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Multi-Dimensional Long-Run Average Problems for Vector Addition Systems with States

Authors Krishnendu Chatterjee , Thomas A. Henzinger, Jan Otop

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Krishnendu Chatterjee
  • IST Austria, Klosterneuburg, Austria
Thomas A. Henzinger
  • IST Austria, Klosterneuburg, Austria
Jan Otop
  • University of Wrocław, Poland


We want to thank the anonymous reviewers for their helpful comments, which contributed to the final version of this paper.

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Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Multi-Dimensional Long-Run Average Problems for Vector Addition Systems with States. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 23:1-23:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


A vector addition system with states (VASS) consists of a finite set of states and counters. A transition changes the current state to the next state, and every counter is either incremented, or decremented, or left unchanged. A state and value for each counter is a configuration; and a computation is an infinite sequence of configurations with transitions between successive configurations. A probabilistic VASS consists of a VASS along with a probability distribution over the transitions for each state. Qualitative properties such as state and configuration reachability have been widely studied for VASS. In this work we consider multi-dimensional long-run average objectives for VASS and probabilistic VASS. For a counter, the cost of a configuration is the value of the counter; and the long-run average value of a computation for the counter is the long-run average of the costs of the configurations in the computation. The multi-dimensional long-run average problem given a VASS and a threshold value for each counter, asks whether there is a computation such that for each counter the long-run average value for the counter does not exceed the respective threshold. For probabilistic VASS, instead of the existence of a computation, we consider whether the expected long-run average value for each counter does not exceed the respective threshold. Our main results are as follows: we show that the multi-dimensional long-run average problem (a) is NP-complete for integer-valued VASS; (b) is undecidable for natural-valued VASS (i.e., nonnegative counters); and (c) can be solved in polynomial time for probabilistic integer-valued VASS, and probabilistic natural-valued VASS when all computations are non-terminating.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Quantitative automata
  • vector addition systems
  • mean-payoff
  • multidimension
  • probabilistic semantics


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