Advertisement

Presburger Arithmetic, Rational Generating Functions, and Quasi-Polynomials

  • Kevin Woods
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)

Abstract

A Presburger formula is a Boolean formula with variables in ℕ that can be written using addition, comparison (≤, =, etc.), Boolean operations (and, or, not), and quantifiers (∀ and ∃). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, if p = (p 1,…,p n ) are a subset of the free variables in a Presburger formula, we can define a counting function g(p) to be the number of solutions to the formula, for a given p. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. In the full version of this paper, we also translate known computational complexity results into this setting and discuss open directions.

Keywords

Free Variable Toric Variety Counting Function Integer Point Numerical Semigroup 
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.

References

  1. 1.
    Barvinok, A.: A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19(4), 769–779 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  3. 3.
    Barvinok, A.: The complexity of generating functions for integer points in polyhedra and beyond. In: International Congress of Mathematicians, vol. III, pp. 763–787. Eur. Math. Soc., Zürich (2006)Google Scholar
  4. 4.
    Barvinok, A., Pommersheim, J.: An algorithmic theory of lattice points in polyhedra. In: New Perspectives in Algebraic Combinatorics (Berkeley, CA, 1996–1997). Math. Sci. Res. Inst. Publ., vol. 38, pp. 91–147. Cambridge Univ. Press, Cambridge (1999)Google Scholar
  5. 5.
    Barvinok, A., Woods, K.: Short rational generating functions for lattice point problems. J. Amer. Math. Soc. 16(4), 957–979 (2003) (electronic)Google Scholar
  6. 6.
    Beck, M.: The partial-fractions method for counting solutions to integral linear systems. Discrete Comput. Geom. 32(4), 437–446 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Beck, M., Robins, S.: Computing the continuous discretely. Undergraduate Texts in Mathematics. Springer, New York (2007); Integer-point enumeration in polyhedrazbMATHGoogle Scholar
  8. 8.
    Berman, L.: The complexity of logical theories. Theoret. Comput. Sci. 11(1), 57, 71–77 (1980); With an introduction “On space, time and alternation”MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Blanco, V., García-Sánchez, P.A., Puerto, J.: Counting numerical semigroups with short generating functions. Internat. J. Algebra Comput. 21(7), 1217–1235 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Boudet, A., Comon, H.: Diophantine equations, Presburger arithmetic and finite automata. In: Kirchner, H. (ed.) CAAP 1996. LNCS, vol. 1059, pp. 30–43. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  11. 11.
    Brion, M.: Points entiers dans les polyèdres convexes. Ann. Sci. École Norm. Sup. 4, 653–663 (1988)MathSciNetGoogle Scholar
  12. 12.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlagen Math. 6, 66–92 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cassels, J.W.S.: An introduction to the geometry of numbers. Classics in Mathematics. Springer, Berlin (1997); Corrected reprint of the 1971 editionzbMATHGoogle Scholar
  14. 14.
    Clauss, P., Loechner, V.: Parametric analysis of polyhedral iteration spaces. Journal of VLSI Signal Processing 19(2), 179–194 (1998)CrossRefGoogle Scholar
  15. 15.
    Cobham, A.: On the base-dependence of sets of numbers recognizable by finite automata. Math. Systems Theory 3, 186–192 (1969)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and Presburger arithmetic. In: Vardi, M.Y. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    Cooper, D.: Theorem proving in arithmetic without multiplication. Machine Intelligence 7, 91–99 (1972)zbMATHGoogle Scholar
  18. 18.
    D’Alessandro, F., Intrigila, B., Varricchio, S.: On some counting problems for semi-linear sets. CoRR abs/0907.3005 (2009)Google Scholar
  19. 19.
    Davis, M.: Hilbert’s tenth problem is unsolvable. Amer. Math. Monthly 80, 233–269 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    De Loera, J., Haws, D., Hemmecke, R., Huggins, P., Sturmfels, B., Yoshida, R.: Short rational functions for toric algebra. To appear in Journal of Symbolic Computation (2004)Google Scholar
  21. 21.
    Ehrhart, E.: Sur les polyèdres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris 254, 616–618 (1962)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Fischer, M., Rabin, M.: Super-exponential complexity of Presburger arithmetic. In: Complexity of Computation. SIAM–AMS Proc., vol. VII, pp. 27–41. Amer. Math. Soc., Providence (1974); Proc. SIAM-AMS Sympos., New York (1973)Google Scholar
  23. 23.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)zbMATHGoogle Scholar
  24. 24.
    Fürer, M.: The complexity of Presburger arithmetic with bounded quantifier alternation depth. Theoret. Comput. Sci. 18(1), 105–111 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Ginsburg, S., Spanier, E.: Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics 16(2), 285–296 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Grädel, E.: Subclasses of Presburger arithmetic and the polynomial-time hierarchy. Theoret. Comput. Sci. 56(3), 289–301 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Guo, A., Miller, E.: Lattice point methods for combinatorial games. Adv. in Appl. Math. 46(1-4), 363–378 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Hoşten, S., Sturmfels, B.: Computing the integer programming gap. To appear in Combinatorics (2004)Google Scholar
  29. 29.
    Kannan, R.: Test sets for integer programs, ∀ ∃ sentences. In: Polyhedral Combinatorics. DIMACS Ser. Discrete Math. Theoret. Comput. Sci, vol. 1, pp. 39–47. Amer. Math. Soc., Providence (1990); Morristown, NJ (1989)Google Scholar
  30. 30.
    Klaedtke, F.: Bounds on the automata size for Presburger arithmetic. ACM Trans. Comput. Log. 9(2), 34 (2008)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lenstra Jr., H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Miller, E., Sturmfels, B.: Combinatorial commutative algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005)Google Scholar
  33. 33.
    Oppen, D.: A superexponential upper bound on the complexity of Presburger arithmetic. J. Comput. System Sci. 16(3), 323–332 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Parker, E., Chatterjee, S.: An automata-theoretic algorithm for counting solutions to presburger formulas. In: Duesterwald, E. (ed.) CC 2004. LNCS, vol. 2985, pp. 104–119. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  35. 35.
    Presburger, M.: On the completeness of a certain system of arithmetic of whole numbers in which addition occurs as the only operation. Hist. Philos. Logic 12(2), 225–233 (1991); Translated from the German and with commentaries by Dale JacquetteMathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Pugh, W.: Counting solutions to presburger formulas: how and why. SIGPLAN Not. 29(6), 121–134 (1994)CrossRefGoogle Scholar
  37. 37.
    Ramírez Alfonsín, J.L.: The Diophantine Frobenius problem. Oxford Lecture Series in Mathematics and its Applications, vol. 30. Oxford University Press, Oxford (2005)zbMATHCrossRefGoogle Scholar
  38. 38.
    Scarf, H.: Test sets for integer programs. Math. Programming Ser. B 79(1-3), 355–368 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Schrijver, A.: Theory of Linear and Integer Programming. Interscience Series in Discrete Mathematics. John Wiley & Sons Ltd., Chichester (1986)zbMATHGoogle Scholar
  40. 40.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)zbMATHGoogle Scholar
  41. 41.
    Stanley, R.P.: Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980); Combinatorial mathematics, optimal designs and their applications. In: Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo. (1978)Google Scholar
  42. 42.
    Sturmfels, B.: On vector partition functions. J. Combin. Theory Ser. A 72(2), 302–309 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  43. 43.
    Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)zbMATHGoogle Scholar
  44. 44.
    Thomas, R.: A geometric Buchberger algorithm for integer programming. Math. Oper. Res. 20(4), 864–884 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Thomas, R.: The structure of group relaxations. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbook of Discrete Optimization (2003)Google Scholar
  46. 46.
    Verdoolaege, S., Woods, K.: Counting with rational generating functions. J. Symbolic Comput. 43(2), 75–91 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Wolper, P., Boigelot, B.: An automata-theoretic approach to Presburger arithmetic constraints. In: Mycroft, A. (ed.) SAS 1995. LNCS, vol. 983, pp. 21–32. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  48. 48.
    Woods, K.: Rational Generating Functions and Lattice Point Sets. PhD thesis, University of Michigan (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kevin Woods
    • 1
  1. 1.Oberlin CollegeOberlinUSA

Personalised recommendations