On the SUP-INF Method for Proving Presburger Formulas
Journal of the ACM (JACM)
An NP-Complete Number-Theoretic Problem
Journal of the ACM (JACM)
Elementary bounds for presburger arithmetic
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
Presburger arithmetic with bounded quantifier alternation
STOC '78 Proceedings of the tenth annual ACM symposium on Theory of computing
Precise bounds for presburger arithmetic and the reals with addition: Preliminary report
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
Complexity and uniformity of elimination in Presburger arithmetic
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
Mixed real-integer linear quantifier elimination
ISSAC '99 Proceedings of the 1999 international symposium on Symbolic and algebraic computation
Deciding linear-trigonometric problems
ISSAC '00 Proceedings of the 2000 international symposium on Symbolic and algebraic computation
Presburger liveness verification of discrete timed automata
Theoretical Computer Science
Quantifier elimination for the reals with a predicate for the powers of two
Theoretical Computer Science
Proof Synthesis and Reflection for Linear Arithmetic
Journal of Automated Reasoning
Effective Quantifier Elimination for Presburger Arithmetic with Infinity
CASC '09 Proceedings of the 11th International Workshop on Computer Algebra in Scientific Computing
Model checking coverability graphs of vector addition systems
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Verifying mixed real-integer quantifier elimination
IJCAR'06 Proceedings of the Third international joint conference on Automated Reasoning
Weak integer quantifier elimination beyond the linear case
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
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We studied formulas of elementary number theory resulting from formulas of Presburger arithmetic PrA (additive elementary theory of integers with order) by substituting for some variables, polynomials and integer values of rational functions in a single new variable y, and quantifying over y. We show that the extension ALA (almost linear arithmetic) of PrA obtained in this way, has essentially the same upper and lower complexity bounds as the original theory. The same applies to the fragments of ALA obtained by restricting the number or type of quantifiers in formulas. We also show new upper complexity bounds for quantifier elimination in PrA and its fragments. The results form a common extension of known complexity bounds for PrA and for the existential almost linear problems studied by Gurari & Ibarra. The method is applicable also to other extensions of PrA without order, related to the famous bounds for binary forms of A. Baker.