Journal of Automated Reasoning

, Volume 47, Issue 4, pp 341–367 | Cite as

An Interpolating Sequent Calculus for Quantifier-Free Presburger Arithmetic

  • Angelo Brillout
  • Daniel Kroening
  • Philipp RümmerEmail author
  • Thomas Wahl


Craig interpolation has become a versatile tool in formal verification, used for instance to generate program assertions that serve as candidates for loop invariants. In this paper, we consider Craig interpolation for quantifier-free Presburger arithmetic (QFPA). Until recently, quantifier elimination was the only available interpolation method for this theory, which is, however, known to be potentially costly and inflexible. We introduce an interpolation approach based on a sequent calculus for QFPA that determines interpolants by annotating the steps of an unsatisfiability proof with partial interpolants. We prove our calculus to be sound and complete. We have extended the Princess theorem prover to generate interpolating proofs, and applied it to a large number of publicly available Presburger arithmetic benchmarks. The results document the robustness and efficiency of our interpolation procedure. Finally, we compare the procedure against alternative interpolation methods, both for QFPA and linear rational arithmetic.


Sequent calculus Presburger arithmetic Craig interpolation 

Mathematics Subject Classification (2010)



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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Angelo Brillout
    • 1
  • Daniel Kroening
    • 2
  • Philipp Rümmer
    • 3
    Email author
  • Thomas Wahl
    • 2
  1. 1.ETH ZurichZurichSwitzerland
  2. 2.Computer Science DepartmentOxford UniversityOxfordUK
  3. 3.Department of Information TechnologyUppsala UniversityUppsalaSweden

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