On The Complexity of Counter Reachability Games

TL;DR

This paper investigates the computational complexity of counter reachability games across different semantics, revealing that complexity often remains consistent but varies in specific cases such as one-dimensional objectives.

Contribution

It provides a comprehensive complexity analysis of counter reachability games under various semantics, highlighting cases where complexity depends on the objective value.

Findings

Complexity is consistent across most semantics.

In one dimension, complexity depends on whether the target value is zero or not.

Most cases have the same complexity regardless of semantics.

Abstract

Counter reachability games are played by two players on a graph with labelled edges. Each move consists in picking an edge from the current location and adding its label to a counter vector. The objective is to reach a given counter value in a given location. We distinguish three semantics for counter reachability games, according to what happens when a counter value would become negative: the edge is either disabled, or enabled but the counter value becomes zero, or enabled. We consider the problem of deciding the winner in counter reachability games and show that, in most cases, it has the same complexity under all semantics. Surprisingly, under one semantics, the complexity in dimension one depends on whether the objective value is zero or any other integer.