Neuro-symbolic artificial intelligence is an emerging area that combines
traditional symbolic techniques with neural networks. In this paper, we
consider its application to sequential decision making under uncertainty. We
introduce neuro-symbolic partially observable Markov decision processes
(NS-POMDPs), which model an agent that perceives a continuous-state environment
using a neural network and makes decisions symbolically, and study the problem
of optimising discounted cumulative rewards. This requires functions over
continuous-state beliefs, for which we propose a novel piecewise linear and
convex representation (P-PWLC) in terms of polyhedra covering the
continuous-state space and value vectors, and extend Bellman backups to this
representation. We prove the convexity and continuity of value functions and
present two value iteration algorithms that ensure finite representability by
exploiting the underlying structure of the continuous-state model and the
neural perception mechanism. The first is a classical (exact) value iteration
algorithm extending $\alpha$-functions of Porta et al (2006) to the P-PWLC
representation for continuous-state spaces. The second is a point-based
(approximate) method called NS-HSVI, which uses the P-PWLC representation and
belief-value induced functions to approximate value functions from below and
above for two types of beliefs, particle-based and region-based. Using a
prototype implementation, we show the practical applicability of our approach
on two case studies that employ (trained) ReLU neural networks as perception
functions, dynamic car parking and an aircraft collision avoidance system, by
synthesising (approximately) optimal strategies. An experimental comparison
with the finite-state POMDP solver SARSOP demonstrates that NS-HSVI is more
robust to particle disturbances.

本文介绍了神经符号人工智能在不确定性的情况下进行序列决策的一种新方法，即神经符号部分可观察马尔可夫决策过程 (NS-POMDPs)；提出了基于多面体的新型连续状态置信度分段线性凸表示 (P-PWLC)，并且扩展将 Bellman 备份应用于此表示，设计了两种值迭代算法，即经典的值迭代算法和基于点的近似方法。通过两个车辆停车和飞行器避碰案例研究，展示了该方法的实际应用。