Symbolic construction of GR(1) contracts for synchronous systems with full information

Abstract: This work proposes a symbolic algorithm for the construction of assume-guarantee specifications that allow multiple agents to cooperate. Each agent is assigned goals expressed in a fragment of linear temporal logic known as generalized reactivity of rank 1 (GR(1)). These goals may be unrealizable, unless additional assumptions are made by each agent about the behavior of the other agents. The proposed algorithm constructs weakly fair assumptions for each agent, to ensure that they can cooperate successfully. A necessary requirement is that the given goals be cooperatively satisfiable. We prove that there exist games for which the GR(1) fragment with liveness properties over states is not sufficient to ensure realizability from any state in the cooperatively winning set. The obstruction is due to circular dependencies of liveness goals. To prevent circularity, we introduce nested games as a formalism to express specifications with conditional assumptions. The algorithm is symbolic, with fixpoint structure similar to the GR(1) synthesis algorithm, implying time complexity polynomial in the number of states, and linear in the number of recurrence goals.