On the difference between one and many

preliminary version
  • Janos Simon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 52)


We examine the following question: ‘Given a problem, is it more difficult to tell how many solutions the problem has than just deciding whether it has a solution?’. We show, that in specific cases, the question can be put into a mathematically meaningful form, namely when we can translate ‘number of solutions’ as ‘number of distinct accepting computations of a nondeterministic Turing machine’ (perhaps with appropriate weights). In this context, as we show, these questions are equivalent to problems about probabilistic machines (in the sense of Gill (9)).

In the first part of the paper we examine time-bounded computations, and justify our claim that this formalization is really the ma thematical form of the question above by exhibiting a unifying model (the treshold machine) which has a special subcases the nondeterministic and the probabilistic machines. We show that natural complete problems exist and prove some elementary properties of the model.

In the second part we examine tape-bounded machines. We show that probabilistic tape-bounded machines may be simulated by deterministic Turing machines with only a polynomial increase in the amount of tape needed. This settles an open problem of Gill's (9).

This is a very powerful and perhaps unexpected result: it is the best known situation in which we are able to show that powerful ‘extras’ like nondeterminism, get us only a polynomial improvement. The result is similar in content to Savitch's celebrated simulation of non-deterministic machines (20). The proof is completely unrelated to Savitch's (his construction does not work in the probabilistic case) and is quite involved, using some powerful recent results in complexity theory (10) (18) (4).


Polynomial Time Span Tree Turing Machine Random Access Machine Probabilistic Machine 
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.


  1. [1]
    Aho, A., J.E. Hopcroft and J.D. Ullman: The design and analysis of computer algorithms. Addison Wesley, Reading 1974.Google Scholar
  2. [2]
    Baker, T., J. Gill and R. Solovay: Relativization of the P=?NP question. SIAM. J. COMP. 4:4, 431–442Google Scholar
  3. [3]
    Cook, S.A.: The complexity of theorem-proving procedures. Proc. 3rd STOC (1971) 151–158Google Scholar
  4. [4]
    Csanky, L.: Fast parallel matrix inversion algorithms. SIAM J. COMP. 5:4, 618–623Google Scholar
  5. [5]
    Eilenberg, S.: Automata, languages and programming Academic Press, N. York (1974) vol. A.Google Scholar
  6. [6]
    Gabow, H. N., S. N. Maheshwari and L. Osterweil: On two problems in the generation of program test paths. IEEE Trans. Software Eng. 3: 2 (sept. 1976) 227–231.Google Scholar
  7. [7]
    Galil, Z.: On direct encodings of nondeterministic Turing machines operating in polynomial time into P-complete problems. SIGACT News 6:1 (1974) 19–23.Google Scholar
  8. [8]
    Garey, M. R., D. S. Johnson and L. Stockmeyer: Some simplified NP-complete problems. Proc. 6 th STOC (1974) 47–63.Google Scholar
  9. [9]
    Gill, J. III: Computational complexity of probabilistic Turing machines. Proc. 6 th STOC (1974) 91–95.Google Scholar
  10. [10]
    Hartmanis, J. and J. Simon: On the power of multiplication in random access machines. Proc. 15 th SWAT (1974) 13–23.Google Scholar
  11. [11]
    Hartmanis, J. and J. Simon: On the structure of feasible computations in M. Rubinoff and M. C. Yovits (eds): Advances in Computers v. 14 Academic Press, N. York (1976) 1–43.Google Scholar
  12. [12]
    Hoperoft, J. E. and J. D. Ullman: Formal languages and their relation to automata. Addison-Wesley, Reading Mass 1969.Google Scholar
  13. [13]
    Hunt, H. B. III: On time and tape complexity of languages. Proc. 5 th STOC (1973) 10–19.Google Scholar
  14. [14]
    Johnson, D.: Algorithms for shortest parths. TR 73–169 Cornell U.Google Scholar
  15. [15]
    Karp, R. M.: Reducibility among combinatorial problems in R. E. Miller and J. W. Thatcher (eds): Complexity of computer computations. Plenum Press NY (1972) 85–104s.Google Scholar
  16. [16]
    Kirchoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Störme gefuhrt wird. Ann. Phys. Chem. 72 (1847) 497–508.Google Scholar
  17. [17]
    Meyer, A. R. and L. J. Stockmeyer: The equivalence of regular expressions with squaring requires exponential space. Proc. 13 th SWAT (1972) 125–129.Google Scholar
  18. [18]
    Pratt, V. R., L. Stockmeyer: A characterization of the power of vector machines. JCSS 12:2 (1976) 198–221.Google Scholar
  19. [19]
    Sahni, S. L.: Some related problems from network flows, game theory and integer programming. Proc. 13 th SWAT (1972) 130–138.Google Scholar
  20. [20]
    Savitch, W. L.: Relationships between nondeterministic an deter ministic tape complexities. JCSS 4:2 (1970) 177–192.Google Scholar
  21. [21]
    Sethi, R.: Complete register allocation problems. SIAM. J. Comp. 4:3 (1975) 226–248.Google Scholar
  22. [22]
    Stockmeyer, L. J.: The polynomial-time hierarchy. IBM Res. Rep. RC 5379.Google Scholar
  23. [23]
    Valiant, L. G.: Unpublished manuscript.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • Janos Simon
    • 1
  1. 1.Dept. C. ComputaçãoUnicampCampinas SPBrasil

Personalised recommendations