Robust Simulations of Turing Machines with Analytic Maps and Flows

In this paper, we show that closed-form analytic maps and flows can simulate Turing machines in an error-robust manner. The maps and ODEs defining the flows are explicitly obtained and the simulation is performed in real time.

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

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

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

[4]  Begnaud Francis Hildebrand,et al.  Introduction to numerical analysis: 2nd edition , 1987 .

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

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

[7]  Alison Connolly.  Human skills (2nd Edn) , 1991 .

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

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

[10]  Marcelo Viana,et al.  Dynamical Systems: Moving into the Next Century , 2001 .

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

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

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

[14]  B. Engquist,et al.  Mathematics Unlimited: 2001 and Beyond , 2001 .

[15]  Celso Grebogi,et al.  Shadowability of Chaotic Dynamical Systems , 2002 .

[16]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[17]  M. Hirsch,et al.  Differential Equations, Dynamical Systems, and Linear Algebra , 1974 .

[18]  Eduardo Sontag,et al.  Computational power of neural networks , 1995 .

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

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

[21]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[22]  S. Pilyugin.  Shadowing in dynamical systems , 1999 .

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

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

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

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

[27]  Arnold Neumaier,et al.  Introduction to Numerical Analysis , 2001 .

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