Supervisory Controller Synthesis for Non-terminating Processes is an Obliging Game

We present a new algorithm to solve the supervisory control problem over non-terminating processes modeled as $\omega$-regular automata. A solution to the problem was obtained by Thistle in 1995 which uses complex manipulations of automata. This algorithm is notoriously hard to understand and, to the best of our knowledge, has never been implemented. We show a new solution to the problem through a reduction to reactive synthesis. A naive, and incorrect, approach reduces the supervisory control problem to a reactive synthesis problem that asks for a control strategy which ensures the given specification if the plant behaves in accordance to its liveness properties. This is insufficient. A correct control strategy might not fulfill the specification but force the plant to invalidate its liveness property. To prevent such solutions, supervisory control additionally requires that the controlled system is non-conflicting: any finite word compliant with the supervisor should be extendable to a word satisfying the plants' liveness properties. To capture this additional requirement, our solution goes through obliging games instead. An obliging game has two requirements: a strong winning condition as in reactive synthesis and a weak winning condition. A strategy is winning if it satisfies the strong condition and additionally, every partial play can be extended to satisfy the weak condition. Obliging games can be reduced to $\omega$-regular reactive synthesis, for which symbolic algorithms exist. We reduce supervisor synthesis to obliging games. The strong condition is an implication: if the plant behaves in accordance with its liveness properties, the specification should also hold. The weak condition is the plants' liveness property.