Kolmogorov Complexity and Computational Complexity

There are many ways to measure the complexity of a given object, but there are two measures of particular importance in the theory of computing: One is Kolmogorov complexity, which measures the amount of information necessary to describe an object. Another is computational complexity, which measures the computational resources necessary to recognize (or produce) an object. The relation between these two complexity measures has been studied since the 1960s. More recently, the more generalized notion of resource bounded Kolmogorov complexity and its relation to computational complexity have received much attention. Now many interesting and deep observations on this topic have been established. This book consists of four survey papers concerning these recent studies on resource bounded Kolmogorov complexity and computational complexity. It also contains one paper surveying several types of Kolmogorov complexity measures. The papers are based on invited talks given at the AAAI Spring Symposium on Minimal-Length Encoding in 1990. The book is the only collection of survey papers on this subject and provides fundamental information for researchers in the field.

[1]  A. A..  Probability, Statistics and Truth , 1940, Nature.

[2]  A. Church.  On the concept of a random sequence , 1940 .

[3]  H. Chernoff.  A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations , 1952 .

[4]  E. F. Moore,et al.  On the Possibilities of Designing Circuits out of Various Elements.On the Synthesis of Contact Circuits.On the Synthesis of Contact Networks , 1959 .

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

[6]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

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

[8]  A. Kolmogorov.  Three approaches to the quantitative definition of information , 1968 .

[9]  Andrei N. Kolmogorov,et al.  Logical basis for information theory and probability theory , 1968, IEEE Trans. Inf. Theory.

[10]  Donald W. Loveland,et al.  A Variant of the Kolmogorov Concept of Complexity , 1969, Information and Control.

[11]  Gregory J. Chaitin,et al.  On the Length of Programs for Computing Finite Binary Sequences: statistical considerations , 1969, JACM.

[12]  L. Levin,et al.  THE COMPLEXITY OF FINITE OBJECTS AND THE DEVELOPMENT OF THE CONCEPTS OF INFORMATION AND RANDOMNESS BY MEANS OF THE THEORY OF ALGORITHMS , 1970 .

[13]  Albert R. Meyer,et al.  The Equivalence Problem for Regular Expressions with Squaring Requires Exponential Space , 1972, SWAT.

[14]  Nancy A. Lynch,et al.  Comparison of polynomial-time reducibilities , 1974, STOC '74.

[15]  Nancy A. Lynch,et al.  On Reducibility to Complex or Sparse Sets , 1975, JACM.

[16]  John Gill,et al.  Relativizations of the P =? NP Question , 1975, SIAM J. Comput..

[17]  Juris Hartmanis,et al.  On isomorphisms and density of NP and other complete sets , 1976, STOC '76.

[18]  Leonard Berman,et al.  On the structure of complete sets: Almost everywhere complexity and infinitely often speedup , 1976, 17th Annual Symposium on Foundations of Computer Science (sfcs 1976).

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

[20]  Vladimir A. Uspensky,et al.  What are the gains of the theory of algorithms: Basis developments connected with the concept of algorithm and with its application in mathematics , 1979, Algorithms in Modern Mathematics and Computer Science.

[21]  Nicholas Pippenger,et al.  On simultaneous resource bounds , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[22]  Richard J. Lipton,et al.  Some connections between nonuniform and uniform complexity classes , 1980, STOC '80.

[23]  Daniel J. Moore,et al.  Completeness, Approximation and Density , 1981, SIAM J. Comput..

[24]  Ravi Kannan,et al.  Circuit-Size Lower Bounds and Non-Reducibility to Sparse Sets , 1982, Inf. Control..

[25]  Manuel Blum,et al.  How to generate cryptographically strong sequences of pseudo random bits , 1982, 23rd Annual Symposium on Foundations of Computer Science (sfcs 1982).

[26]  Michael Sipser,et al.  A complexity theoretic approach to randomness , 1983, STOC.

[27]  J. Hartigan.  Theories of Probability , 1983 .

[28]  Juris Hartmanis,et al.  Generalized Kolmogorov complexity and the structure of feasible computations , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[29]  Vijay V. Vazirani,et al.  Trapdoor pseudo-random number generators, with applications to protocol design , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[30]  Péter Gács,et al.  On the relation between descriptional complexity and algorithmic probability , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[31]  Juris Hartmanis,et al.  Computation Times of NP Sets of Different Densities , 1983, ICALP.

[32]  Robert E. Wilber.  Randomness and the density of hard problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[33]  Boris A. Trakhtenbrot,et al.  A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms , 1984, Annals of the History of Computing.

[34]  Timothy J. Long,et al.  Quantitative Relativizations of Complexity Classes , 1984, SIAM J. Comput..

[35]  Leonid A. Levin,et al.  Randomness Conservation Inequalities; Information and Independence in Mathematical Theories , 1984, Inf. Control..

[36]  Ker-I Ko,et al.  Continuous optimization problems and a polynomial hierarchy of real functions , 1985, J. Complex..

[37]  Timothy J. Long.  On Restricting the Size of Oracles Compared with Restricting Access to Oracles , 1985, SIAM J. Comput..

[38]  Ker-I Ko,et al.  On Circuit-Size Complexity and the Low Hierarchy in NP , 1985, SIAM J. Comput..

[39]  Yacov Yacobi,et al.  Hard-core theorems for complexity classes , 1985, JACM.

[40]  Ding-Zhu Du.  Generalized complexity cores and levelability of intractable sets , 1985 .

[41]  Leslie G. Valiant,et al.  NP is as easy as detecting unique solutions , 1985, STOC '85.

[42]  L. Longpré.  Resource Bounded Kolmogorov Complexity, A Link between Computational Complexity & Information Theory , 1986 .

[43]  Dung T. Huynh,et al.  Resource- Bounded Kolmogorov Complexity of Hard Languages , 1986, SCT.

[44]  José L. Balcázar,et al.  The polynomial-time hierarchy and sparse oracles , 1986, JACM.

[45]  Uwe Schöning.  Complexity and Structure , 1986, Lecture Notes in Computer Science.

[46]  Ker-I Ko,et al.  On the Notion of Infinite Pseudorandom Sequences , 1986, Theor. Comput. Sci..

[47]  José L. Balcázar,et al.  Sparse Sets, Lowness and Highness , 1986, SIAM J. Comput..

[48]  Leonid A. Levin,et al.  Average Case Complete Problems , 1986, SIAM J. Comput..

[49]  Timothy J. Long,et al.  Relativizing complexity classes with sparse oracles , 1986, JACM.

[50]  Klaus Ambos-Spies,et al.  Randomness, Relativizations, and Polynomial Reducibilities , 1986, SCT.

[51]  Ding-Zhu Du,et al.  The existence and density of generalized complexity cores , 1987, JACM.

[52]  Dung T. Huynh.  On Solving Hard Problems by Polynomial-Size Circuits , 1987, Inf. Process. Lett..

[53]  Ker-I Ko,et al.  On Sets Truth-Table Reducible to Sparse Sets , 1988, SIAM J. Comput..

[54]  A. Kolmogorov,et al.  ALGORITHMS AND RANDOMNESS , 1988 .

[55]  Osamu Watanabe,et al.  Kolmogorov complexity and degrees of tally sets , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[56]  Ding-Zhu Du,et al.  On polynomial and generalized complexity cores , 1988, [1988] Proceedings. Structure in Complexity Theory Third Annual Conference.

[57]  Noam Nisan,et al.  Hardness vs. randomness , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[58]  Uwe Schöning.  Probabilistic Complexity Classes and Lowness , 1989, J. Comput. Syst. Sci..

[59]  Eric Allender,et al.  Some consequences of the existence of pseudorandom generators , 1987, J. Comput. Syst. Sci..

[60]  Ding-Zhu Du,et al.  On Inefficient Special Cases of NP-Complete Problems , 1989, Theor. Comput. Sci..

[61]  Osamu Watanabe,et al.  Kolmogorov Complexity and Degrees of Tally Sets , 1990, Inf. Comput..

[62]  P. Vitányi,et al.  Applications of Kolmogorov Complexity in the Theory of Computation , 1990 .

[63]  Jack H. Lutz,et al.  Category and Measure in Complexity Classes , 1990, SIAM J. Comput..

[64]  Mitsunori Ogihara,et al.  On one query, self-reducible sets , 1991, [1991] Proceedings of the Sixth Annual Structure in Complexity Theory Conference.

[65]  Jack H. Lutz,et al.  An Upward Measure Separation Theorem , 1991, Theor. Comput. Sci..

[66]  Ming Li,et al.  Average Case Complexity Under the Universal Distribution Equals Worst-Case Complexity , 1992, Inf. Process. Lett..

[67]  Jack H. Lutz.  Almost Everywhere High Nonuniform Complexity , 1992, J. Comput. Syst. Sci..

[68]  Eric Allender,et al.  Lower bounds for the low hierarchy , 1992, JACM.

[69]  Osamu Watanabe,et al.  Instance complexity , 1994, JACM.