Symmetry Reduction Enables Model Checking of More Complex Emergent Behaviours of Swarm Navigation Algorithms

Abstract

The emergent global behaviours of robotic swarms are important for them to achieve their navigation task goals. These emergent behaviours can be verified to assess their correctness, through techniques like model checking. Model checking exhaustively explores all possible behaviours, based on a discrete model of the system, such as a swarm in a grid. A common problem in model checking is the state-space explosion that arises when the states of the model are numerous. We propose a novel implementation of symmetry reduction, in the form of encoding navigation algorithms relatively with respect to a reference, exploiting the symmetrical properties of swarms in grids. We applied the relative encoding to a swarm navigation algorithm, Alpha, modelled for the NuSMV model checker. A comparison of the state-space and verification results with an absolute (or global) and a relative encoding of the Alpha algorithm highlights the advantages of our approach, allowing model checking both larger grid sizes and higher numbers of robots, and consequently verifying more complex emergent behaviours. For example, a property was verified for a grid with 3 robots and a maximum allowed size of $8 \times 8$ cells in a global encoding, whereas this size was increased to $16 \times 16$ using a relative encoding. Also, the time to verify a property for a swarm of 3 robots in a $6 \times 6$ grid was reduced from almost 10 hours to only 7 minutes. Our approach is transferable to other swarm navigation algorithms.