Linear Equations with Ordered Data

Authors Piotr Hofman , Slawomir Lasota

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Piotr Hofman
  • University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland
Slawomir Lasota
  • University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland

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Piotr Hofman and Slawomir Lasota. Linear Equations with Ordered Data. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Following a recently considered generalization of linear equations to unordered data vectors, we perform a further generalization to ordered data vectors. These generalized equations naturally appear in the analysis of vector addition systems (or Petri nets) extended with ordered data. We show that nonnegative-integer solvability of linear equations is computationally equivalent (up to an exponential blowup) to the reachability problem for (plain) vector addition systems. This high complexity is surprising, and contrasts with NP-completeness for unordered data vectors. This also contrasts with our second result, namely polynomial time complexity of the solvability problem when the nonnegative-integer restriction on solutions is relaxed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parallel computing models
  • Theory of computation → Timed and hybrid models
  • Theory of computation → Automata over infinite objects
  • Linear equations
  • Petri nets
  • Petri nets with data
  • vector addition systems
  • sets with atoms
  • orbit-finite sets


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