Theoretical Foundations for Abstraction-Based Probabilistic Planning

TL;DR

This paper introduces a formal framework for probabilistic planning using affine-operators and affine-trees, providing theoretical foundations for action projection and abstraction under uncertainty.

Contribution

It develops a general theoretical framework for probabilistic planning based on affine-operators, unifying various abstraction techniques and deriving projection rules with proven correctness.

Findings

Three projection rules with correctness proofs

Affine-operator properties enabling probabilistic action modeling

Manifestation of existing action abstraction types within the framework

Abstract

Modeling worlds and actions under uncertainty is one of the central problems in the framework of decision-theoretic planning. The representation must be general enough to capture real-world problems but at the same time it must provide a basis upon which theoretical results can be derived. The central notion in the framework we propose here is that of the affine-operator, which serves as a tool for constructing (convex) sets of probability distributions, and which can be considered as a generalization of belief functions and interval mass assignments. Uncertainty in the state of the worlds is modeled with sets of probability distributions, represented by affine-trees while actions are defined as tree-manipulators. A small set of key properties of the affine-operator is presented, forming the basis for most existing operator-based definitions of probabilistic action projection and action abstraction. We derive and prove correct three projection rules, which vividly illustrate the precision-complexity tradeoff in plan projection. Finally, we show how the three types of action abstraction identified by Haddawy and Doan are manifested in the present framework.