A Survey on Analog Models of Computation

We present a survey on analog models of computations. Analog can be understood both as computing by analogy, or as working on the continuum. We consider both approaches, often intertwined, with a point of view mostly oriented by computation theory.

[1]  Jehoshua Bruck,et al.  Programmability of Chemical Reaction Networks , 2009, Algorithmic Bioprocesses.

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

[3]  Artiom Alhazov,et al.  Membrane Computing , 2013, Lecture Notes in Computer Science.

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

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

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

[7]  G. J. Chaitin,et al.  The Halting Probability Omega: Irreducible Complexity in Pure Mathematics , 2006 .

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

[9]  Dominic Scalise,et al.  Emulating cellular automata in chemical reaction–diffusion networks , 2016, Natural Computing.

[10]  Matthew Cook,et al.  Unsupervised learning of digit recognition using spike-timing-dependent plasticity , 2015, Front. Comput. Neurosci..

[11]  Scott Aaronson,et al.  Closed timelike curves make quantum and classical computing equivalent , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  Jonathan W. Mills,et al.  The nature of the Extended Analog Computer , 2008 .

[13]  Mark Braverman,et al.  Noise vs computational intractability in dynamics , 2012, ITCS '12.

[14]  Ulf Hashagen,et al.  The first computers: history and architectures , 2000 .

[15]  Amaury Pouly,et al.  On the Functions Generated by the General Purpose Analog Computer , 2016, Inf. Comput..

[16]  James Aspnes,et al.  An Introduction to Population Protocols , 2007, Bull. EATCS.

[17]  Hajnal Andréka,et al.  Logical Axiomatizations of Space-Time. Samples from the Literature , 2006 .

[18]  Warren D. Smith,et al.  Church's Thesis Meets Quantum Mechanics , 1999 .

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

[20]  G. Church,et al.  Next-Generation Digital Information Storage in DNA , 2012, Science.

[21]  Simone Tini,et al.  Taylor approximation for hybrid systems , 2005, Inf. Comput..

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

[23]  Cameron Beebe Model-Based Computation , 2016, UCNC.

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

[25]  Hava T. Siegelmann,et al.  The Super-Turing Computational Power of plastic Recurrent Neural Networks , 2014, Int. J. Neural Syst..

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

[27]  J. V. Tucker,et al.  Computability of Operators on Continuous and Discrete Time Streams , 2014, Comput..

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

[29]  Michael J. Todd,et al.  Polynomial Algorithms for Linear Programming , 1988 .

[30]  N. C. A. da Costa,et al.  Dynamical system where proving chaos is equivalent to proving Fermat's conjecture , 1993 .

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

[32]  François Fages,et al.  Strong Turing Completeness of Continuous Chemical Reaction Networks and Compilation of Mixed Analog-Digital Programs , 2017, CMSB.

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

[34]  Bernard Chazelle,et al.  The Convergence of Bird Flocking , 2009, JACM.

[35]  Bruce J. MacLennan,et al.  The promise of analog computation , 2014, Int. J. Gen. Syst..

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

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

[38]  Construction of a universal ordinary differential equation C∞ of order 3 , 2016, Bulletin des Sciences Mathématiques.

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

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

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

[42]  José Félix Costa,et al.  The ARNN model relativises P=NP and P!=NP , 2013, Theor. Comput. Sci..

[43]  B. Poizat Les petits cailloux : une approche modèle-théorique de l'algorithmie , 1995 .

[44]  Jacob Beal,et al.  Organizing the Aggregate: Languages for Spatial Computing , 2012, ArXiv.

[45]  Bruce J. MacLennan,et al.  Natural computation and non-Turing models of computation , 2004, Theor. Comput. Sci..

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

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

[48]  Dominic Scalise,et al.  Designing modular reaction-diffusion programs for complex pattern formation , 2014 .

[49]  W. Stemmer The Evolution of Molecular Computation , 1995, Science.

[50]  G. Dowek Les métamorphoses du calcul : une étonnante histoire de mathématiques , 2007 .

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

[52]  Heekuck Oh,et al.  Neural Networks for Pattern Recognition , 1993, Adv. Comput..

[53]  André Platzer,et al.  The Complete Proof Theory of Hybrid Systems , 2012, 2012 27th Annual IEEE Symposium on Logic in Computer Science.

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

[55]  Paul G. Spirakis,et al.  Passively mobile communicating machines that use restricted space , 2011, FOMC '11.

[56]  Apostolos Syropoulos,et al.  Hypercomputation: Computing Beyond the Church-Turing Barrier , 2008 .

[57]  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.

[58]  André Platzer,et al.  Logical Foundations of Cyber-Physical Systems , 2018, Springer International Publishing.

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

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

[61]  Pietro Milici,et al.  Tractional Motion Machines: Tangent-Managing Planar Mechanisms as Analog Computers and Educational Artifacts , 2012, UCNC.

[62]  Joël Ouaknine,et al.  On Recurrent Reachability for Continuous Linear Dynamical Systems , 2015, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[63]  Maël Pégny,et al.  How to Make a Meaningful Comparison of Models: The Church–Turing Thesis Over the Reals , 2016, Minds and Machines.

[64]  Jerzy Mycka,et al.  The Euclid Abstract Machine: Trisection of the Angle and the Halting Problem , 2006, UC.

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

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

[67]  Drew Endy,et al.  Amplifying Genetic Logic Gates , 2013, Science.

[68]  J. V. Tucker,et al.  Computable functions and semicomputable sets on many-sorted algebras , 2001, Logic in Computer Science.

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

[70]  David Doty,et al.  Timing in chemical reaction networks , 2013, SODA.

[71]  Hava T. Siegelmann,et al.  The Computational Power of Interactive Recurrent Neural Networks , 2012, Neural Computation.

[72]  Hava T. Siegelmann,et al.  Evolving recurrent neural networks are super-Turing , 2011, The 2011 International Joint Conference on Neural Networks.

[73]  Mahesh Viswanathan The Church-Turing Thesis , 2016 .

[74]  Christine Gaßner,et al.  The Separation of Relativized Versions of P and DNP for the Ring of the Reals , 2010, J. Univers. Comput. Sci..

[75]  Christine Gaßner Strong Turing Degrees for Additive BSS RAM's , 2013, Log. Methods Comput. Sci..

[76]  D. Saari,et al.  Off to infinity in finite time , 1995 .

[77]  Shang Chen,et al.  On the decidability and complexity of problems for restricted hierarchical hybrid systems , 2016, Theor. Comput. Sci..

[78]  Lulu Qian,et al.  Efficient Turing-Universal Computation with DNA Polymers , 2010, DNA.

[79]  Hojjat Adeli,et al.  Spiking Neural Networks , 2009, Int. J. Neural Syst..

[80]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[81]  Yiannis N. Moschovakis,et al.  Abstract recursion as a foundation for the theory of algorithms , 1984 .

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

[83]  Emmanuel Jeandel Topological Automata , 2006, Theory of Computing Systems.

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

[85]  Timothy Bourke,et al.  Non-standard semantics of hybrid systems modelers , 2012, J. Comput. Syst. Sci..

[86]  Alessandro E. P. Villa,et al.  Expressive power of first-order recurrent neural networks determined by their attractor dynamics , 2016, J. Comput. Syst. Sci..

[87]  Ulf Hashagen Rechner für die Wissenschaft: „Scientific Computing“ und Informatik im deutschen Wissenschaftssystem 1870–1970 , 2017 .

[88]  François Fages,et al.  Abstract interpretation and types for systems biology , 2008, Theor. Comput. Sci..

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

[90]  Amir Ali Ahmadi,et al.  On Complexity of Lyapunov Functions for Switched Linear Systems , 2014 .

[91]  Bruce J. MacLennan,et al.  Analog Computation , 2009, Encyclopedia of Complexity and Systems Science.

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

[93]  Nachum Dershowitz,et al.  Three Paths to Effectiveness , 2010, Fields of Logic and Computation.

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

[95]  Paul G. Spirakis,et al.  New Models for Population Protocols , 2011, Synthesis Lectures on Distributed Computing Theory.

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

[97]  Amir Ali Ahmadi,et al.  Complexity of ten decision problems in continuous time dynamical systems , 2012, 2013 American Control Conference.

[98]  Michael J. Fischer,et al.  Stably Computable Properties of Network Graphs , 2005, DCOSS.

[99]  Gilles Dowek,et al.  The physical Church thesis as an explanation of the Galileo thesis , 2012, Natural Computing.

[100]  Diogo Poças,et al.  Fixed Point Techniques in Analog Systems , 2016 .

[101]  Michael J. Fischer,et al.  Self-stabilizing population protocols , 2005, TAAS.

[102]  Sergio Gomez Colmenarejo,et al.  Hybrid computing using a neural network with dynamic external memory , 2016, Nature.

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

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

[105]  Benedikt Löwe,et al.  New Computational Paradigms , 2005 .

[106]  Zoltán Toroczkai,et al.  Optimization hardness as transient chaos in an analog approach to constraint satisfaction , 2011, ArXiv.

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

[108]  Giuseppe Longo,et al.  The mathematics of computing between logic and physics , 2011 .

[109]  Luca Cardelli,et al.  Turing universality of the Biochemical Ground Form , 2010, Mathematical Structures in Computer Science.

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

[111]  A. Lesne Discrete vs continuous controversy in physics , 2022 .

[112]  Richard Whyman Physical Computation, P/poly and P/log , 2016, PC.

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

[114]  Michael J. Fischer,et al.  Computation in networks of passively mobile finite-state sensors , 2004, PODC '04.

[115]  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.

[116]  Narendra Karmarkar,et al.  A new polynomial-time algorithm for linear programming , 1984, Comb..

[117]  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 .

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

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

[120]  Rachid Guerraoui,et al.  When Birds Die: Making Population Protocols Fault-Tolerant , 2006, DCOSS.

[121]  Richard McGehee,et al.  Solutions of the collinear four body problem which become unbounded in finite time , 1975 .

[122]  J.H. van Schuppen,et al.  Observability of hybrid systems and turing machines , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

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

[124]  David Hiebeler Dynamics and Resistance to Neighborhood Perturbations of Discrete-and Continuous-Time Cellular Automata , 2006, J. Cell. Autom..

[125]  Symeon Bozapalidis Extending Stochastic and Quantum Functions , 2002, Theory of Computing Systems.

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

[127]  Klaus Meer,et al.  Real Interactive Proofs for VPSPACE , 2016, MFCS.

[128]  André Platzer,et al.  Differential Dynamic Logic for Hybrid Systems , 2008, Journal of Automated Reasoning.

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

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

[131]  George J. Pappas,et al.  Discrete abstractions of hybrid systems , 2000, Proceedings of the IEEE.

[132]  Alessandro E. P. Villa,et al.  The expressive power of analog recurrent neural networks on infinite input streams , 2012, Theor. Comput. Sci..

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

[134]  Joël Ouaknine,et al.  Model-Checking for Real-Time Systems , 1995, FCT.

[135]  Serafim Rodrigues,et al.  Turing Computation with Recurrent Artificial Neural Networks , 2015, CoCo@NIPS.

[136]  Kurt Mehlhorn,et al.  Physarum Can Compute Shortest Paths: Convergence Proofs and Complexity Bounds , 2013, ICALP.

[137]  Benjamin W. Wah,et al.  Trace-Based Methods for Solving Nonlinear Global Optimization and Satisfiability Problems , 1997, J. Glob. Optim..

[138]  Karl Svozil,et al.  Scale-invariant cellular automata and self-similar Petri nets , 2006, nlin/0606007.

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

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

[141]  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..

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

[143]  Luca Cardelli,et al.  Programmable chemical controllers made from DNA. , 2013, Nature nanotechnology.

[144]  Andrew Adamatzky,et al.  Physarum Machine: Implementation of a Kolmogorov-Uspensky Machine on a Biological substrate , 2007, Parallel Process. Lett..

[145]  Harvey M. Friedman,et al.  Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory , 1971 .

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

[147]  Alessandro E. P. Villa,et al.  Computational capabilities of recurrent neural networks based on their attractor dynamics , 2015, 2015 International Joint Conference on Neural Networks (IJCNN).

[148]  Saugata Basu,et al.  Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem , 2008, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[149]  Jerzy Mycka,et al.  A foundation for real recursive function theory , 2009, Ann. Pure Appl. Log..

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

[151]  SimonsenJakob Grue,et al.  Programming in Biomolecular Computation , 2010 .

[152]  Hava T. Siegelmann,et al.  Neural networks and analog computation - beyond the Turing limit , 1999, Progress in theoretical computer science.

[153]  Amir Pnueli,et al.  Low dimensional hybrid systems - decidable, undecidable, don't know , 2012, Inf. Comput..

[154]  Paul G. Spirakis,et al.  On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols , 2015, ICALP.

[155]  Martin D. Davis The Myth of Hypercomputation , 2004 .

[156]  Ho-Lin Chen,et al.  Deterministic function computation with chemical reaction networks , 2012, Natural Computing.

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

[158]  Rachid Guerraoui,et al.  Names Trump Malice: Tiny Mobile Agents Can Tolerate Byzantine Failures , 2009, ICALP.

[159]  Charles H. Bennett,et al.  The thermodynamics of computation—a review , 1982 .

[160]  Pekka Orponen,et al.  Universal computation by finite two-dimensional coupled map lattices , 1996 .

[161]  Erik Winfree,et al.  Physical principles for DNA tile self-assembly. , 2017, Chemical Society reviews.

[162]  Klaus Meer,et al.  Periodic Generalized Automata over the Reals , 2016, LATA.

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

[164]  Arnold Schönhage Storage Modification Machines , 1980, SIAM J. Comput..

[165]  Luca Cardelli,et al.  Strand algebras for DNA computing , 2009, Natural Computing.

[166]  Gérard Berry,et al.  The chemical abstract machine , 1989, POPL '90.

[167]  Jürgen Schmidhuber,et al.  Long Short-Term Memory , 1997, Neural Computation.

[168]  David Soloveichik,et al.  Robustness of Expressivity in Chemical Reaction Networks , 2016, DNA.

[169]  Nachum Dershowitz,et al.  A Natural Axiomatization of Computability and Proof of Church's Thesis , 2008, Bulletin of Symbolic Logic.

[170]  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.

[171]  Amir M. Ben-Amram Mortality of iterated piecewise affine functions over the integers: Decidability and complexity , 2015, Comput..

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

[173]  S. Sastry,et al.  Is there life after Zeno? Taking executions past the breaking (Zeno) point , 2006, 2006 American Control Conference.

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

[175]  N. C. A. da Costa,et al.  Undecidable hopf bifurcation with undecidable fixed point , 1994 .

[176]  Amir Ali Ahmadi,et al.  Stability of Polynomial Differential Equations: Complexity and Converse Lyapunov Questions , 2013, ArXiv.

[177]  Edwin J. Beggs,et al.  An Analogue-Digital Model of Computation: Turing Machines with Physical Oracles , 2017 .

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

[180]  G. Seelig,et al.  DNA as a universal substrate for chemical kinetics , 2010, Proceedings of the National Academy of Sciences.

[181]  Johanne Cohen,et al.  Homonym Population Protocols , 2017, Theory of Computing Systems.

[182]  Rupak Majumdar,et al.  Verification of population protocols , 2016, Acta Informatica.

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

[184]  François Fages,et al.  Cells as Machines: Towards Deciphering Biochemical Programs in the Cell , 2014, ICDCIT.

[185]  Alessandro E. P. Villa,et al.  The Super-Turing Computational Power of Interactive Evolving Recurrent Neural Networks , 2013, ICANN.

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

[187]  Nachum Dershowitz,et al.  Axiomatizing Analog Algorithms , 2016, CiE.

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

[189]  Michael R. Williams,et al.  A history of computing technology , 1985 .

[190]  Xiaobo Sharon Hu,et al.  Efficient Analog Circuits for Boolean Satisfiability , 2016, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

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

[192]  Gheorghe Paun,et al.  A guide to membrane computing , 2002, Theor. Comput. Sci..

[193]  Joël Ouaknine,et al.  On the Skolem Problem for Continuous Linear Dynamical Systems , 2015, ICALP.

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

[195]  Mark Priestley,et al.  ENIAC in Action: Making and Remaking the Modern Computer , 2016 .

[196]  Bernard Chazelle,et al.  Natural algorithms , 2009, SODA.

[197]  C. Lee Giles,et al.  The Neural Network Pushdown Automaton: Model, Stack and Learning Simulations , 2017, ArXiv.

[198]  L. Rubel A universal differential equation , 1981 .

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

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

[201]  B. Jack Copeland,et al.  Hypercomputation: philosophical issues , 2004, Theor. Comput. Sci..

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

[203]  W. Paul Cockshott,et al.  Physical constraints on hypercomputation , 2008, Theor. Comput. Sci..

[204]  Joël Ouaknine,et al.  On Reachability for Hybrid Automata over Bounded Time , 2011, ICALP.

[205]  Florent Franchette La thèse de l'hyper-calcul : enjeux et problèmes philosophiques , 2012 .

[206]  Mark Braverman,et al.  Space-bounded Church-Turing thesis and computational tractability of closed systems , 2015, Physical review letters.

[207]  Mark Braverman,et al.  Tight space-noise tradeoffs in computing the ergodic measure , 2015, ArXiv.

[208]  B. Chazelle An Algorithmic Approach to Collective Behavior , 2015 .

[209]  Grzegorz Rozenberg,et al.  Handbook of Natural Computing , 2011, Springer Berlin Heidelberg.

[210]  D. M. V. Hesteren Evolutionary Game Theory , 2017 .

[211]  James Renegar,et al.  A mathematical view of interior-point methods in convex optimization , 2001, MPS-SIAM series on optimization.

[212]  Cristopher Moore,et al.  The Nature of Computation , 2011 .

[213]  Andrew Adamatzky,et al.  Atlas of Physarum Computing , 2015 .

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

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

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

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

[218]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

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

[220]  Dan ie l T. Gil lespie A rigorous derivation of the chemical master equation , 1992 .

[221]  Xiaoyuan Li,et al.  Real-Time Computability of Real Numbers by Chemical Reaction Networks , 2017, UCNC.

[222]  Alessandro E. P. Villa,et al.  An Attractor-Based Complexity Measurement for Boolean Recurrent Neural Networks , 2014, PloS one.

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

[224]  J. V. Tucker,et al.  Experimental computation of real numbers by Newtonian machines , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[225]  Andrew S. Cassidy,et al.  A million spiking-neuron integrated circuit with a scalable communication network and interface , 2014, Science.

[226]  David Soloveichik,et al.  Probability 1 computation with chemical reaction networks , 2014, Natural Computing.

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

[228]  Walid Gomaa,et al.  On The Complexity of Bounded Time Reachability for Piecewise Affine Systems , 2014, RP.

[229]  Maël Pégny Les deux formes de la thèse de Church-Turing et l’épistémologie du calcul , 2012 .

[230]  Tetsuya Asai,et al.  Reaction-diffusion computers , 2005 .

[231]  Edward William Blakey,et al.  A model-independent theory of computational complexity : from patience to precision and beyond , 2010 .

[232]  Masahiro Nagamatu,et al.  On the stability of Lagrange programming neural networks for satisfiability problems of propositional calculus , 1996, Neurocomputing.

[233]  Klaus Meer,et al.  On Ladner's result for a class of real machines with restricted use of constants , 2012, Inf. Comput..

[234]  Saugata Basu A Complex Analogue of Toda’s Theorem , 2012, Found. Comput. Math..

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

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

[237]  A. Tero,et al.  A mathematical model for adaptive transport network in path finding by true slime mold. , 2007, Journal of theoretical biology.

[238]  Amaury Pouly,et al.  A Universal Ordinary Differential Equation , 2017, ICALP.

[239]  Bruce J. MacLennan,et al.  Field Computation: A Theoretical Framework for Massively Parallel Analog Computation Parts I -- IV , 1990 .

[240]  Matthew Cook,et al.  Computation with finite stochastic chemical reaction networks , 2008, Natural Computing.

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

[242]  André Platzer Logic & Proofs for Cyber-Physical Systems , 2016, IJCAR.

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

[244]  Charles Care Technology for Modelling - Electrical Analogies, Engineering Practice, and the Development of Analogue Computing , 2010, History of Computing.

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

[246]  D. Gillespie Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

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

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

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

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

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

[252]  N. Margolus Physics-like models of computation☆ , 1984 .

[253]  David Eisenstat,et al.  The computational power of population protocols , 2006, Distributed Computing.

[254]  J. Hofbauer,et al.  Evolutionary game dynamics , 2011 .

[255]  Jérôme Olivier Durand-Lose,et al.  Abstract Geometrical Computation 1: Embedding Black Hole Computations with Rational Numbers , 2006, Fundam. Informaticae.

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

[257]  Bruce J. MacLennan,et al.  Field Computation in Natural and Artificial Intelligence , 1999, Encyclopedia of Complexity and Systems Science.

[258]  Annick Lesne,et al.  Discrete-time and continuous-time modelling: some bridges and gaps , 2007, Mathematical Structures in Computer Science.

[259]  Indranil Gupta,et al.  The design of novel distributed protocols from differential equations , 2007, Distributed Computing.

[260]  J. V. Tucker,et al.  A hierarchy for BPP// log? based on counting calls to an oracle , 2017 .

[261]  Jeffrey C. Lagarias,et al.  Karmarkar's linear programming algorithm and Newton's method , 1991, Math. Program..

[262]  Michael A. Saunders,et al.  On projected newton barrier methods for linear programming and an equivalence to Karmarkar’s projective method , 1986, Math. Program..

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

[264]  Ho-Lin Chen,et al.  Rate-independent computation in continuous chemical reaction networks , 2014, ITCS.

[265]  Tanya Sienko Molecular Computing , 2005, Genetic Programming and Evolvable Machines.

[266]  André Platzer,et al.  A Uniform Substitution Calculus for Differential Dynamic Logic , 2015, CADE.

[267]  Erik Winfree,et al.  Complexity of Self-Assembled Shapes , 2004, SIAM J. Comput..

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

[269]  Larry Owens,et al.  Vannevar Bush and the Differential Analyzer: The Text and Context of an Early Computer , 1991 .

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

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

[272]  Scott Messick Compactness in the Theory of Continuous Automata , 2016, LFCS.

[273]  Klaus Meer,et al.  Some Aspects of a Complexity Theory for Continuous Time Systems , 2007, CiE.

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

[275]  Nachum Dershowitz,et al.  Towards an Axiomatization of Simple Analog Algorithms , 2012, TAMC.

[276]  Bernard Chazelle,et al.  Diffusive Influence Systems , 2015, SIAM J. Comput..

[277]  Amaury Pouly,et al.  Polynomial Time Corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length , 2016, J. ACM.

[278]  Robin Gandy,et al.  Church's Thesis and Principles for Mechanisms , 1980 .

[279]  Edwin J. Beggs,et al.  Computational complexity with experiments as oracles , 2008, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[280]  Julien Cervelle,et al.  Constructing Continuous Systems from Discrete Cellular Automata , 2013, CiE.

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

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

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

[284]  Andrew Adamatzky,et al.  Physarum Machines: Computers from Slime Mould , 2010 .

[285]  Kurt M. Anstreicher,et al.  Linear programming and the newton barrier flow , 1988, Math. Program..

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

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

[288]  Pierre Fraigniaud,et al.  Computing with Large Populations Using Interactions , 2012, MFCS.

[289]  Phil Blunsom,et al.  Learning to Transduce with Unbounded Memory , 2015, NIPS.

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

[291]  André Platzer,et al.  Logical Analysis of Hybrid Systems - Proving Theorems for Complex Dynamics , 2010 .

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

[293]  A.D. Ames,et al.  Characterization of Zeno behavior in hybrid systems using homological methods , 2005, Proceedings of the 2005, American Control Conference, 2005..

[294]  Nachum Dershowitz,et al.  The Church-Turing Thesis over Arbitrary Domains , 2008, Pillars of Computer Science.

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

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

[297]  Alex Graves,et al.  Neural Turing Machines , 2014, ArXiv.

[298]  Klaus Meer,et al.  Generalized finite automata over real and complex numbers , 2014, Theor. Comput. Sci..

[299]  Pietro Milici Tractional Motion Machines Extend GPAC-generable Functions , 2012, Int. J. Unconv. Comput..

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

[301]  Amir Ali Ahmadi,et al.  Switched stability of nonlinear systems via SOS-convex Lyapunov functions and semidefinite programming , 2013, 52nd IEEE Conference on Decision and Control.

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

[303]  Jun Gu,et al.  On optimizing the satisfiability (SAT) problem , 1999, Journal of Computer Science and Technology.

[304]  Yuri Gurevich,et al.  Sequential abstract-state machines capture sequential algorithms , 2000, TOCL.

[305]  Klaus Meer,et al.  An Algebraic Proof of the Real Number PCP Theorem , 2015, MFCS.

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