The P \(\mathop = \limits^? \) NP-Problem: A View from the 1990s

  • A. A. Razborov

Keywords

Boolean Function Turing Machine Complexity Theory Combinatorial Problem Computable Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. [Aro94]
    S. Arora. Probabilistic checking of proofs and hardness of approximation problems. Electron. Colloq. Comp. Complexity, 1994, 1. Available via http://www.eccc.uni-trier.de/eccc-local/ECCC-Books/sanjeev_book_readme.htmGoogle Scholar
  2. [Aro95]
    S. Arora. Reductions, codes, PCPs and inapproximability. In: 36th Annual IEEE Symposium on Foundations of Computer Science (Milwaukee, WI, 1995). Los Alamitos, CA: IEEE Comp. Soc. Press, 1995, 404–413.Google Scholar
  3. [BF98]
    R. Beigel, Bin Fu. Solving intractable problems with DNA computing. In: Thirteenth Annual IEEE Conference on Computational Complexity (Buffalo, NY, 1998). Los Alamitos, CA: IEEE Comp. Soc. Press, 1998, 154–168.CrossRefGoogle Scholar
  4. [Blu67]
    M. Blum. A machine-independent theory of the complexity of recursive functions. J. Assoc. Comp. Machin., 1967, 14, 322–336.MATHMathSciNetGoogle Scholar
  5. [Cob64]
    A. Cobham. The intrinsic computational difficulty of functions. In: Proceedings of the 1964 International Congress for Logic, Methodology, and the Philosophy of Science. Amsterdam: North-Holland, 1964, 24–30.Google Scholar
  6. [Coo71a]
    S. A. Cook. Characterizations of pushdown machines in terms of time-bounded computers. J. Assoc. Comp. Machin., 1971, 18(1), 4–18.MATHGoogle Scholar
  7. [Coo71b]
    S. A. Cook. The complexity of theorem proving procedures. In: Proceedings of the 3rd Annual ACM Symposium on the Theory of Computing. 1971, 151–158.Google Scholar
  8. [Coo90]
    S. A. Cook. Computational complexity of higher type functions. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Kyoto, 1990). Tokyo: Math. Soc. Japan, 1991, 55–69.Google Scholar
  9. [Dan60]
    G. B. Dantzig. On the significance of solving linear programming problems with some integer variables. Econometrics, 1960, 28(1), 31–44.MathSciNetGoogle Scholar
  10. [Edm62]
    J. Edmonds. Covers and packings in a family of sets. Bull. Amer. Math. Soc., 1962, 68, 494–499.MATHMathSciNetCrossRefGoogle Scholar
  11. [Edm65a]
    J. Edmonds. Minimum partition of a matroid into independent sets. J. Res. Nat. Bureau Stand. (B), 1965, 69, 67–72.MATHMathSciNetGoogle Scholar
  12. [Edm65b]
    J. Edmonds. Paths, trees, and flowers. Canad. J. Math., 1965, 17, 449–467.MATHMathSciNetGoogle Scholar
  13. [Gim65]
    J. F. Gimpel. A method of producing a Boolean function having an arbitrarily prescribed implicant table. IEEE Trans. Comp., 1965, 14, 485–488.MATHGoogle Scholar
  14. [GJ79]
    M. R. Garey, D. S. Johnson. Computers and Intractability. A Guide to the Theory of NP -Completeness. San Francisco, CA: W.H. Freeman, 1979.Google Scholar
  15. [Hen65]
    F. C. Hennie. One-tape, off-line Turing machine computations. Information and Control, 1965, 8(6), 553–578.MathSciNetCrossRefGoogle Scholar
  16. [HS65]
    J. Hartmanis, R. E. Stearns. On the computational complexity of algorithms. Trans. Amer. Math. Soc., 1965, 117, 285–306.MATHMathSciNetCrossRefGoogle Scholar
  17. [HS66]
    F. C. Hennie, R. E. Stearns. Two-tape simulation of multitape Turing machines. J. Assoc. Comp. Machin., 1966, 13(4), 533–546.MATHMathSciNetGoogle Scholar
  18. [Kar72]
    R. M. Karp. Reducibility among combinatorial problems. In: Complexity of Computer Computations (Yorktown Heights, NY, 1972). New York: Plenum Press, 1972, 85–103.Google Scholar
  19. [Lev73]
    L. A. Levin. Universal brute-force search problems. Problemy Peredachi Informatsii, 1973, 9(3), 115–116 (Russian).MATHMathSciNetGoogle Scholar
  20. [Nel55]
    R. J. Nelson. Review of [Qui52]. J. Symbolic Logic, 1955, 20, 105–108.MATHCrossRefGoogle Scholar
  21. [Qui52]
    W. V. Quine. The problem of simplifying truth functions. Amer. Math. Monthly, 1952, 59, 521–531.MATHMathSciNetCrossRefGoogle Scholar
  22. [Rab66]
    M. Rabin. Mathematical theory of automata. In: Proceedings of the 19th ACM Symposium in Applied Mathematics. Providence, RI: Amer. Math. Soc., 1967, 153–175.Google Scholar
  23. [Rab77]
    M. Rabin. Decidable theories. In: Handbook of Mathematical Logic (ed. J. Barwise). Amsterdam: North-Holland, 1977, Chapter C.3.Google Scholar
  24. [Rit63]
    R. W. Ritchie. Classes of predictably computable functions. Trans. Amer. Math. Soc., 1963, 106, 139–173.MATHMathSciNetCrossRefGoogle Scholar
  25. [Sho97]
    P. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Computing, 1997, 26(5), 1484–1509.MATHMathSciNetCrossRefGoogle Scholar
  26. [Sip92]
    M. Sipser. The history of the P versus NP problem. In: Proceedings of the 24th ACM Symposium on the Theory of Computing. 1992, 603–618.Google Scholar
  27. [Sli99]
    A. Slissenko. Leningrad/St. Petersburg (1961–1998): From logic to complexity and further. In: People and Ideas in Theoretical Computer Science (ed. C. S. Calude). Singapore: Springer, 1999, 274–313.Google Scholar
  28. [Sma98]
    S. Smale. Mathematical problems for the next century. Math. Intelligencer, 1998, 20(2), 7–15.MATHMathSciNetCrossRefGoogle Scholar
  29. [Tra64]
    B. A. Trakhtenbrot. Turing computations with logarithmic retardation. Algebra i Logika, 1964, 3(4) 33–48 (Russian).MATHMathSciNetGoogle Scholar
  30. [Tra84]
    B. A. Trakhtenbrot. A survey of Russian approaches to perebor (brute-force search) algorithms. Ann. Hist. Computing, 1984, 6(4), 384–400.MATHMathSciNetGoogle Scholar
  31. [Yabl59a]
    S. V. Yablonskii. On the impossibility of eliminating the brute-force search of all functions in P 2 when solving certain problems of circuit theory. Dokl. Akad. Nauk SSSR, 1959, 124(1), 44–47 (Russian).MATHMathSciNetGoogle Scholar
  32. [Yabl59b]
    S. V. Yablonskii. On the algorithmic difficulty of the design of minimal contact circuits. In: Problems of Cybernetics (ed. A. A. Lyapunov), No. 2. Moscow: Fizmatgiz, 1959, 75–121 (Russian).Google Scholar
  33. [Yano59]
    S. A. Yanovskaya. Mathematical logic and the foundations of mathematics. In: Mathematics in the USSR after 40 Years. 1917–1957, Vol. 1. Moscow: Fizmatgiz, 1959, 13–120 (pp. 44–45: papers by G. S. Tseitin) (Russian).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • A. A. Razborov

There are no affiliations available

Personalised recommendations