Theoretical Computer Science.

[54]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[55]  Maurice Margenstern,et al.  Register Cellular Automata in the Hyperbolic Plane , 2004, Fundam. Informaticae.

[56]  I. I. Artobolevskiĭ Mechanisms for the generation of plane curves , 1964 .

[57]  Wolfgang Maass,et al.  Spiking neurons and the induction of finite state machines , 2002, Theor. Comput. Sci..

[58]  M. B. Pour-El,et al.  The wave equation with computable initial data such that its unique solution is not computable , 1981 .

[59]  Igor Potapov,et al.  Computation in One-Dimensional Piecewise Maps and Planar Pseudo-Billiard Systems , 2005, UC.

[60]  Keijo Ruohonen,et al.  Decidability and complexity of event detection problems for ODEs , 1997, Complex..

[61]  Thomas Brihaye A note on the undecidability of the reachability problem for o-minimal dynamical systems , 2006, Math. Log. Q..

[62]  Olivier Bournez,et al.  Elementarily computable functions over the real numbers and R-sub-recursive functions , 2005, Theor. Comput. Sci..

[63]  Thomas Brihaye,et al.  On the expressiveness and decidability of o-minimal hybrid systems , 2005, J. Complex..

[64]  Cristopher Moore,et al.  An Analog Characterization of the Grzegorczyk Hierarchy , 2002, J. Complex..

[65]  Terrence J. Sejnowski,et al.  New Directions in Statistical Signal Processing: From Systems to Brains (Neural Information Processing) , 2006 .

[66]  Pravin Varaiya,et al.  What's decidable about hybrid automata? , 1995, STOC '95.

[67]  Vannevar Bush,et al.  The differential analyzer. A new machine for solving differential equations , 1931 .

[68]  Jan H. van Schuppen,et al.  Observability of Piecewise-Affine Hybrid Systems , 2004, HSCC.

[69]  Eduardo D. Sontag,et al.  Computational Aspects of Feedback in Neural Circuits , 2006, PLoS Comput. Biol..

[70]  Amir Pnueli,et al.  Reachability Analysis of Dynamical Systems Having Piecewise-Constant Derivatives , 1995, Theor. Comput. Sci..

[71]  José Félix Costa,et al.  Analog computers and recursive functions over the reals , 2003, J. Complex..

[72]  L. Rubel,et al.  A differentially algebraic replacement theorem, and analog computability , 1987 .

[73]  Eugene Asarin,et al.  Challenges in Timed Languages: from applied theory to basic theory (Column: Concurrency) , 2004, Bull. EATCS.

[74]  Warren D. Smith Plane Mechanisms and the \downhill Principle" , 1998 .

[75]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[76]  Christopher J. Bishop,et al.  Pulsed Neural Networks , 1998 .

[77]  U. Helmke,et al.  Optimization and Dynamical Systems , 1994, Proceedings of the IEEE.

[78]  Cristopher Moore,et al.  Iteration, Inequalities, and Differentiability in Analog Computers , 2000, J. Complex..

[79]  Patricia Bouyer,et al.  Are Timed Automata Updatable? , 2000, CAV.

[80]  Pekka Orponen,et al.  A Survey of Continous-Time Computation Theory , 1997, Advances in Algorithms, Languages, and Complexity.

[81]  Klaus Weihrauch,et al.  Computable Analysis: An Introduction , 2014, Texts in Theoretical Computer Science. An EATCS Series.

[82]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and an Introduction to Chaos , 2003 .

[83]  Daniel S. Graça,et al.  Robust Simulations of Turing Machines with Analytic Maps and Flows , 2005, CiE.

[84]  Máté Lengyel,et al.  Computing with spikes , 2006 .

[85]  A. Church An Unsolvable Problem of Elementary Number Theory , 1936 .

[86]  Sebastian A. Wills,et al.  Computation with Spiking Neurons , 2004 .

[87]  Rajeev Alur,et al.  A Theory of Timed Automata , 1994, Theor. Comput. Sci..

[88]  Pekka Orponen,et al.  General-Purpose Computation with Neural Networks: A Survey of Complexity Theoretic Results , 2003, Neural Computation.

[89]  Benedikt Löwe,et al.  Logical Approaches to Computational Barriers: CiE 2006 , 2007, J. Log. Comput..

[90]  Moore,et al.  Unpredictability and undecidability in dynamical systems. , 1990, Physical review letters.

[91]  Wolfgang Maass,et al.  On the Computational Power of Noisy Spiking Neurons , 1995, NIPS.

[92]  Cristopher Moore,et al.  Dynamical Recognizers: Real-Time Language Recognition by Analog Computers , 1998, Theor. Comput. Sci..

[93]  Max Black Achilles and the Tortoise , 1951 .

[94]  Alan Bundy,et al.  Constructing Induction Rules for Deductive Synthesis Proofs , 2006, CLASE.

[95]  Keijo Ruohonen,et al.  Chomskian Hierarchies of Families of Sets of Piecewise Continuous Functions , 2004, Theory of Computing Systems.

[96]  Nathan Deckard,et al.  Extended Analog Computers : A Unifying Paradigm for VLSI , Plastic and Colloidal Computing Systems , 2005 .

[97]  Michael S. Branicky,et al.  Studies in hybrid systems: modeling, analysis, and control , 1996 .

[98]  Gheorghe Paun,et al.  Membrane Computing , 2002, Natural Computing Series.

[99]  Michel Cosnard,et al.  Computability with Low-Dimensional Dynamical Systems , 1994, Theor. Comput. Sci..

[100]  Eugene Asarin,et al.  Widening the Boundary between Decidable and Undecidable Hybrid Systems , 2002, CONCUR.

[101]  Shouchuan Hu Differential equations with discontinuous right-hand sides☆ , 1991 .

[102]  Hava T. Siegelmann,et al.  Analog computation via neural networks , 1993, [1993] The 2nd Israel Symposium on Theory and Computing Systems.

[103]  Jerzy Mycka,et al.  Computability on reals, infinite limits and differential equations , 2007, Appl. Math. Comput..

[104]  James D. Murray Mathematical Biology: I. An Introduction , 2007 .

[105]  R. Brockett Dynamical Systems and Their Associated Automata , 1994 .

[106]  M. Hogarth Does general relativity allow an observer to view an eternity in a finite time? , 1992 .

[107]  Pekka Orponen,et al.  Exponential transients in continuous-time Liapunov systems , 2003, Theor. Comput. Sci..

[108]  Ning Zhong,et al.  The Wave Equation with Computable Initial Data Whose Unique Solution Is Nowhere Computable , 1996, Math. Log. Q..

[109]  Pascal Koiran The topological entropy of iterated piecewise affine maps is uncomputable , 2001, Discret. Math. Theor. Comput. Sci..

[110]  Marian Boykan Pour-El,et al.  Computability in analysis and physics , 1989, Perspectives in Mathematical Logic.

[111]  T. Head Formal language theory and DNA: an analysis of the generative capacity of specific recombinant behaviors. , 1987, Bulletin of mathematical biology.

[112]  Nicolai Vorobjov,et al.  Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Hybrid Systems , 2006, CiE.

[113]  Anuj Puri Dynamical Properties of Timed Automata , 2000, Discret. Event Dyn. Syst..

[114]  István Németi,et al.  Non-Turing Computations Via Malament–Hogarth Space-Times , 2001 .

[115]  Hervé Abdi,et al.  A NEURAL NETWORK PRIMER , 1994 .

[116]  Jérôme Olivier Durand-Lose Abstract Geometrical Computation: Turing-Computing Ability and Undecidability , 2005, CiE.

[117]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

[118]  Manuel Lameiras Campagnolo,et al.  The Methods of Approximation and Lifting in Real Computation , 2007, Electron. Notes Theor. Comput. Sci..

[119]  Wulfram Gerstner,et al.  Spiking neurons , 1999 .

[120]  Wolfgang Maass,et al.  On Computation with Pulses , 2000, Electron. Colloquium Comput. Complex..

[121]  Pekka Orponen,et al.  The Computational Power of Discrete Hopfield Nets with Hidden Units , 1996, Neural Computation.

[122]  Jerzy Mycka,et al.  A new conceptual framework for analog computation , 2007, Theor. Comput. Sci..

[123]  Mark D. Bowles,et al.  U.S. Technological Enthusiasm and British Technological Skepticism in the Age of the Analog Brain , 1996, IEEE Ann. Hist. Comput..

[124]  Ann Copestake The Differential Analyser , 1940, Nature.

[125]  Marco Bernardo,et al.  Formal methods for the design of real-time systems : International School on Formal Methods for the Design of Computer, Communication and Software Systems, SFM-RT 2004, Bertinoro, Italy, September 13-18, 2004 : Revised lectures , 2004 .

[126]  Keijo Ruohonen Undecidable Event Detection Problems for Odes of Dimension One and Two , 1997, RAIRO Theor. Informatics Appl..

[127]  W. Maass,et al.  What makes a dynamical system computationally powerful ? , 2022 .

[128]  Cristopher Moore,et al.  Recursion Theory on the Reals and Continuous-Time Computation , 1996, Theor. Comput. Sci..

[129]  Daniel Graça,et al.  The general purpose analog computer and recursive functionsover the reals , 2002 .

[130]  Tien D. Kieu Hypercomputation with quantum adiabatic processes , 2004, Theor. Comput. Sci..

[131]  Thomas A. Henzinger,et al.  The Algorithmic Analysis of Hybrid Systems , 1995, Theor. Comput. Sci..

[132]  B. Dickinson,et al.  The complexity of analog computation , 1986 .

[133]  Hava T. Siegelmann,et al.  On probabilistic analog automata , 2003, Theor. Comput. Sci..

[134]  Wolfgang Maass,et al.  Spiking Neurons , 1998, NC.

[135]  Michael R. Williams About This Issue , 1997, IEEE Ann. Hist. Comput..

[136]  Paul Caspi,et al.  A Kleene theorem for timed automata , 1997, Proceedings of Twelfth Annual IEEE Symposium on Logic in Computer Science.

[137]  Martin Davis,et al.  The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions , 2004 .

[138]  Alexander Moshe Rabinovich,et al.  Automata over continuous time , 2003, Theor. Comput. Sci..

[139]  E. B. Davies Building Infinite Machines , 2001, The British Journal for the Philosophy of Science.

[140]  Jerzy Mycka,et al.  Real recursive functions and their hierarchy , 2004, J. Complex..

[141]  D. R. HARTREE,et al.  The Differential Analyser , 1935, Nature.

[142]  William Thomson IV. On an instrument for calculating (∫φ(x) ψ (x)dx), the integral of the product of two given functions , 1876, Proceedings of the Royal Society of London.

[143]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[144]  Cristian S. Calude,et al.  Coins, Quantum Measurements, and Turing's Barrier , 2002, Quantum Inf. Process..

[145]  H. Siegelmann,et al.  Analog computation with dynamical systems , 1998 .

[146]  Hajnal Andréka,et al.  New Physics and Hypercomputation , 2006, SOFSEM.

[147]  A. Turing On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[148]  Thomas J. Naughton,et al.  An optical model of computation , 2005, Theor. Comput. Sci..

[149]  Olivier Bournez Complexite algorithmique des systemes dynamiques continus et hybrides , 1999 .

[150]  Ahmed Bouajjani,et al.  Perturbed Turing machines and hybrid systems , 2001, Proceedings 16th Annual IEEE Symposium on Logic in Computer Science.

[151]  Pekka Orponen,et al.  Computational complexity of neural networks: a survey , 1994 .

[152]  L. Faybusovich Hamiltonian structure of dynamical systems which solve linear programming problems , 1991 .

[153]  Olivier Bournez,et al.  Recursive Analysis Characterized as a Class of Real Recursive Functions , 2006, Fundam. Informaticae.

[154]  Peter Clote,et al.  Computation Models and Function Algebras , 1999, Handbook of Computability Theory.

[155]  Michael Ghil,et al.  Boolean difference equations. I - Formulation and dynamic behavior , 1984 .

[156]  Michael Casey Correction to Proof That Recurrent Neural Networks Can Robustly Recognize Only Regular Languages , 1998, Neural Computation.

[157]  Wolfgang Maass,et al.  Networks of Spiking Neurons: The Third Generation of Neural Network Models , 1996, Electron. Colloquium Comput. Complex..

[158]  R J Lipton,et al.  DNA solution of hard computational problems. , 1995, Science.

[159]  Keijo Ruohonen Event Detection for ODEs and Nonrecursive Hierarchies , 1994, Results and Trends in Theoretical Computer Science.

[160]  Jerzy Mycka,et al.  The P ne NP conjecture in the context of real and complex analysis , 2006, J. Complex..

[161]  Martin Fränzle,et al.  Analysis of Hybrid Systems: An Ounce of Realism Can Save an Infinity of States , 1999, CSL.

[162]  P. D. Welch The Extent of Computation in Malament–Hogarth Spacetimes , 2008, The British Journal for the Philosophy of Science.

[163]  Cristopher Moore,et al.  Generalized shifts: unpredictability and undecidability in dynamical systems , 1991 .

[164]  István Németi,et al.  Relativistic computers and the Turing barrier , 2006, Appl. Math. Comput..

[165]  Wolfgang Maass,et al.  A model for fast analog computations with noisy spiking neurons , 1997 .

[166]  Cristopher Moore,et al.  Closed-for Analytic Maps in One and Two Dimensions can Simulate Universal Turing Machines , 1999, Theor. Comput. Sci..

[167]  Hava T. Siegelmann,et al.  Probabilistic analysis of a differential equation for linear programming , 2001, J. Complex..

[168]  Per A. Holst Analog computer , 2003 .

[169]  Pekka Orponen,et al.  On the Effect of Analog Noise in Discrete-Time Analog Computations , 1996, Neural Computation.

[170]  Thomas A. Henzinger,et al.  Robust Timed Automata , 1997, HART.

[171]  Joseph Sifakis,et al.  An Approach to the Description and Analysis of Hybrid Systems , 1992, Hybrid Systems.

[172]  Hava T. Siegelmann,et al.  A Theory of Complexity for Continuous Time Systems , 2002, J. Complex..

[173]  Olivier Bournez Achilles and the Tortoise Climbing up the Hyper-Arithmetical Hierarchy , 1999, Theor. Comput. Sci..

[174]  Daniel Silva Graça,et al.  Some recent developments on Shannon's General Purpose Analog Computer , 2004, Math. Log. Q..

[175]  Karlis Cerans,et al.  Deciding Reachability for Planar Multi-polynomial Systems , 1996, Hybrid Systems.

[176]  Nicolai Vorobjov,et al.  Pfaffian Hybrid Systems , 2004, CSL.

[177]  M. B. Pour-El,et al.  Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers) , 1974 .

[178]  S. Omohundro Modelling cellular automata with partial differential equations , 1984 .

[179]  Thomas A. Henzinger,et al.  Robust Undecidability of Timed and Hybrid Systems , 2000, HSCC.

[180]  Jerzy Mycka,et al.  The New Promise of Analog Computation , 2007, CiE.

[181]  Lee A. Rubel,et al.  A survey of transcendentally transcendental functions , 1989 .

[182]  L. Miles,et al.  2000 , 2000, RDH.

[183]  Marian Boylan Pour-el,et al.  A computable ordinary differential equation which possesses no computable solution , 1979 .

[184]  Eduardo D. Sontag,et al.  Analog Neural Nets with Gaussian or Other Common Noise Distributions Cannot Recognize Arbitrary Regular Languages , 1999, Neural Computation.

[185]  Ker-I Ko,et al.  On the Computational Complexity of Ordinary Differential Equations , 1984, Inf. Control..

[186]  S. Shankar Sastry,et al.  O-Minimal Hybrid Systems , 2000, Math. Control. Signals Syst..

[187]  Claude E. Shannon,et al.  Mathematical Theory of the Differential Analyzer , 1941 .

[188]  Scott Aaronson,et al.  Guest Column: NP-complete problems and physical reality , 2005, SIGA.

[189]  G. SILYN ROBERTS,et al.  Achilles and the Tortoise , 1944, Nature.

[190]  Hava T. Siegelmann,et al.  On the Computational Power of Neural Nets , 1995, J. Comput. Syst. Sci..

[191]  Mark Hogarth Non-Turing Computers are the New Non-Euclidean Geometries , 2009, Int. J. Unconv. Comput..

[192]  Klaus Meer,et al.  A Step towards a Complexity Theory for Analog Systems , 2002, Math. Log. Q..

[193]  Boris A. Trakhtenbrot Origins and metamorphoses of the Trinity: logic, nets, automata , 1995, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science.

[194]  Terrence J. Sejnowski,et al.  What Makes a Dynamical System Computationally Powerful , 2007 .

[195]  Olivier Finkel On the Shuffle of Regular Timed Languages , 2006, Bull. EATCS.

[196]  Warren D. Smith Church's thesis meets the N-body problem , 2006, Appl. Math. Comput..

[197]  J. Earman,et al.  Forever Is a Day: Supertasks in Pitowsky and Malament-Hogarth Spacetimes , 1993, Philosophy of Science.

[198]  Michael A. Arbib,et al.  The handbook of brain theory and neural networks , 1995, A Bradford book.

[199]  Danièle Beauquier Pumping Lemmas for Timed Automata , 1998, FoSSaCS.

[200]  Paul Caspi,et al.  Timed regular expressions , 2002, JACM.

[201]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[202]  J. J. Hopfield,et al.  “Neural” computation of decisions in optimization problems , 1985, Biological Cybernetics.

[203]  Alexander Moshe Rabinovich,et al.  From Finite Automata toward Hybrid Systems (Extended Abstract) , 1997, FCT.

[204]  Scott Aaronson,et al.  NP-complete Problems and Physical Reality , 2005, Electron. Colloquium Comput. Complex..

[205]  L.-S. Lee,et al.  A continuous-time optical neural network , 1988, IEEE 1988 International Conference on Neural Networks.

[206]  Keijo Ruohonen Undecidability of Event Detection for ODEs , 1993, J. Inf. Process. Cybern..

[207]  Eugene Asarin,et al.  Achilles and the Tortoise Climbing Up the Arithmetical Hierarchy , 1995, J. Comput. Syst. Sci..

[208]  Patricia Bouyer,et al.  Expressiveness of Updatable Timed Automata , 2000, MFCS.

[209]  Edward R. Griffor Handbook of Computability Theory , 1999, Handbook of Computability Theory.

[210]  Pekka Orponen,et al.  A Continuous-Time Hopfield Net Simulation of Discrete Neural Networks , 2000 .

[211]  Daniel S. Graça,et al.  Computability with polynomial differential equations , 2008, Adv. Appl. Math..

[212]  Eugene Asarin,et al.  Noisy Turing Machines , 2005, ICALP.

[213]  John N. Tsitsiklis,et al.  Complexity of stability and controllability of elementary hybrid systems , 1999, Autom..

[214]  Hava T. Siegelmann,et al.  Random matrix theory for the analysis of the performance of an analog computer: a scaling theory , 2004 .

[215]  George J. Pappas,et al.  Hybrid Systems: Computation and Control: 7th International Workshop, Hscc 2004, Philadelphia, Pa, Usa, March 2004: Proceedings (Lecture Notes in Computer Science, 2993) , 2004 .

[216]  S. Kleene General recursive functions of natural numbers , 1936 .

[217]  D. Deutsch Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[218]  Mathieu Hoyrup Dynamical systems: stability and simulability , 2007, Math. Struct. Comput. Sci..

[219]  Jean-Charles Delvenne,et al.  Computational Universality in Symbolic Dynamical Systems , 2004, MCU.