ABSTRACT
We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by “short” we mean sentences with a bounded number of variables, quantifers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan’s partition can be found in polynomial time, the satisfability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time.
Supplemental Material
- {BP99} A. Barvinok and J. E. Pommersheim, An algorithmic theory of lattice points in polyhedra, in New Perspectives in Algebraic Combinatorics, Cambridge Univ. Press, Cambridge, 1999, 91–147.Google Scholar
- {BW03} A. Barvinok and K. Woods, Short rational generating functions for lattice point problems, Jour. AMS 16 (2003), 957–979.Google Scholar
- {BV07} N. Berline and M. Vergne, Local Euler–Maclaurin formula for polytopes, Mosc. Math. J. 7 (2007), 355–386.Google ScholarCross Ref
- {CH16} D. Chistikov and C. Haase, The taming of the semi-linear set, in Proc. ICALP 2016, 127:1–127:13.Google Scholar
- {DHTY04} J. A. De Loera, R. Hemmecke, J. Tauzer and R. Yoshida, Effective lattice point counting in rational convex polytopes, J. Symbolic Comput. 38 (2004), 1273–1302.Google ScholarCross Ref
- {DK97} M. Dyer and R. Kannan, On Barvinok’s algorithm for counting lattice points in fixed dimension, Math. Oper. Res. 22 (1997), 545–549. {Eis03} F. Eisenbrand, Fast integer programming in fixed dimension, in Proc. 11th ESA, Springer, Berlin, 2003, 196–207. {Eis10} F. Eisenbrand, Integer programming and algorithmic geometry of numbers, in 50 years of Integer Programming, Springer, Berlin, 2010, 505–560.Google ScholarDigital Library
- {EH12} F. Eisenbrand and N. Hähnle, Minimizing the number of lattice points in a translated polygon, in Proc. 24th SODA, SIAM, Philadelphia, PA, 2012, 1123–1130. Google ScholarDigital Library
- {ES08} F. Eisenbrand and G. Shmonin, Parametric integer programming in fixed dimension, Math. Oper. Res. 33 (2008), 839–850.Google ScholarCross Ref
- {FR74} M. J. Fischer and M. O. Rabin, Super-Exponential Complexity of Presburger Arithmetic, in Proc. SIAM-AMS Symposium in Applied Mathematics, AMS, Providence, RI, 1974, 27–41.Google Scholar
- {FT87} A. Frank and É. Tardos, An application of simultaneous Diophantine approximation in combinatorial optimization, Combinatorica 7 (1987), 49–65. {Für82} M. Fürer, The complexity of Presburger arithmetic with bounded quantifier alternation depth, Theoret. Comput. Sci. 18 (1982), 105–111. {Grä87} E. Grädel, The complexity of subclasses of logical theories, Dissertation, Universität Basel, 1987. Google ScholarDigital Library
- {Kan90} R. Kannan, Test sets for integer programs, ∀ ∃ sentences, in Polyhedral Combinatorics, AMS, Providence, RI, 1990, 39–47 {Kan92} R. Kannan, Lattice translates of a polytope and the Frobenius problem, Combinatorica 12 (1992), 161–177. {Köp07} M. Köppe, A primal Barvinok algorithm based on irrational decompositions, SIAM J. Discrete Math. 21 (2007), 220–236. Google ScholarDigital Library
- {KV08} M. Köppe and S. Verdoolaege, Computing parametric rational generating functions with a primal Barvinok algorithm, Electron. J. Combin. 15 (2008), no. 1, RP 16, 19 pp. {Len83} H. Lenstra, Integer programming with a fixed number of variables, Math. Oper. Res. 8 (1983), 538–548. Google ScholarDigital Library
- {MS05} E. Miller and B. Sturmfels, Combinatorial commutative algebra, Springer, New York, 2005.Google Scholar
- {NP17a} D. Nguyen and I. Pak, Enumeration of integer points in projections of unbounded polyhedra, extended abstract to appear in Proc. IPCO 2017 ; arXiv:1612.08030. {NP17b} D. Nguyen and I. Pak, The computational complexity of integer programming with alternations; arXiv:1702.08662.Google Scholar
- {NP17c} D. Nguyen and I. Pak, Complexity of short generating functions; arXiv:1702.08660. {Opp78} D. C. Oppen, A 2 22 pn upper bound on the complexity of Presburger arithmetic, J. Comput. System Sci. 16 (1978), 323–332. {Pak02} I. Pak, On sampling integer points in polyhedra, in Foundations of Computational Mathematics, World Sci., River Edge, NJ, 2002, 319–324.Google Scholar
- {PP15} I. Pak and G. Panova, On the complexity of computing Kronecker coefficients, to appear in Computational Complexity; arXiv:1404.0653. Google ScholarDigital Library
- {Pre29} M. Presburger, Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt (in German), in Comptes Rendus du I congrès de Mathématiciens des Pays Slaves, Warszawa, 1929, 92–101; English transltion in Hist. Philos. Logic 12 (1991), 225–233.Google Scholar
- {RL78} C. R. Reddy and D. W. Loveland, Presburger arithmetic with bounded quantifier alternation, Proc. 10th STOC (1978), 320-325. {Sca84} B. Scarpellini, Complexity of subcases of Presburger arithmetic, Trans. AMS 284 (1984), 203–218. {Sch86} A. Schrijver, Theory of linear and integer programming, John Wiley, Chichester, 1986. Google ScholarDigital Library
- {Sch97} U. Schöning, Complexity of Presburger arithmetic with fixed quantifier dimension, Theory Comput. Syst. 30 (1997), 423–428. {V+07} S. Verdoolaege, R. Seghir, K. Beyls, V. Loechner and M. Bruynooghe, Counting integer points in parametric polytopes using Barvinok’s rational functions, Algorithmica 48 (2007), 37–66. {Woo04} K. Woods, Rational Generating Functions and Lattice Point Sets, Ph.D. thesis, University of Michigan, 2004, 112 pp. {Woo15} K. Woods, Presburger arithmetic, rational generating functions, and quasipolynomials, J. Symb. Log. 80 (2015), 433–449.Google ScholarCross Ref
Index Terms
- Complexity of short Presburger arithmetic
Recommendations
Bounds on the automata size for Presburger arithmetic
Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by the automata-based approach for Presburger arithmetic. In this article, we give an ...
Deciding Boolean Algebra with Presburger Arithmetic
We describe an algorithm for deciding the first-order multisorted theory BAPA, which combines Boolean algebras of sets of uninterpreted elements (BA) and Presburger arithmetic operations (PA). BAPA can express the relationship between integer variables ...
Generic Complexity of Presburger Arithmetic
Special Issue: Symposium on Computer Science, Guest Editors: Sergei Artemov, Volker Diekert and Dima GrigorievFischer and Rabin proved in (Proceedings of the SIAM-AMS Symposium in Applied Mathematics, vol. 7, pp. 27–41, 1974) that the decision problem for Presburger Arithmetic has at least double exponential worst-case complexity (for deterministic and for ...
Comments