Safety first: a two-stage algorithm for the synthesis of reactive systems

Abstract

In the game-theoretic approach to the synthesis of reactive systems, specifications are often expressed as ω-regular languages. Computing a winning strategy to an infinite game whose winning condition is an ω-regular language is then the main step in obtaining an implementation. Conjoining all the properties of a specification to obtain a monolithic game suffers from the doubly exponential determinization that is required. Despite the success of symbolic algorithms, the monolithic approach is not practical. Existing techniques achieve efficiency by imposing restrictions on the ω-regular languages they deal with. In contrast, we present an approach that achieves improvement in performance through the decomposition of the problem while still accepting the full set of ω-regular languages. Each property is translated into a deterministic ω-regular automaton explicitly while the two-player game defined by the collection of automata is played symbolically. Safety and persistence properties usually make up the majority of a specification. We take advantage of this by solving the game incrementally. Each safety and persistence property is used to gradually construct the parity game. Optimizations are applied after each refinement of the graph. This process produces a compact symbolic encoding of the parity game. We then compose the remaining properties and solve one final game after possibly solving smaller games to further optimize the graph. An implementation is finally derived from the winning strategies computed. We compare the results of our tool to those of the synthesis tool Anzu.