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Neuro-symbolic approaches to artificial intelligence, which combine neuralnetworks with classical symbolic techniques, are growing in prominence,necessitating formal approaches to reason about their correctness. We propose anovel modelling formalism called neuro-symbolic concurrent stochastic games(NS-CSGs), which comprise probabilistic finite-state agents interacting in ashared continuous-state environment observed through perception mechanismsimplemented as neural networks (NNs). We focus on the class of NS-CSGs withBorel state spaces and prove the existence and measurability of the valuefunction for zero-sum discounted cumulative rewards under piecewise-constantrestrictions on the components of this class of models. To compute values andsynthesise strategies, we present, for the first time, implementable valueiteration (VI) and policy iteration (PI) algorithms to solve a class ofcontinuous-state CSGs. These require a finite representation of the pre-imageof the environment's NN perception mechanism and rely on finite abstractrepresentations of value functions and strategies closed under VI or PI. First,we introduce a Borel measurable piecewise-constant (B-PWC) representation ofvalue functions, extend minimax backups to this representation and proposeB-PWC VI. Second, we introduce two novel representations for the valuefunctions and strategies, constant-piecewise-linear (CON-PWL) andconstant-piecewise-constant (CON-PWC) respectively, and proposeMinimax-action-free PI by extending a recent PI method based on alternatingplayer choices for finite state spaces to Borel state spaces, which does notrequire normal-form games to be solved. We illustrate our approach with adynamic vehicle parking example by generating approximately optimal strategiesusing a prototype implementation of the B-PWC VI algorithm.

Strategy Synthesis for Zero-Sum Neuro-Symbolic Concurrent Stochastic Games