# Long-Run Average Behavior of Vector Addition Systems with States

## TL;DR

This paper investigates the long-run average behavior of vector addition systems with states (VASS), providing complexity results and decidability for various cases, and establishing a connection to quadratic Diophantine inequalities.

## Contribution

It offers the first polynomial-time decidability result for uniform cost functions on integer-valued VASS and explores the complexity of the problem for natural-valued and general cost functions.

## Key findings

- Polynomial-time decidability for uniform cost functions on integer-valued VASS
- NP-completeness for general cost functions on integer-valued VASS
- Undecidability results for natural-valued VASS with general cost functions

## Abstract

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.