A Computable Economist’s Perspective on Computational Complexity

A computable economist.s view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called "Post's Program of Research for Higher Recursion Theory". Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix.

[1]  Avi Wigderson,et al.  P , NP and mathematics – a computational complexity perspective , 2006 .

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

[3]  Michael Sipser,et al.  Borel sets and circuit complexity , 1983, STOC.

[4]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences , 1966, JACM.

[5]  Karen A. Frenkel.  Complexity and parallel processing: an interview with Richard Karp , 1986, CACM.

[6]  室 章治郎.  Michael R.Garey/David S.Johnson 著, "COMPUTERS AND INTRACTABILITY A guide to the Theory of NP-Completeness", FREEMAN, A5判変形判, 338+xii, \5,217, 1979 , 1980 .

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

[8]  R. Soare.  Recursively enumerable sets and degrees , 1987 .

[9]  P. Odifreddi.  The theory of functions and sets of natural numbers , 1989 .

[10]  Vaughan R. Pratt,et al.  Every Prime has a Succinct Certificate , 1975, SIAM J. Comput..

[11]  F. Richman,et al.  Varieties of Constructive Mathematics: CONSTRUCTIVE ALGEBRA , 1987 .

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

[13]  G. Lorentz.  Approximation of Functions , 1966 .

[14]  Kenneth Steiglitz,et al.  Combinatorial Optimization: Algorithms and Complexity , 1981 .

[15]  Michael Sipser,et al.  The history and status of the P versus NP question , 1992, STOC '92.

[16]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[17]  K. Velupillai.  Economic Dynamics and Computation-Resurrecting the Icarus Tradition , 2004 .

[18]  B. Cipra,et al.  What's happening in the mathematical sciences , 1998 .

[19]  V. Uspensky.  Complexity and Entropy: An Introduction to the Theory of Kolmogorov Complexity , 1992 .

[20]  J. Borwein,et al.  On the complexity of familiar functions and numbers , 1988 .

[21]  A. N. Kolmogorov.  Combinatorial foundations of information theory and the calculus of probabilities , 1983 .

[22]  R. Goodwin.  Iteration, Automatic Computers, and Economic Dynamics , 1982 .

[23]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part I , 1964, Inf. Control..

[24]  W. R. Howard.  Gödel's Theorem: An Incomplete Guide to its Use and Abuse , 2006 .

[25]  J. Shepherdson.  Computational Complexity of Real Functions , 1985 .

[26]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[27]  J. Kelly.  Computability and Logic , 1988 .

[28]  Some rapidly growing functions , 1980 .

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

[30]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[31]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[32]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[33]  XVI. On the relation to Physics of the notion of convergence of series , 1926 .

[34]  Michiel van Lambalgen,et al.  Independence, randomness and the axiom of choice , 1992, Journal of Symbolic Logic.

[35]  Herbert S. Wilf,et al.  Algorithms and Complexity , 1994, Lecture Notes in Computer Science.

[36]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[37]  R. Lucas.  Making a miracle , 1993 .

[38]  Martin D. Davis,et al.  Computability and Unsolvability , 1959, McGraw-Hill Series in Information Processing and Computers.

[39]  Eric Schechter,et al.  Handbook of Analysis and Its Foundations , 1996 .

[40]  K. A. Frenkel.  Complexity and parallel processing , 1986 .

[41]  Alexander Shen,et al.  Relations between varieties of kolmogorov complexities , 1996 .

[42]  J. Barkley Rosser,et al.  Extensions of some theorems of Gödel and Church , 1936, Journal of Symbolic Logic.

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

[44]  Ray J. Solomonoff,et al.  A Formal Theory of Inductive Inference. Part II , 1964, Inf. Control..

[45]  Jr. Hartley Rogers.  Theory of Recursive Functions and Effective Computability , 1969 .

[46]  Andrew Donald Booth,et al.  Theory of the transmission and processing of information , 1961 .

[47]  V. Tikhomirov,et al.  A.N. Kolmogorov and approximation theory , 1989 .

[48]  Fred Richman,et al.  Intuitionism As Generalization , 1990 .

[49]  W. W. Ball,et al.  Mathematical Recreations and Essays , 1905, Nature.

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

[51]  S. C. Kleene,et al.  Introduction to Metamathematics , 1952 .

[52]  George Boolos,et al.  Computability and logic , 1974 .

[53]  Stephen Smale.  Theory of computation , 1992 .

[54]  F. N. Cole.  On the factoring of large numbers , 1903 .

[55]  Emil L. Post.  Recursively enumerable sets of positive integers and their decision problems , 1944 .

[56]  Ernst Specker,et al.  Nicht konstruktiv beweisbare Sätze der Analysis , 1949, Journal of Symbolic Logic.

[57]  G. Malajovich.  Ko Ker-I. Complexity theory of real functions. Progress in theoretical computeer science. Birkhäuser, Boston, Basel, and Berlin, 1991, viii + 309 pp. , 1992 .

[58]  J. Rosser.  Computational and Dynamic Complexity in Economics , 2009 .

[59]  R. Feynman.  Simulating physics with computers , 1999 .

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

[61]  Carlos Casacuberta,et al.  Mathematical research today and tomorrow : viewpoints of seven Fields Medalists , 1992 .

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

[63]  Raymond M. Smullyan,et al.  Godel's Incompleteness Theorems , 1992 .