Advertisement

Foundations of Computational Mathematics

, Volume 10, Issue 4, pp 429–454 | Cite as

Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

  • Saugata BasuEmail author
  • Thierry Zell
Article

Abstract

Toda (in SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines in Bull. Am. Math. Soc. (NS) 21(1): 1–46, 1989) has been missing so far. In this paper we formulate and prove a real analogue of Toda’s theorem. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques, we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry: the problem of deciding sentences in the first-order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result may be of independent interest to researchers in algorithmic semi-algebraic geometry.

Keywords

Polynomial hierarchy Betti numbers Semi-algebraic sets Toda’s theorem 

Mathematics Subject Classification (2000)

14P10 14P25 68W30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Basu, On bounding the Betti numbers and computing the Euler characteristic of semi-algebraic sets, Discrete Comput. Geom. 22(1), 1–18 (1999). zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Basu, Computing the first few Betti numbers of semi-algebraic sets in single exponential time, J. Symb. Comput. 41(1), 1125–1154 (2006). MR 2262087 (2007k:14120). zbMATHCrossRefGoogle Scholar
  3. 3.
    S. Basu, Algorithmic semi-algebraic geometry and topology—recent progress and open problems, in Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemporary Mathematics, vol. 453 (American Mathematical Society, Providence, 2008), pp. 139–212. Google Scholar
  4. 4.
    S. Basu, R. Pollack, M.-F. Roy, On the combinatorial and algebraic complexity of quantifier elimination, J. ACM 43(6), 1002–1045 (1996). MR 98c:03077. zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    S. Basu, R. Pollack, M.-F. Roy, Computing roadmaps of semi-algebraic sets on a variety, J. Am. Math. Soc. 13(1), 55–82 (2000). MR 1685780 (2000h:14048). zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    S. Basu, R. Pollack, M.-F. Roy, Algorithms in Real Algebraic Geometry, 2nd edn., Algorithms and Computation in Mathematics, vol. 10 (Springer, Berlin, 2006). MR 1998147 (2004g:14064). zbMATHGoogle Scholar
  7. 7.
    S. Basu, R. Pollack, M.-F. Roy, Computing the first Betti number of a semi-algebraic set, Found. Comput. Math. 8(1), 97–136 (2008). zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    A. Bjorner, M.L. Wachs, V. Welker, Poset fiber theorems, Trans. Am. Math. Soc. 357(5), 1877–1899 (2004). MathSciNetGoogle Scholar
  9. 9.
    L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and Real Computation (Springer, New York, 1998). With a foreword by Richard M. Karp. MR 99a:68070. Google Scholar
  10. 10.
    L. Blum, M. Shub, S. Smale, On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Am. Math. Soc. (NS) 21(1), 1–46 (1989). MR 90a:68022. zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Bochnak, M. Coste, M.-F. Roy, Géométrie Algébrique Réelle (second edition in English: Real Algebraic Geometry), Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 12(36) (Springer, Berlin, 1987 (1998)). MR 949442 (90b:14030). Google Scholar
  12. 12.
    P. Bürgisser, F. Cucker, Variations by complexity theorists on three themes of Euler, Bézout, Betti, and Poincaré, in Complexity of Computations and Proofs, Quad. Mat., vol. 13 (Dept. Math., Seconda Univ. Napoli, Caserta, 2004), pp. 73–151. MR 2131406 (2006c:68053). Google Scholar
  13. 13.
    P. Bürgisser, F. Cucker, Counting complexity classes for numeric computations. II. Algebraic and semialgebraic sets, J. Complex. 22(2), 147–191 (2006). MR 2200367 (2007b:68059). zbMATHCrossRefGoogle Scholar
  14. 14.
    P. Bürgisser, F. Cucker, M. Lotz, Counting complexity classes for numeric computations. III. Complex projective sets, Found. Comput. Math. 5(4), 351–387 (2005). MR 2189543 (2006h:68039). zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. Canny, Computing road maps in general semi-algebraic sets, Comput. J. 36, 504–514 (1993). zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    R. Cluckers, F. Loeser, Constructible motivic functions and motivic integration, Invent. Math. 173(1), 23–121 (2008). MR 2403394 (2009g:14018). zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    H. Delfs, M. Knebusch, Locally Semialgebraic Spaces, Lecture Notes in Mathematics, vol. 1173 (Springer, Berlin, 1985). MR 819737 (87h:14019). zbMATHGoogle Scholar
  18. 18.
    P. Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43, 273–307 (1974). MR 0340258 (49 #5013). CrossRefMathSciNetGoogle Scholar
  19. 19.
    P. Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52, 137–252 (1980). MR 601520 (83c:14017). zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    B. Dwork, On the rationality of the zeta function of an algebraic variety, Am. J. Math. 82(3), 631–648 (1960). zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    M. Edmundo, N. Peatfield, o-minimal Čech cohomology, Q. J. Math. 59(2), 213–220 (2008). MR 2428077. zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    A. Gabrielov, N. Vorobjov, T. Zell, Betti numbers of semialgebraic and sub-Pfaffian sets, J. Lond. Math. Soc. (2) 69(1), 27–43 (2004). MR 2025325 (2004k:14105). zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    A. Gabrielov, N. Vorobjov, Approximation of definable sets by compact families, and upper bounds on homotopy and homology, J. Lond. Math. Soc. (2) 80(1), 35–54 (2009). MR 2520376. zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    L. Gournay, J.J. Risler, Construction of roadmaps of semi-algebraic sets, Appl. Algebra Eng. Commun. Comput. 4(4), 239–252 (1993). zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    D. Grigoriev, Complexity of deciding Tarski algebra, J. Symb. Comput. 5(1–2), 65–108 (1988). MR 90b:03054. CrossRefGoogle Scholar
  26. 26.
    D. Grigoriev, N. Vorobjov, Counting connected components of a semi-algebraic set in subexponential time, Comput. Complex. 2(2), 133–186 (1992). CrossRefGoogle Scholar
  27. 27.
    J. Matoušek, Using the Borsuk–Ulam Theorem. Universitext (Springer, Berlin, 2003). Lectures on topological methods in combinatorics and geometry, Written in cooperation with Anders Björner and Günter M. Ziegler. MR 1988723 (2004i:55001). zbMATHGoogle Scholar
  28. 28.
    K. Meer, Counting problems over the reals, Theoret. Comput. Sci. 242(1–2), 41–58 (2000). MR 1769145 (2002g:68041). zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    C. Papadimitriou, Computational Complexity (Addison-Wesley, San Diego, 1994). zbMATHGoogle Scholar
  30. 30.
    J. Renegar, On the computational complexity and geometry of the first-order theory of the reals. I-III, J. Symb. Comput. 13(2), 255–352 (1992). zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    U. Schöning, Probabilistic complexity classes and lowness, J. Comput. Syst. Sci. 39(1), 84–100 (1989) MR 1013721 (91b:68041a). zbMATHCrossRefGoogle Scholar
  32. 32.
    M. Shub, S. Smale, On the intractability of Hilbert’s Nullstellensatz and an algebraic version of “\(\mathrm{NP}\not=\mathrm{P}\)?”, Duke Math. J. 81(1), 47–54 (1995). A celebration of John F. Nash, Jr. MR 1381969 (97h:03067). zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    E.H. Spanier, Algebraic Topology (McGraw-Hill Book, New York, 1966). MR 0210112 (35 #1007). zbMATHGoogle Scholar
  34. 34.
    L. Stockmeyer, The polynomial-time hierarchy, Theoret. Comput. Sci. 3(1), 1–22 (1977). MR 0438810 (55 #11716). zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    A. Tarski, A Decision Method for Elementary Algebra and Geometry, 2nd edn. (University of California Press, Berkeley/Los Angeles, 1951). MR 13,423a. zbMATHGoogle Scholar
  36. 36.
    S. Toda, PP is as hard as the polynomial-time hierarchy, SIAM J. Comput. 20(5), 865–877 (1991). MR 1115655 (93a:68047). zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    L.G. Valiant, V.V. Vazirani, NP is as easy as detecting unique solutions, Theoret. Comput. Sci. 47(1), 85–93 (1986). MR 871466 (88i:68021). zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    A. Weil, Number of solutions of equations over finite fields, Bull. Am. Math. Soc. 55, 497–508 (1949). zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    D.Yu. Grigoriev, N.N. Vorobjov Jr., Solving systems of polynomial inequalities in subexponential time, J. Symb. Comput. 5(1–2), 37–64 (1988). MR 949112 (89h:13001). CrossRefGoogle Scholar

Copyright information

© SFoCM 2010

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.School of Mathematics and Computing SciencesLenoir-Rhyne UniversityHickoryUSA

Personalised recommendations