Strategy Synthesis for Zero-Sum Neuro-Symbolic Concurrent Stochastic Games
Rui Yan, Gabriel Santos, Gethin Norman, David Parker, Marta Kwiatkowska
TL;DR
This work introduces neuro-symbolic concurrent stochastic games (NS-CSGs), a framework where two neural-perception-equipped agents reason symbolically about a shared continuous environment. It establishes formal foundations under Borel/BFCP assumptions, proving the existence and measurability of the zero-sum discounted value $V^\star$ and providing practical finite abstractions. The paper then develops two computationally tractable algorithms, a B-PWC value iteration and a minimax-action-free policy iteration, both leveraging preimage and image operations over BFCPs to guarantee finite representability and convergence. A prototype application in dynamic vehicle parking demonstrates strategy synthesis under neural-perception, highlighting the approach’s potential for verifiable neuro-symbolic control. Overall, the contribution integrates neural perception with symbolic decision-making in stochastic games, delivering rigorous existence results and scalable, convergent algorithms for zero-sum objectives in uncountable state spaces.
Abstract
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 two probabilistic finite-state agents interacting in a shared continuous-state environment. Each agent observes the environment using a neural perception mechanism, which converts inputs such as images into symbolic percepts, and makes decisions symbolically. We focus on the class of NS-CSGs with Borel state spaces and prove the existence and measurability of the value function for zero-sum discounted cumulative rewards under piecewise-constant restrictions on the components of this class of models. To compute values and synthesise strategies, we present, for the first time, practical value iteration (VI) and policy iteration (PI) algorithms to solve this new subclass of continuous-state CSGs. These require a finite decomposition of the environment induced by the neural perception mechanisms of the agents and rely on finite abstract representations of value functions and strategies closed under VI or PI. First, we introduce a Borel measurable piecewise-constant (B-PWC) representation of value functions, extend minimax backups to this representation and propose a value iteration algorithm called B-PWC VI. Second, we introduce two novel representations for the value functions and strategies, constant-piecewise-linear (CON-PWL) and constant-piecewise-constant (CON-PWC) respectively, and propose Minimax-action-free PI by extending a recent PI method based on alternating player choices for finite state spaces to Borel state spaces, which does not require normal-form games to be solved.