Analog computation beyond the Turing limit

The main purpose of this paper is quite uncontroversial. First, we recall some models of analog computations (including these allowed to perform Turing uncomputable tasks). Second, we support the suggestions that such hypercomputable capabilities of such systems can be explained by the use of infinite limits. Additionally, the inner restrictions of analog models of computations are indicated.

[1]  David V. Forrest The Physics of Immortality: Modern Cosmology, God, and Resurrection , 1996 .

[2]  N. Costa,et al.  Undecidability and incompleteness in classical mechanics , 1991 .

[3]  Michael Barr,et al.  The Emperor's New Mind , 1989 .

[4]  Olivier Bournez,et al.  An Analog Characterization of Elementarily Computable Functions over the Real Numbers , 2004, ICALP.

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

[6]  H. Stowell The emperor's new mind R. Penrose, Oxford University Press, New York (1989) 466 pp. $24.95 , 1990, Neuroscience.

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

[8]  P. Odifreddi Classical recursion theory , 1989 .

[9]  Lee A. Rubel,et al.  Some mathematical limitations of the general-purpose analog computer , 1988 .

[10]  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 .

[11]  The Structure of Time. , 1980 .

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

[13]  C. Moore,et al.  Closed-form Analytic Maps in One and Two DimensionsCan Simulate Turing Machines , 1996 .

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

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

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

[17]  A. Grzegorczyk On the definitions of computable real continuous functions , 1957 .

[18]  Gregory J. Chaitin,et al.  The Limits of Mathematics , 1995, J. Univers. Comput. Sci..

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

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

[21]  Jerzy Mycka,et al.  Undecidability over Continuous Time , 2006, Log. J. IGPL.

[22]  Lee A. Rubel,et al.  Digital simulation of analog computation and Church's thesis , 1989, Journal of Symbolic Logic.

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

[24]  Oron Shagrir,et al.  Physical Hypercomputation and the Church–Turing Thesis , 2003, Minds and Machines.

[25]  Jerzy Mycka,et al.  Real Recursive Functions and Baire Classes , 2004, Fundam. Informaticae.

[26]  Yurii Rogozhin,et al.  Small Universal Turing Machines , 1996, Theor. Comput. Sci..

[27]  Frank J. Tipler,et al.  Book-Review - the Physics of Immortality - Modern Cosmology God and the Resurrection of the Dead , 1994 .

[28]  N. Cutland Computability: An Introduction to Recursive Function Theory , 1980 .

[29]  Ian Stewart,et al.  Deciding the undecidable , 1991, Nature.

[30]  Klaus Weihrauch,et al.  The Arithmetical Hierarchy of Real Numbers , 1999, MFCS.

[31]  Vyvyan Evans,et al.  The Structure of Time , 2003 .

[32]  Jerzy Mycka mu -Recursion and infinite limits , 2003, Theor. Comput. Sci..

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

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

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

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

[37]  Zhihong Xia,et al.  The existence of noncollision singularities in newtonian systems , 1992 .