The Complexity of Computations

This chapter provides a survey of the main concepts and results of the computational complexity theory. We start with basic concepts such as time and space complexities, complexity classes P and NP, and Boolean circuits. We discuss how the use of randomness and interaction enables one to compute more efficiently. In this context we also mention some basic concepts from theoretical cryptography. We then discuss two related ways to make computations more efficient: the use of processors working in parallel and the use of quantum circuits. Among other things, we explain Shor’s quantum algorithm for factoring integers. The topic of the last section is important for the foundations of mathematics, although it is less related to computational complexity; it is about algorithmic complexity of finite strings of bits.

[1]  M. Tsfasman,et al.  Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound , 1982 .

[2]  Andrei N. Kolmogorov,et al.  On Tables of Random Numbers (Reprinted from "Sankhya: The Indian Journal of Statistics", Series A, Vol. 25 Part 4, 1963) , 1998, Theor. Comput. Sci..

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

[4]  N. Koblitz.  A Course in Number Theory and Cryptography , 1987 .

[5]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[6]  Claude E. Shannon,et al.  The synthesis of two-terminal switching circuits , 1949, Bell Syst. Tech. J..

[7]  P. Erdös.  Some remarks on the theory of graphs , 1947 .

[8]  A. Razborov.  Lower bounds on the size of bounded depth circuits over a complete basis with logical addition , 1987 .

[9]  Norbert Blum.  A Boolean Function Requiring 3n Network Size , 1984, Theor. Comput. Sci..

[10]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[11]  Volker Strassen,et al.  A Fast Monte-Carlo Test for Primality , 1977, SIAM J. Comput..

[12]  Lajos Rónyai,et al.  Norm-graphs and bipartite turán numbers , 1996, Comb..

[13]  Russell Impagliazzo,et al.  Derandomizing Polynomial Identity Tests Means Proving Circuit Lower Bounds , 2003, STOC '03.

[14]  D. Ulig.  On the synthesis of self-correcting schemes from functional elements with a small number of reliable elements , 1974 .

[15]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[16]  Alexander A. Razborov,et al.  On the method of approximations , 1989, STOC '89.

[17]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[18]  Pavel Pudlák.  Quantum deduction rules , 2009, Ann. Pure Appl. Log..

[19]  R. Impagliazzo,et al.  P=BPP unless E has sub-exponential circuits: Derandomizing the XOR Lemma , 2002 .

[20]  Miklós Ajtai,et al.  ∑11-Formulae on finite structures , 1983, Ann. Pure Appl. Log..

[21]  Gregory J. Chaitin,et al.  On the Simplicity and Speed of Programs for Computing Infinite Sets of Natural Numbers , 1969, J. ACM.

[22]  Leslie G. Valiant,et al.  On Time Versus Space , 1977, JACM.

[23]  Kristoffer Arnsfelt Hansen,et al.  Tight Bounds on Computing Error-Correcting Codes by Bounded-Depth Circuits With Arbitrary Gates , 2012, IEEE Transactions on Information Theory.

[24]  R. Solovay,et al.  Relativizations of the $\mathcal{P} = ?\mathcal{NP}$ Question , 1975 .

[25]  M. Rabin.  DIGITALIZED SIGNATURES AND PUBLIC-KEY FUNCTIONS AS INTRACTABLE AS FACTORIZATION , 1979 .

[26]  Charles H. Bennett,et al.  Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. , 1992, Physical review letters.

[27]  Charles H. Bennett,et al.  Logical reversibility of computation , 1973 .

[28]  Qi Cheng.  Straight-line programs and torsion points on elliptic curves , 2003, computational complexity.

[29]  Ketan Mulmuley,et al.  Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems , 2002, SIAM J. Comput..

[30]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[31]  Richard E. Ladner,et al.  On the Structure of Polynomial Time Reducibility , 1975, JACM.

[32]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[33]  Adi Shamir,et al.  A method for obtaining digital signatures and public-key cryptosystems , 1978, CACM.

[34]  Pavel Pudlák.  Complexity Theory and Genetics: The Computational Power of Crossing Over , 2001, Inf. Comput..

[35]  Leslie G. Valiant,et al.  The Complexity of Computing the Permanent , 1979, Theor. Comput. Sci..

[36]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[37]  Noam Nisan,et al.  Hardness vs Randomness , 1994, J. Comput. Syst. Sci..

[38]  Walter J. Savitch,et al.  Time Bounded Random Access Machines with Parallel Processing , 1979, JACM.

[39]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[40]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[41]  Sanjeev Arora,et al.  Probabilistic checking of proofs: a new characterization of NP , 1998, JACM.

[42]  Stefan Reisch,et al.  Hex ist PSPACE-vollständig , 1981, Acta Informatica.

[43]  Arnold Schönhage,et al.  Schnelle Multiplikation großer Zahlen , 1971, Computing.

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

[45]  Greg Kuperberg.  A Subexponential-Time Quantum Algorithm for the Dihedral Hidden Subgroup Problem , 2005, SIAM J. Comput..

[46]  László Babai,et al.  Trading group theory for randomness , 1985, STOC '85.

[47]  J. Hartmanis,et al.  On the Computational Complexity of Algorithms , 1965 .

[48]  R. Solomonoff.  A PRELIMINARY REPORT ON A GENERAL THEORY OF INDUCTIVE INFERENCE , 2001 .

[49]  Alexander A. Razborov,et al.  Natural Proofs , 1997, J. Comput. Syst. Sci..

[50]  Gary L. Miller.  Riemann's Hypothesis and Tests for Primality , 1976, J. Comput. Syst. Sci..

[51]  L. Vaidman,et al.  Quantum mechanical interaction-free measurements , 1993, hep-th/9305002.

[52]  S. Smale,et al.  On a theory of computation and complexity over the real numbers; np-completeness , 1989 .

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

[54]  A. Harrow,et al.  Quantum algorithm for linear systems of equations. , 2008, Physical review letters.

[55]  R. Soare,et al.  Π⁰₁ classes and degrees of theories , 1972 .

[56]  David Hilbert.  Ueber die Endlichkeit des Invariantensystems für binäre Grundformen , 1888 .