Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
156 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
45 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

An efficient quantifier elimination procedure for Presburger arithmetic (2405.01183v1)

Published 2 May 2024 in cs.LO, cs.FL, and math.LO

Abstract: All known quantifier elimination procedures for Presburger arithmetic require doubly exponential time for eliminating a single block of existentially quantified variables. It has even been claimed in the literature that this upper bound is tight. We observe that this claim is incorrect and develop, as the main result of this paper, a quantifier elimination procedure eliminating a block of existentially quantified variables in singly exponential time. As corollaries, we can establish the precise complexity of numerous problems. Examples include deciding (i) monadic decomposability for existential formulas, (ii) whether an existential formula defines a well-quasi ordering or, more generally, (iii) certain formulas of Presburger arithmetic with Ramsey quantifiers. Moreover, despite the exponential blowup, our procedure shows that under mild assumptions, even NP upper bounds for decision problems about quantifier-free formulas can be transferred to existential formulas. The technical basis of our results is a kind of small model property for parametric integer programming that generalizes the seminal results by von zur Gathen and Sieveking on small integer points in convex polytopes.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (33)
  1. Algorithmic analysis of programs with well quasi-ordered domains. Inform. and Comput., 160(1–2):109–127, 2000. doi:10.1006/inco.1999.2843.
  2. Ramsey quantifiers in linear arithmetics. In Proc. POPL 2024, pages 1–32, 2024. doi:10.1145/3632843.
  3. Ramsey quantifiers in linear arithmetics, 2023. arXiv:2311.04031, doi:10.48550/arXiv.2311.04031.
  4. Bounds on positive integral solutions of linear Diophantine equations. P. Am. Math. Soc., 55:299–304, 1976. doi:10.1090/S0002-9939-1976-0396605-3.
  5. Sensitivity theorems in integer linear programming. Math. Program., 34:251–264, 1986. doi:10.1007/BF01582230.
  6. D. C. Cooper. Theorem proving in arithmetic without multiplication. In Bernard Meltzer and Donald Michie, editors, Proceedings of the Seventh Annual Machine Intelligence Workshop, Edinburgh, 1971, volume 7, pages 91–99. Edinburgh University Press, 1972.
  7. Alain Finkel. A generalization of the procedure of Karp and Miller to well structured transition systems. In Proc. ICALP 1987, volume 267 of Lecture Notes in Computer Science, pages 499–508. Springer, 1987. doi:10.1007/3-540-18088-5_43.
  8. The well structured problem for Presburger counter machines. In Proc. FSTTCS 2019, volume 150 of LIPIcs, pages 41:1–41:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. doi:10.4230/LIPICS.FSTTCS.2019.41.
  9. The well structured problem for Presburger counter machines. CoRR, abs/1910.02736, 2019. arXiv:1910.02736, doi:10.48550/arXiv.1910.02736.
  10. Well-structured transition systems everywhere! Theor. Comput. Sci., 256(1–2):63–92, 2001. doi:10.1016/S0304-3975(00)00102-X.
  11. Super-exponential complexity of Presburger arithmetic. In Bob F. Caviness and Jeremy R. Johnson, editors, Quantifier Elimination and Cylindrical Algebraic Decomposition, pages 122–135, Vienna, 1998. Springer Vienna. doi:10.1007/978-3-7091-9459-1_5.
  12. Martin Fürer. The complexity of Presburger arithmetic with bounded quantifier alternation depth. Theor. Comput. Sci., 18:105–111, 1982. doi:10.1016/0304-3975(82)90115-3.
  13. Bounded regular sets. P. Am. Math. Soc., 17(5):1043–1049, 1966. doi:10.1090/S0002-9939-1966-0201310-3.
  14. Erich Grädel. Dominoes and the complexity of subclasses of logical theories. Ann. Pure Appl. Log., 43(1):1–30, 1989. doi:10.1016/0168-0072(89)90023-7.
  15. Spatio-temporal data handling with constraints. GeoInformatica, 5(1):95–115, 2001. doi:10.1023/A:1011464022461.
  16. On the existential theories of Büchi arithmetic and linear p𝑝pitalic_p-adic fields. In Proc. LICS 2019, pages 1–10. IEEE, 2019. doi:10.1109/LICS.2019.8785681.
  17. Christoph Haase. Subclasses of Presburger arithmetic and the weak EXP hierarchy. In Proc. CSL-LICS 2014, pages 47:1–47:10. ACM, 2014. doi:10.1145/2603088.2603092.
  18. Presburger arithmetic with stars, rational subsets of graph groups, and nested zero tests. In Proc. LICS 2019, pages 1–14. IEEE, 2019. doi:10.1109/LICS.2019.8785850.
  19. Jacques Hadamard. Rèsolution d’une question relative aux dèterminants. B. Sci. Math., 2(17):240–246, 1893.
  20. Monadic decomposition in integer linear arithmetic. In Proc. IJCAR 2020, volume 12166 of Lecture Notes in Computer Science, pages 122–140. Springer, 2020. doi:10.1007/978-3-030-51074-9_8.
  21. Constraint Databases. Springer, 2000.
  22. Mohan Nair. On Chebyshev-type inequalities for primes. Am. Math. Mon., 89(2):126–129, 1982. doi:10.2307/2320934.
  23. Derek C. Oppen. A 222p⁢nsuperscript2superscript2superscript2𝑝𝑛2^{2^{2^{pn}}}2 start_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT italic_p italic_n end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT upper bound on the complexity of Presburger arithmetic. J. Comput. Syst. Sci., 16(3):323–332, 1978. doi:10.1016/0022-0000(78)90021-1.
  24. Loïc Pottier. Minimal solutions of linear diophantine systems: Bounds and algorithms. In Proc. RTA 1991, volume 488 of Lecture Notes in Computer Science, pages 162–173. Springer, 1991. doi:10.1007/3-540-53904-2_94.
  25. Mojżesz Presburger. Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In Comptes Rendus du I congres de Mathematiciens des Pays Slaves, pages 92–101. Ksiaznica Atlas, 1929.
  26. Presburger arithmetic with bounded quantifier alternation. In Proc. STOC 1978, pages 320–325, New York, NY, USA, 1978. ACM. doi:10.1145/800133.804361.
  27. Approximate formulas for some functions of prime numbers. Illinois J. Math., 6(1):64 – 94, 1962. doi:10.1215/ijm/1255631807.
  28. Sasha Rubin. Automata presenting structures: A survey of the finite string case. Bull. Symb. Log., 14(2):169–209, 2008. doi:10.2178/BSL/1208442827.
  29. Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1986.
  30. Monadic decomposition. J. ACM, 64(2):14:1–14:28, 2017. doi:10.1145/3040488.
  31. Joachim von zur Gathen and Malte Sieveking. A bound on solutions of linear integer equalities and inequalities. Proceedings of the American Mathematical Society, 72(1):155–158, 1978. doi:10.1090/S0002-9939-1978-0500555-0.
  32. Volker Weispfenning. The complexity of almost linear Diophantine problems. J. Symb. Comput., 10(5):395–403, 1990. doi:10.1016/S0747-7171(08)80051-X.
  33. Volker Weispfenning. Complexity and uniformity of elimination in Presburger arithmetic. In Proc. ISSAC 1997, pages 48–53. ACM, 1997. doi:10.1145/258726.258746.
Citations (4)
View on Semantic Scholar

Summary

We haven't generated a summary for this paper yet.

Ai Sparkles Streamline Icon: https://streamlinehq.com Summarize Now