We provide a generic decision procedure for energy games with energy-bounded attacker and reachability objective, moving beyond vector-valued energies and vector-addition updates. All we demand is that energies form well-founded bounded join-semilattices, and that energy updates have an upward-closed domain and can be "undone" through a Galois-connected function. We instantiate these Galois energy games to common energy games, declining energy games, multi-weighted reachability games, coverability on vector addition systems with states, and shortest path problems, supported by an Isabelle-formalization and two implementations. For the instantiations, our simple algorithm is polynomial w.r.t. game graph size and exponential w.r.t. dimension.

We present the first game characterization of contrasimilarity, the weakest form of bisimilarity. It corresponds to an elegant modal characterization of nested trees of impossible future behavior. The game is exponential but finite for finite-state systems and can thus be used for contrasimulation equivalence checking, of which no tool has been capable to date. By reduction from weak trace equivalence, we establish that contrasimilarity is PSPACE-complete. A machine-checked Isabelle/HOL formalization backs our work and enables further use of contrasimilarity in verification contexts.

Abstract We characterize all common notions of behavioral equivalence by one 6-dimensional energy game, where energies bound capabilities of an attacker trying to tell processes apart. The defender-winning initial credits exhaustively determine which preorders and equivalences from the (strong) linear-time–branching-time spectrum relate processes. The time complexity is exponential, which is optimal due to trace equivalence being covered. This complexity improves drastically on our previous approach for deciding groups of equivalences where exponential sets of distinguishing HML formulas are constructed on top of a super-exponential reachability game. In experiments using the VLTS benchmarks, the algorithm performs on par with the best similarity algorithm.

We introduce a generalization of the bisimulation game that finds distinguishing Hennessy-Milner logic formulas from every finitary, subformula-closed language in van Glabbeek's linear-time--branching-time spectrum between two finite-state processes. We identify the relevant dimensions that measure expressive power to yield formulas belonging to the coarsest distinguishing behavioral preorders and equivalences; the compared processes are equivalent in each coarser behavioral equivalence from the spectrum. We prove that the induced algorithm can determine the best fit of (in)equivalences for a pair of processes.

Abstract We introduce a generalization of the bisimulation game that can be employed to find all relevant distinguishing Hennessy–Milner logic formulas for two compared finite-state processes. By measuring the use of expressive powers, we adapt the formula generation to just yield formulas belonging to the coarsest distinguishing behavioral preorders/equivalences from the linear-time–branching-time spectrum. The induced algorithm can determine the best fit of (in)equivalences for a pair of processes.

Coupled similarity is a notion of equivalence for systems with internal actions. It has outstanding applications in contexts where internal choices must transparently be distributed in time or space, for example, in process calculi encodings or in action refinements. No tractable algorithms for the computation of coupled similarity have been proposed up to now. Accordingly, there has not been any tool support. We present a game-theoretic algorithm to compute coupled similarity , running in cubic time and space with respect to the number of states in the input transition system. We show that one cannot hope for much better because deciding the coupled simulation preorder is at least as hard as deciding the weak simulation preorder. Our results are backed by an Isabelle/HOL formalization, as well as by a parallelized implementation using the Apache Flink framework. Data or code related to this paper is available at: [ 2 ].