# Topological Approximate Dynamic Programming under Temporal Logic Constraints

## TL;DR

This paper introduces a Topological Approximate Dynamic Programming method for planning in stochastic systems with high-level temporal logic constraints, improving efficiency and scalability.

## Contribution

It presents a novel decomposition approach based on automaton topology and extends model-free ADP to handle LTL constraints in MDPs.

## Key findings

- Complexity does not grow polynomially with the product MDP size.
- Algorithm is correct and efficient in robotic motion planning.
- Decomposition reduces computational burden in temporal logic planning.

## Abstract

In this paper, we develop a Topological Approximate Dynamic Programming (TADP) method for planningin stochastic systems modeled as Markov Decision Processesto maximize the probability of satisfying high-level systemspecifications expressed in Linear Temporal Logic (LTL). Ourmethod includes two steps: First, we propose to decompose theplanning problem into a sequence of sub-problems based on thetopological property of the task automaton which is translatedfrom the LTL constraints. Second, we extend a model-freeapproximate dynamic programming method for value iterationto solve, in an order reverse to a causal dependency of valuefunctions, one for each state in the task automaton. Particularly,we show that the complexity of the TADP does not growpolynomially with the size of the product Markov DecisionProcess (MDP). The correctness and efficiency of the algorithmare demonstrated using a robotic motion planning example.