A Reduction from Unbounded Linear Mixed Arithmetic Problems into Bounded Problems

  • Martin BrombergerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10900)


We present a combination of the Mixed-Echelon-Hermite transformation and the Double-Bounded reduction for systems of linear mixed arithmetic that preserve satisfiability and can be computed in polynomial time. Together, the two transformations turn any system of linear mixed constraints into a bounded system, i.e., a system for which termination can be achieved easily. Existing approaches for linear mixed arithmetic, e.g., branch-and-bound and cuts from proofs, only explore a finite search space after application of our two transformations. Instead of generating a priori bounds for the variables, e.g., as suggested by Papadimitriou, unbounded variables are eliminated through the two transformations. The transformations orient themselves on the structure of an input system instead of computing a priori (over-)approximations out of the available constants. Experiments provide further evidence to the efficiency of the transformations in practice. We also present a polynomial method for converting certificates of (un)satisfiability from the transformed to the original system.


Linear arithmetic Integer arithmetic Mixed arithmetic SMT Linear Transformations Constraint solving 


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Authors and Affiliations

  1. 1.Max Planck Institute for Informatics and Saarland University, Saarland Informatics CampusSaarbrückenGermany
  2. 2.Graduate School of Computer Science, Saarland Informatics CampusSaarbrückenGermany

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