Synthesizing Reactive Test Environments for Autonomous Systems: Testing Reach-Avoid Specifications with Multi-Commodity Flows

We study automated test generation for testing discrete decision-making modules in autonomous systems. Linear temporal logic is used to encode the system specification - requirements of the system under test - and the test specification, which is unknown to the system and describes the desired test behavior. The reactive test synthesis problem is to find constraints on system actions such that in a test execution, both the system and test specifications are satisfied. To do this, we use the specifications and their corresponding Büchi automata to construct the specification product automaton. Then, a virtual product graph representing all possible test executions of the system is constructed from the transition system and the specification product automaton. The main result of this paper is framing the test synthesis problem as a multi-commodity network flow optimization. This optimization is used to derive reactive constraints on system actions, which constitute the test environment. The resulting test environment ensures that the system meets the test specification while also satisfying the system specification. We illustrate this framework in simulation using grid world examples and demonstrate it on hardware with the Unitree A1 quadruped, where we test dynamic locomotion behaviors reactively.