Linear Arithmetic with Stars

  • Ruzica Piskac
  • Viktor Kuncak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5123)


We consider an extension of integer linear arithmetic with a “star” operator takes closure under vector addition of the solution set of a linear arithmetic subformula. We show that the satisfiability problem for this extended language remains in NP (and therefore NP-complete). Our proof uses semilinear set characterization of solutions of integer linear arithmetic formulas, as well as a generalization of a recent result on sparse solutions of integer linear programming problems. As a consequence of our result, we present worst-case optimal decision procedures for two NP-hard problems that were previously not known to be in NP. The first is the satisfiability problem for a logic of sets, multisets (bags), and cardinality constraints, which has applications in verification, interactive theorem proving, and description logics. The second is the reachability problem for a class of transition systems whose transitions increment the state vector by solutions of integer linear arithmetic formulas.


Decision Procedure Regular Expression Atomic Formula Integer Vector Star Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ruzica Piskac
    • 1
  • Viktor Kuncak
    • 1
  1. 1.School of Computer and Communication SciencesEPFLSwitzerland

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