We consider how to construct state abstractions compatible with a given set of abstract actions, to obtain a well-formed abstract Markov decision process (MDP). We show that the Bellman equation suggests that abstract states should represent distributions over states in the ground MDP; we characterize the conditions under which the resulting process is Markov and approximately model-preserving, derive an algorithm for constructing the abstract MDP, and apply it to visual chain and maze tasks. We generalize these results to the factored actions case, characterize the conditions that lead to factored abstract states, and apply the resulting algorithm to a visual grid and Montezuma's Revenge. These results provide a principled, powerful framework for learning neurosymbolic abstract Markov decision processes.

Current approaches using sequential networks have shown promise in estimating field variables for dynamical systems, but they are often limited by high rollout errors. The unresolved issue of rollout error accumulation results in unreliable estimations as the network predicts further into the future, with each step's error compounding and leading to an increase in inaccuracy. Here, we introduce the State-Exchange Attention (SEA) module, a novel transformer-based module enabling information exchange between encoded fields through multi-head cross-attention. The cross-field multidirectional information exchange design enables all state variables in the system to exchange information with one another, capturing physical relationships and symmetries between fields. Additionally, we introduce an efficient ViT-like mesh autoencoder to generate spatially coherent mesh embeddings for a large number of meshing cells. The SEA integrated transformer demonstrates the state-of-the-art rollout error compared to other competitive baselines. Specifically, we outperform PbGMR-GMUS Transformer-RealNVP and GMR-GMUS Transformer, with a reduction in error of 88% and 91%, respectively. Furthermore, we demonstrate that the SEA module alone can reduce errors by 97% for state variables that are highly dependent on other states of the system. The repository for this work is available at: https://github.com/ParsaEsmati/SEA

We also demonstrate its effectiveness in generating robotic tasks.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

In this paper, we propose polynomial forms to represent distributions of state variables over time for discrete-time stochastic dynamical systems. This problem arises in a variety of applications in areas ranging from biology to robotics. Our approach allows us to rigorously represent the probability distribution of state variables over time, and provide guaranteed bounds on the expectations, moments and probabilities of tail events involving the state variables. First, we recall ideas from interval arithmetic, and use them to rigorously represent the state variables at time t as a function of the initial state variables and noise symbols that model the random exogenous inputs encountered before time t. Next, we show how concentration of measure inequalities can be employed to prove rigorous bounds on the tail probabilities of these state variables. We demonstrate interesting applications that demonstrate how our approach can be useful in some situations to establish mathematically guaranteed bounds that are of a different nature from those obtained through simulations with pseudo-random numbers.