Theory of linear and integer programming
Theory of linear and integer programming
Subclasses of Presburger arithmetic and the polynomial-time hierarchy
Theoretical Computer Science
Counting solutions to Presburger formulas: how and why
PLDI '94 Proceedings of the ACM SIGPLAN 1994 conference on Programming language design and implementation
A polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed
Mathematics of Operations Research
Journal of Combinatorial Theory Series A
A geometric Buchberger algorithm for integer programming
Mathematics of Operations Research
Test sets for integer programs
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
Parametric Analysis of Polyhedral Iteration Spaces
Journal of VLSI Signal Processing Systems - Special issue on application specific systems, architectures and processors
An Automata-Theoretic Approach to Presburger Arithmetic Constraints (Extended Abstract)
SAS '95 Proceedings of the Second International Symposium on Static Analysis
Diophantine Equations, Presburger Arithmetic and Finite Automata
CAAP '96 Proceedings of the 21st International Colloquium on Trees in Algebra and Programming
Multiple Counters Automata, Safety Analysis and Presburger Arithmetic
CAV '98 Proceedings of the 10th International Conference on Computer Aided Verification
Bounds on the automata size for Presburger arithmetic
ACM Transactions on Computational Logic (TOCL)
Counting with rational generating functions
Journal of Symbolic Computation
Computing the integer programming gap
Combinatorica
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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=(p1,…,pn) 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.