A Compositional Approach to Reactive Games under Temporal Logic Specifications

We study the problem of compositional synthesis of controllers for reactive games with linear temporal logic (LTL) specifications. A reactive game is an abstraction of the interaction between a controllable system and its uncontrolled and dynamic environment. A centralized control design for such systems under complex specifications can be computationally expensive. Instead, a compositional approach aims to synthesize a controller for a complex specification by composing the solutions for its component sub-specifications. This mitigates the issue of scalability and has the advantages of being modular and flexible. This paper solves the problem of reactive game synthesis using the compositional approach in two steps. First, we use the notion of randomized permissive strategy to reduce the strategy synthesis problem to that of identifying only the winning region for the controlled agent against the uncontrolled environment. Then, we exploit the inherent compositional nature of LTL formulas to compose the independently computed winning regions of two sub-games into a superset of the composed-game winning region. We make use of elementary set operations to construct this superset. Finally, we introduce an iterative algorithm to extract the exact winning region from the superset. We prove the correctness of our proposed method and illustrate the solution using a toy-problem and a robot motion planning example.