Counting to Complexity: Analyzing Reachability in Counter Automata
A new analysis defines the limits of determining reachable states in vector addition systems with integer counters, offering insights into the inherent computational difficulty of these systems.



 = (0.5, -1)[/latex] to either [latex](\mu, \Delta) = (1.5, 1)[/latex] or [latex](0.8, 1)[/latex] - exhibit distinct behaviors dependent on the number of modes satisfying the condition [latex]\cos(2\delta\theta_{k_c}) = 0[/latex]; specifically, the resulting dynamical mode energy [latex]\tilde{\lambda}_{k_n}(t)[/latex] at time [latex]t_c[/latex] varies with the parameter [latex]m_2[/latex], highlighting a nuanced relationship between quench protocols, mode excitation, and system energy.](https://arxiv.org/html/2603.05284v1/2603.05284v1/x2.png)


