Affine Extensions of Integer Vector Addition Systems with States

Authors Michael Blondin , Christoph Haase , Filip Mazowiecki

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Author Details

Michael Blondin
  • Technische Universität München, Germany
Christoph Haase
  • University of Oxford, United Kingdom
Filip Mazowiecki
  • LaBRI, Université de Bordeaux, France

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Michael Blondin, Christoph Haase, and Filip Mazowiecki. Affine Extensions of Integer Vector Addition Systems with States. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We study the reachability problem for affine Z-VASS, which are integer vector addition systems with states in which transitions perform affine transformations on the counters. This problem is easily seen to be undecidable in general, and we therefore restrict ourselves to affine Z-VASS with the finite-monoid property (afmp-Z-VASS). The latter have the property that the monoid generated by the matrices appearing in their affine transformations is finite. The class of afmp-Z-VASS encompasses classical operations of counter machines such as resets, permutations, transfers and copies. We show that reachability in an afmp-Z-VASS reduces to reachability in a Z-VASS whose control-states grow polynomially in the size of the matrix monoid. Our construction shows that reachability relations of afmp-Z-VASS are semilinear, and in particular enables us to show that reachability in Z-VASS with transfers and Z-VASS with copies is PSPACE-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Logic and verification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Complexity classes
  • Vector addition systems
  • affine transformations
  • reachability
  • semilinear sets
  • computational complexity


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