New Computational Paradigms: Changing Conceptions of What is Computable

In recent years, classical computability has expanded beyond its original scope to address issues related to computability and complexity in algebra, analysis, and physics. The deep interconnection between "computation" and "proof" has originated much of the most significant work in constructive mathematics and mathematical logic of the last 70 years. Moreover, the increasingly compelling necessity to deal with computability in the real world (such as computing on continuous data, biological computing, and physical models) has brought focus to new paradigms of computation that are based on biological and physical models. These models address questions of efficiency in a radically new way and even threaten to move the so-called Turing barrier, i.e. the line between the decidable and the un-decidable. This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Andrew Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computations allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory. New paradigms of computation arising from biology and quantum physics are also discussed, as well as the computability of the real numbers and its related issues. This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.

[1]  S. S. Goncharov,et al.  Constructive models of complete solvable theories , 1973 .

[2]  Ulrich Kohlenbach,et al.  On the Computational Content of the Krasnoselski and Ishikawa Fixed Point Theorems , 2000, CCA.

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

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

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

[6]  Yuri L. Ershov,et al.  Theory of Numberings , 1999, Handbook of Computability Theory.

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

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

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

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

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

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

[13]  W. G. Dotson,et al.  On the Mann iterative process , 1970 .

[14]  Christopher J. Ash.  Categoricity in hyperarithmetical degrees , 1987, Ann. Pure Appl. Log..

[15]  Ulrich Kohlenbach,et al.  Arithmetizing proofs in analysis , 1998 .

[16]  U. Kohlenbach.  Analysing proofs in analysis , 1996 .

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

[18]  Charles N. Delzell.  Case Distinctions are Necessary for Representing polynomials as Sums of Squares , 1982 .

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

[20]  Andrea Sorbi,et al.  Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy , 2006 .

[21]  Ulrich Kohlenbach.  Some Logical Metatheorems with Applications in Functional Analysis , 2003 .

[22]  Lawrence Feiner,et al.  Hierarchies of Boolean algebras , 1970, Journal of Symbolic Logic.

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

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

[25]  J. Hopfield.  Neurons withgraded response havecollective computational properties likethoseoftwo-state neurons , 1984 .

[26]  Georg Kreisel,et al.  Constructive Logic Versus Algebraization I , 1982 .

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

[28]  Andreas Weiermann,et al.  A CLASSIFICATION OF RAPIDLY GROWING RAMSEY FUNCTIONS , 1976 .

[29]  William A. Kirk,et al.  A FIXED POINT THEOREM FOR ASYMPTOTICALLY NONEXPANSIVE MAPPINGS , 1972 .

[30]  Rohit Parikh,et al.  Existence and feasibility in arithmetic , 1971, Journal of Symbolic Logic.

[31]  C. W. Groetsch,et al.  A Note on Segmenting Mann Iterates , 1972 .

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

[33]  T. Coquand,et al.  Generating non-Noetherian modules constructively , 2004 .

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

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

[36]  Serikzhan Badaev,et al.  Minimal Coverings in the Rogers Semilattices of∑-Computable Numberings , 2002 .

[37]  Wolfgang Gehlen,et al.  Regular Article: On a Conjecture Concerning Strong Unicity Constants , 1999 .

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

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

[40]  Stephen G. Simpson,et al.  Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[41]  J. A. Clarkson.  Uniformly convex spaces , 1936 .

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

[43]  Myron S Henry,et al.  Continuity theorems for the rational product approximation operator , 1977 .

[44]  Serikzhan Badaev,et al.  COMPUTABLE NUMBERINGS IN THE HIERARCHY OF ERSHOV , 2006 .

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

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

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

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

[49]  Wataru Takahashi,et al.  A convexity in metric space and nonexpansive mappings, I , 1970 .

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

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

[52]  Paulo Oliva,et al.  Proof Mining: A Systematic Way of Analysing Proofs in Mathematics , 2002 .

[53]  S. Badaev.  ELEMENTARY PROPERTIES OF ROGERS SEMILATTICES OF ARITHMETICAL NUMBERINGS , 2004 .

[54]  Pavel Pudlák,et al.  Cuts, consistency statements and interpretations , 1985, Journal of Symbolic Logic.

[55]  S. S. Goncharov,et al.  Generalized computable numerations and nontrivial rogers semilattices , 1997 .

[56]  Christopher J. Ash,et al.  Pairs of computable structures , 1990 .

[57]  W. A. Kirk,et al.  Iteration processes for nonexpansive mappings , 1983 .

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

[59]  Ulrich Kohlenbach,et al.  Approximation , 2018, Passive, Active, and Digital Filters.

[60]  S. Ishikawa.  Fixed points and iteration of a nonexpansive mapping in a Banach space , 1976 .

[61]  Jürgen Schu,et al.  Iterative construction of fixed points of asymptotically nonexpansive mappings , 1991 .

[62]  U. Kohlenbach,et al.  Bounds on Iterations of Asymptotically Quasi-Nonexpansive Mappings , 2003 .

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

[64]  Jeff B. Paris,et al.  A Note on the Undefinability of Cuts , 1983, J. Symb. Log..

[65]  W. A. Kirk,et al.  Topics in Metric Fixed Point Theory , 1990 .

[66]  Georg Kreisel,et al.  On the interpretation of non-finitist proofs—Part I , 1951, Journal of Symbolic Logic.

[67]  Peter Clote,et al.  Bounded Arithmetic for NC, ALogTIME, L and NL , 1992, Ann. Pure Appl. Log..

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

[69]  Wolfgang Maass,et al.  Computation with spiking neurons , 2003 .

[70]  Philipp Gerhardy.  A Quantitative Version of Kirk's Fixed Point Theorem for Asymptotic Contractions , 2004 .

[71]  Tomonari Suzuki,et al.  Fixed-point theorem for asymptotic contractions of Meir–Keeler type in complete metric spaces , 2006 .

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

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

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

[75]  Serikzhan A. Badaev.  On Rogers Semilattices , 2006, TAMC.

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

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

[78]  Julia F. Knight,et al.  Enumerations in computable structure theory , 2005, Ann. Pure Appl. Log..

[79]  Andrea Sorbi,et al.  On elementary theories and isomorphism types of Rogers semilattices , 2006 .

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

[81]  M. Edelstein.  On Fixed and Periodic Points Under Contractive Mappings , 1962 .

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

[83]  Charles N. Delzell.  Continuous Sums of Squares of Forms , 1982 .

[84]  Ulrich Kohlenbach,et al.  The approximate fixed point property in product spaces , 2005, math/0510563.

[85]  Philipp Gerhardy,et al.  Strongly uniform bounds from semi-constructive proofs , 2004, Ann. Pure Appl. Log..

[86]  U. Kohlenbach.  A QUANTITATIVE VERSION OF A THEOREM DUE TO BORWEIN-REICH-SHAFRIR , 2001 .

[87]  S. S. Goncharov,et al.  Computability and Computable Models , 2007 .

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

[89]  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 .

[90]  Andrea Sorbi,et al.  Algebraic Properties of Rogers Semilattices of Arithmetical Numberings , 2003 .

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

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

[93]  V. D. Dzgoev.  Constructive enumeration of Boolean lattices , 1988 .

[94]  Ulrich Kohlenbach,et al.  Applied Proof Theory - Proof Interpretations and their Use in Mathematics , 2008, Springer Monographs in Mathematics.

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

[96]  András Kroó,et al.  On the uniform modulus of continuity of the operator of best approximation in the space of periodic functions , 1979 .

[97]  Ulrich Kohlenbach,et al.  Some computational aspects of metric fixed-point theory , 2005 .

[98]  Keijo Ruohonen.  Decidability and complexity of event detection problems for ODEs , 1997 .

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

[100]  U. Kohlenbach.  Foundational and Mathematical Uses of Higher Types , 1999 .

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

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

[103]  Ulrich Kohlenbach,et al.  Effective Moduli from Ineffective Uniqueness Proofs. An Unwinding of de La Vallée Poussin's Proof for Chebycheff Approximation , 1993, Ann. Pure Appl. Log..

[104]  B. Rhoades,et al.  A comparison of various definitions of contractive mappings , 1977 .

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

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

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

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

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

[110]  Steffen Lempp,et al.  Friedberg Numberings of Families of n-Computably Enumerable Sets , 2002 .

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

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

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

[114]  E. Rakotch,et al.  A note on contractive mappings , 1962 .

[115]  Itai Shafrir,et al.  Nonexpansive iterations in hyperbolic spaces , 1990 .

[116]  S. Buss,et al.  An Application of Boolean Complexity to Separation Problems in Bounded Arithmetic , 1994 .

[117]  S. S. Goncharov.  Computable single-valued numerations , 1980 .

[118]  C. Spector.  Provably recursive functionals of analysis: a consistency proof of analysis by an extension of princ , 1962 .

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

[120]  Wolfgang Maass,et al.  Paradigms for Computing with Spiking Neurons , 2002 .

[121]  Laurentiu Leustean.  A quadratic rate of asymptotic regularity for CAT(0)-spaces , 2005 .

[122]  M. Edelstein,et al.  Nonexpansive Mappings, Asymptotic Regularity and Successive Approximations , 1978 .

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

[124]  S. S. Goncharov,et al.  Problem of the number of non-self-equivalent constructivizations , 1980 .

[125]  Mircea-Dan Hernest.  Synthesis of Moduli of Uniform Continuity by the Monotone Dialectica Interpretation in the Proof-system MinLog , 2007, Electron. Notes Theor. Comput. Sci..

[126]  András Kroó,et al.  On the continuity of best approximations in the space of integrable functions , 1978 .

[127]  Chung-Chih Li,et al.  Computability in Europe 2007 Computation and Logic in the Real World , 2007 .

[128]  Ulrich Kohlenbach,et al.  Mann iterates of directionally nonexpansive mappings in hyperbolic spaces , 2002 .

[129]  Brailey Sims.  Examples of Fixed Point Free Mappings , 2001 .

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

[131]  Ulrich Kohlenbach,et al.  Mathematically strong subsystems of analysis with low rate of growth of provably recursive functionals , 1996, Arch. Math. Log..

[132]  William A. Kirk,et al.  Approximate fixed points for nonexpansive mappings in uniformly convex spaces , 1990 .

[133]  William A. Kirk,et al.  Nonexpansive mappings and asymptotic regularity , 2000 .

[134]  Laurentiu Leustean.  Proof Mining in R-trees and Hyperbolic Spaces , 2006, Electron. Notes Theor. Comput. Sci..

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

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

[137]  Branimir Lambov.  Rates of Convergence of Recursively Defined Sequences , 2005, Electron. Notes Theor. Comput. Sci..

[138]  Von Kurt Gödel,et al.  ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES , 1958 .

[139]  Paulo Oliva,et al.  Proof mining in L1-approximation , 2003, Ann. Pure Appl. Log..

[140]  H. Luckhardt,et al.  Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken , 1989, Journal of Symbolic Logic.

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

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

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

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

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

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

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

[148]  S. S. Goncharov,et al.  The quantity of nonautoequivalent constructivizations , 1977 .

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

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

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

[152]  D. Jackson,et al.  Note on a class of polynomials of approximation , 1921 .

[153]  Ulrich Kohlenbach,et al.  Proof theory and computational analysis , 1997, COMPROX.

[154]  Michael R. Williams,et al.  About This Issue , 1998, IEEE Ann. Hist. Comput..

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

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

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

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

[159]  Andreas Weiermann,et al.  Phasenübergänge in Logik und Kombinatorik , 2005 .

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

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

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

[163]  F. Browder,et al.  The solution by iteration of nonlinear functional equations in Banach spaces , 1966 .

[164]  Ker-I Ko.  On the computational complexity of best Chebyshev approximations , 1986, J. Complex..

[165]  S. S. Goncharov,et al.  Rogers Semilattices of Families of Arithmetic Sets , 2001 .

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

[167]  S. Goncharov,et al.  Computable Structure and Non-Structure Theorems , 2002 .

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

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

[170]  A. Lachlan.  ON THE LATTICE OF RECURSIVELY ENUMERABLE SETS , 1968 .

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

[172]  Elliott Ward Cheney,et al.  An Elementary Proof of Jackson's Theorem on Mean-Approximation , 1965 .

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

[174]  Richard M. Friedberg,et al.  Three theorems on recursive enumeration. I. Decomposition. II. Maximal set. III. Enumeration without duplication , 1958, Journal of Symbolic Logic.

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

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

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

[178]  Paulo Oliva.  On the Computational Complexity of Best L1-approximation , 2002, Math. Log. Q..

[179]  Jerzy Mycka,et al.  What Lies Beyond the Mountains? , 2005, Bull. EATCS.

[180]  Andrea Sorbi,et al.  Isomorphism Types and Theories of Rogers Semilattices of Arithmetical Numberings , 2003 .

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

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

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

[184]  Thierry Coquand.  Sur un théorème de Kronecker concernant les variétés algébriques , 2004 .

[185]  Michel Coste,et al.  Dynamical method in algebra: effective Nullstellensätze , 2001, Ann. Pure Appl. Log..

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

[187]  Harold S. Shapiro,et al.  Some theorems on Čebyšev approximation , 1963 .

[188]  William A. Kirk,et al.  Fixed points of asymptotic contractions , 2003 .

[189]  Ulrich Kohlenbach,et al.  A Logical Uniform Boundedness Principle for Abstract Metric and Hyperbolic Spaces , 2006, WoLLIC.

[190]  Liu Qihou.  Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings with Error Member☆ , 2001 .

[191]  Ulrich Kohlenbach.  Uniform Asymptotic Regularity for Mann Iterates , 2002 .

[192]  Richard J. Lipton,et al.  Model theoretic aspects of computational complexity , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

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

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