Fair simulation

The simulation preorder for labeled transition systems is defined locally, and operationally, as a game that relates states with their immediate successor states. Simulation enjoys many appealing properties. First, simulation has a denotational characterization: System S simulates system I iff every computation tree embedded in the unrolling of I can be embedded also in the unrolling of S. Second, simulation has a logical characterization: S simulates I iff every universal branching-time formula satisfied by S is satisfied also by I. It follows that simulation is a suitable notion of implementation, and it is the coarsest abstraction of a system that preserves universal branching-time properties. Third, based on its local definition, simulation between finite-state systems can be checked in polynomial time. Finally, simulation implies trace containment, which cannot be defined locally and requires polynomial space for verification. Hence simulation is widely used both in manual and in automatic verification. Liveness assumptions about transition systems are typically modeled using fairness constraints. Existing notions of simulation for fair transition systems, however, are not local, and as a result, many appealing properties of the simulation preorder are lost. We propose a new view of fair simulation by extending the local definition of simulation to account for fairness: System S fairly simulates system I iff in the simulation game, there is a strategy that matches with each fair computation of I a fair computation of S. Our definition enjoys a denotational characterization and has a logical characterization: S fairly simulates I iff every fair computation tree (whose infinite paths are fair) embedded in the unrolling of I can be embedded also in the unrolling of S or, equivalently, iff every Fair-∀AFMC formula satisfied by S is satisfied also by I (∀AFMC is the universal fragment of the alternation-free μ-calculus). The locality of the definition leads us to a polynomial-time algorithm for checking fair simulation for finite-state systems with weak and strong fairness constraints. Finally, fair simulation implies fair trace containment and is therefore useful as an efficiently computable local criterion for proving linear-time abstraction hierarchies of fair systems.