Strategy Synthesis for Zero-sum Neuro-symbolic Concurrent Stochastic Games

Neuro-symbolic approaches to artificial intelligence, which combine neural networks with classical symbolic techniques, are growing in prominence, necessitating formal approaches to reason about their correctness. We propose a novel modelling formalism called neuro-symbolic concurrent stochastic games (NS-CSGs), which comprise a set of probabilistic finite-state agents interacting in a shared continuous-state environment, observed through perception mechanisms implemented as neural networks. Since the environment state space is continuous, we focus on the class of NS-CSGs with Borel state spaces and Borel measurability restrictions on the components of the model. We consider the problem of zero-sum discounted cumulative reward, proving that NS-CSGs are determined and therefore have a value which corresponds to a unique fixed point. From an algorithmic perspective, existing methods to compute values and optimal strategies for CSGs focus on finite state spaces. We present, for the first time, value iteration and policy iteration algorithms to solve a class of uncountable state space CSGs, and prove their convergence. Our approach works by formulating piecewise linear or constant representations of the value functions and strategies of NS-CSGs. We validate the approach with a prototype implementation applied to a dynamic vehicle parking example.