Hierarchical Multi-Contingency Planning

• Hierarchical Multi-Contingency Planning is a framework that decomposes decision-making into abstract symbolic tasks and concrete geometric sub-tasks to manage uncertainty.
• It employs abstraction functions, HTN decompositions, and probabilistic outcome modeling to navigate complex branching scenarios with high computational efficiency.
• Anytime algorithms like HPlan and HQCP integrate risk assessment and belief updates to provide provable performance and practical applicability in stochastic and partially observable domains.

Hierarchical Multi-Contingency Planning addresses the synthesis of robust, high-quality plans or policies in domains where uncertainty or partial observability leads to branching execution-time contingencies. In contrast to flat planning approaches, these frameworks exploit hierarchical abstractions—often at the level of discrete high-level tasks versus continuous geometric/motion subtasks, or through expressive decompositional hierarchies such as HTN (Hierarchical Task Networks)—to efficiently reason about and resolve multiple possible outcomes. This approach is critical for robotics, autonomous systems, and stochastic domains where the curse of dimensionality and combinatorial blow-up of contingencies are acute.

1. Formal Problem Setting

Hierarchical Multi-Contingency Planning extends classical planning formalisms to explicitly encode and resolve branches arising from stochastic actions or observations. The fully observable stochastic case adopts the stochastic shortest path (SSP) model augmented with continuous hybrid actions: $$P_\ell = \langle S,\;A,\;\Omega,\;T,\;C,\;\gamma=1,\;H \rangle$$ where $S$ is the (possibly infinite) hybrid state space, $A$ partitions into symbolic and motion-planning actions $A = A_{\mathrm{sym}} \cup A_{\mathrm{mp}}$, $\Omega$ is the set of possible stochastic outcomes per action, $T$ is a transition kernel, $C$ assigns costs or risks, and $H$ is the finite horizon. The objective is to maximize the probability of attaining a goal region $S_g$ in at most $H$ steps:

$\pi^* = \arg\max_\pi P_{\mathrm{success}}(\pi)$

where $P_{\mathrm{success}}(\pi)$ accumulates the measure of successful trajectories induced by policy $\pi$ (Shah et al., 2021).

In partially observable contexts, the POMDP model is central. A belief $b_t$ is maintained over latent states, repeatedly updated via Bayes' theorem in response to latest observations $o_t$ and actions $a_t$. For hierarchical task decomposition, the HTN paradigm generalizes the planning tuple to

$P = (s_0,\,b_0,\,w_0,\,D,\,C)$

incorporating an initial belief state $b_0$ and an explicit cost model, allowing for actuation operators with probabilistic effects and sensing operators tied to observation outcomes (Zhao, 2023). A contingent plan is then represented as a tree, branching on possible outcomes of sensing actions or stochastic transitions.

2. Hierarchical Abstraction and Decomposition

Hierarchical abstraction is foundational to tractable multi-contingency planning. Two-level hierarchies are prominent: abstract planning is carried out over a symbolic high-level SSP or HTN model, while lower levels handle geometric or continuous refinements.

• Abstraction Function $\alpha: S \to S_h$ compresses the concrete state by omitting continuous or difficult-to-compute fluents, yielding an abstract state $s_h$.
• Concretization Operator $\Gamma$ generates sets of concrete refinements (e.g., trajectory samples) to realize an abstract action at the geometric level, potentially invoking sampling-based motion planners (Shah et al., 2021).

In the HTN context, methods decompose compound tasks into partially ordered subtasks, extended to allow for sensing actions that explicitly branch the task network by observation. Each plan node may then correspond to multiple successor branches, reflecting all realized contingencies (Zhao, 2023).

This hierarchical modeling reduces the search cost by focusing on symbolic planning for wide regions of the contingency space, and invokes costly geometric or belief updates only as needed for likely branches.

3. Policy and Plan Representation

Policies in hierarchical multi-contingency settings are typically encoded as rooted directed acyclic graphs (or policy trees), with each node annotated by an abstract (or belief) state. Edges are labeled by actions and, for stochastic or sensing actions, further decorated by realized outcomes or observations.

• Policy Representation (SSP context):

$\pi = (V, E, \text{cond})$

where $V$ is the set of policy nodes (abstract states), $E$ is a set of action-labeled edges, and $\text{cond}$ maps each outgoing edge to an outcome in $\Omega_h$. Execution semantically follows root-to-leaf paths according to realized outcomes; branches are "resolved" only once all abstract actions on the path are concretized (Shah et al., 2021).

• HTN-Contingency Tree: For belief-aware domains, each sensing action node forks $k$ branches, each corresponding to an observation outcome with probability $p_i$. Belief update functions and branching structure are algebraically defined, with plan cost and probability mass inductively calculated from action and observation probabilities (Zhao, 2023).

This explicit branching structure supports rich contingency handling, but also induces combinatorial complexity, motivating further abstraction and prioritization.

4. Algorithmic Frameworks and Anytime Properties

A core contribution of hierarchical multi-contingency planning research is the design of integrated, anytime algorithms that coordinate abstract planning, selective refinement, and efficient contingency coverage.

• HPlan Algorithm (Editor’s term): At each iteration, the planner either (a) refines the symbolic policy by concretizing a high-probability, unresolved branch (employing geometric generators), or (b) updates the abstraction when infeasibility is detected (adding new preconditions and recomputing the abstract policy). Branch selection leverages a probability-to-cost ($p/c$) heuristic, ensuring focus on the branches with greatest impact per computation (Shah et al., 2021).
• HTN-Based HQCP Algorithm: Interleaves HTN decomposition, belief-state updating, and contingency branching. Admissible cost-to-go heuristics $h(t)$ prune low-quality decompositions, and the search proceeds recursively over (compound/primitive) tasks and their sub-networks, only expanding successful branches (Zhao, 2023).
• Hierarchical Controller for Autonomous Driving: Low-level controllers are deep Q-networks (π*, π₁) trained for nominal and contingency scenarios; a high-level planner actively estimates risk via belief sampling and switches between controllers based on estimated failure probability. Policy switching is resolved at each timestep via online risk assessment (Lecerf et al., 2022).

Anytime properties are theoretically quantified: HPlan achieves a 2-approximation to optimal coverage of contingency space as a function of wall-clock time, and is probabilistically complete in the limit of infinite sampling when the generator measures are nonzero (Shah et al., 2021).

5. Heuristics, Complexity, and Trade-Offs

Hierarchical multi-contingency algorithms employ admissible heuristics and abstraction strategies to contain computational complexity.

• Cost-to-Go Heuristics: $\Delta(t)$ and admissible $h(t)$ are computed to estimate the minimum cost of completing a task $t$, preserving optimality by pruning provably suboptimal branches (Zhao, 2023).
• Prioritization: The $p/c$ heuristic directs refinement effort towards high-probability, low-cost paths. This is proven to achieve a greedy 2-approximation for branch coverage, cast as a fractional knapsack problem (Shah et al., 2021).
• Computational Complexity: Flat planning over all hybrid contingencies is intractable due to exponential branching. Hierarchical decomposition isolates subgoals, reducing planner time to polynomial in symbolic state/action/outcome space (for the symbolic part), while geometric refinement remains exponential in the path length of each refined branch. The space overhead is minimized by maintaining only a single symbolic policy per policy graph node and monotonic abstraction updates.
• Trade-offs and Cutoffs: Planners allow terminating once a prescribed cumulative probability mass (coverage) $\rho$ is achieved, quantifying the residual risk versus precomputation budget (Shah et al., 2021).

6. Empirical Evaluation and Applications

Hierarchical multi-contingency planners have demonstrated practical viability across challenging domains:

Domain	Key Features / Challenges	Result Highlights
Cluttered Table	Uncertain grasp outcomes, combinatorial object branches	80% mass covered in ≈30% of wall-clock time
Aircraft Inspection	UAV navigation with stochastic faults	Scales to $4^H$ branching, solved within 4,000 s
Keva-Plank Structures	Bimodal supply yields $2^n$ stochastic branches	Robust completion with partial refinement
Dining Table Setup	Stochastic item breakage, must avoid dead-ends	Efficient dead-end avoidance, real robot demo
Find the Can	Partial observability, sensor branching	All outcomes covered, fast plan synthesis

In autonomous driving (left-turn intersection), hierarchical contingency planning attained zero collisions across all stochastic scenarios by combining nominal and contingency-trained DQNs within a belief-aware model-based planner (Lecerf et al., 2022). In HTN contingent domains (e.g., medical diagnosis or ZenoTravel), HQCP significantly outperformed PC-SHOP baselines in CPU time and plan quality, always maintaining 100% contingency coverage (Zhao, 2023).

7. Limitations and Prospective Directions

Known limitations include assumptions of finite domains (e.g., discrete belief spaces, no task network cycles), reliance on domain-provided or exhaustively enumerated cost models, and restricted observation models (typically discrete and exact). Potential avenues for future research encompass:

• Extension of hierarchical planning frameworks to continuous or high-dimensional belief/state spaces.
• Incorporation of richer, potentially learned or dynamically parameterized observation models.
• Learning or adaptation of cost-model parameters online under uncertainty.

A plausible implication is that advances in sampling-based generative planning, online cost learning, and abstraction discovery algorithms could strengthen the scalability and generality of hierarchical multi-contingency planning architectures.

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Hierarchical Multi-Contingency Planning, as operationalized in algorithms such as HPlan and HQCP, provides a principled and empirically validated approach to integrated symbolic–geometric, belief-aware, or multi-policy planning under stochastic and partially observable scenarios, with provable guarantees on completeness, anytime performance, and practical applicability across challenging robotic, control, and HTN task domains (Shah et al., 2021, Lecerf et al., 2022, Zhao, 2023).