Long-Run Average Behavior of 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 thank Petr Novotný for helpful discussion on the literature.

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Krishnendu Chatterjee, Thomas A. Henzinger, and Jan Otop. Long-Run Average Behavior of Vector Addition Systems with States. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


A vector addition system with states (VASS) consists of a finite set of states and counters. A configuration is a state and a value for each counter; a transition changes the state and each counter is incremented, decremented, or left unchanged. While qualitative properties such as state and configuration reachability have been studied for VASS, we consider the long-run average cost of infinite computations of VASS. The cost of a configuration is for each state, a linear combination of the counter values. In the special case of uniform cost functions, the linear combination is the same for all states. The (regular) long-run emptiness problem is, given a VASS, a cost function, and a threshold value, if there is a (lasso-shaped) computation such that the long-run average value of the cost function does not exceed the threshold. For uniform cost functions, we show that the regular long-run emptiness problem is (a) decidable in polynomial time for integer-valued VASS, and (b) decidable but nonelementarily hard for natural-valued VASS (i.e., nonnegative counters). For general cost functions, we show that the problem is (c) NP-complete for integer-valued VASS, and (d) undecidable for natural-valued VASS. Our most interesting result is for (c) integer-valued VASS with general cost functions, where we establish a connection between the regular long-run emptiness problem and quadratic Diophantine inequalities. The general (nonregular) long-run emptiness problem is equally hard as the regular problem in all cases except (c), where it remains open.

Subject Classification

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


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