Projection Abstractions in Planning Under the Lenses of Abstractions for MDPs

TL;DR

Projection abstractions unify planning and discounted MDP abstractions by showing how a planning projection can be represented within WFAs, ARMDPs, or ABPMDPs. The work analyzes conditions under which planning projections produce deterministic, connection-preserving abstractions, notably when actions lack conditional effects. It provides a detailed mapping via examples (e.g., Logistics) and highlights the trade-offs between projection-based planning heuristics and MDP-based abstractions, as well as the computational costs. It also identifies gaps for probabilistic planning and SSP frameworks and outlines directions for extending abstractions beyond the no-conditional-effects assumption.

Abstract

The concept of abstraction has been independently developed both in the context of AI Planning and discounted Markov Decision Processes (MDPs). However, the way abstractions are built and used in the context of Planning and MDPs is different even though lots of commonalities can be highlighted. To this day there is no work trying to relate and unify the two fields on the matter of abstractions unraveling all the different assumptions and their effect on the way they can be used. Therefore, in this paper we aim to do so by looking at projection abstractions in Planning through the lenses of discounted MDPs. Starting from a projection abstraction built according to Classical or Probabilistic Planning techniques, we will show how the same abstraction can be obtained under the abstraction frameworks available for discounted MDPs. Along the way, we will focus on computational as well as representational advantages and disadvantages of both worlds pointing out new research directions that are of interest for both fields.

Figures (6)

• Figure 1: Relationship between Planning and MDP w.r.t. the original problem description and the abstraction. The functions $\varphi$ and $\zeta$ transform a problem into its abstraction in the planning and MDP domains, respectively.
• Figure 2: Non-deterministic transitions induced in the abstraction's transition graph by actions with conditional effects.
• Figure 3: An abstraction that cannot be represented through WFA. The aggregation function retains only $p_1$ and $p_2$, the original initial state is $p_4$, and the goal is $\neg p_1 \wedge p_2 \wedge p_3$. Action $a$ requires $\omega_{\Bar{s}_1}(s_6) + \omega_{\Bar{s}_1}(s_8) = 1$, action $b$ requires $\omega_{\Bar{s}_1}(s_6) = 1$, action $h$ requires $\omega_{\Bar{s}_1}(s_8) = 1$, that yield a contradiction.
• Figure 4: Relations among projection abstractions in Planning and abstarctions in MDPs.
• Figure 5: The Logistic transition graph for the running example and its abstraction. A node represents a state. The first component of the state is the location of the package ($L,R,A,B$). The second component is the location of truck $A$ ($L,R$), and the last component is the location of truck $B$ ($L,R$). The dashed nodes are abstracted states merging together states that share the same location of the package. The green dashed node aggregates all the goals of the problem. A SAS+ representation has been used for space constraints. Any state (e.g., $I=LRR$) can be represented in its equivalent propositional (STRIPS) version (e.g., $I = 10000101$).

Theorems & Definitions (16)

• Definition 1: Projection Abstractions
• Definition 2: Weighting Function Abstractions
• Definition 3: Abstract Robust MDPs
• Definition 4: Abstract Bounded Parameter MDPs
• Proposition 0: Absence of ambiguity in the abstraction transition graph
• proof
• Definition 5: Connection Preserving
• Definition 6: Deterministic Abstraction
• Proposition 0: Connection Preserving WFAs
• proof