Probabilistic Lifted Ordering Graph

• p-LOG is a statistical data structure that captures ordering relationships between parameterized landmarks in planning domains.
• It aggregates ordering frequencies across tasks, assigning empirical probabilities for reusable, domain-level planning heuristics.
• The structure enhances planning efficiency by transferring abstract ordering knowledge to guide instance-specific task instantiation.

A probabilistic lifted ordering graph (p-LOG) is a statistically grounded, compact, and parameterized data structure designed to represent weighted abstractions of ordering relationships between parameterized landmarks in automated planning domains. It extends the notion of classical landmark graphs by lifting landmarks and their orderings—replacing objects with variables—to capture domain-level regularities. The weights are interpreted as probabilities, reflecting how frequently an ordering appears across multiple planning tasks. Although these orderings are not always strictly necessary (hence "probabilistic"), p-LOGs retain essential domain knowledge and enable efficient re-use, alignment, and instantiation for new instances, substantially improving instance-agnostic planning heuristics and order extraction (Pérez-Corral et al., 21 Sep 2025).

1. Definition and Function

A probabilistic lifted ordering graph encapsulates statistical information about ordering relationships between lifted (parameterized) landmarks observed across a collection of planning problems within a specific domain. Here:

• Landmarks are conjunctive predicates that must be satisfied at some point in every valid plan.
• Ordering relationships represent necessary or frequently occurring precedence constraints between landmarks (i.e., one landmark must be achieved before another).
• Lifted/parameterized means that landmarks and orderings are represented using variables, not fixed objects, supporting abstraction across different planning tasks.

Edges in the p-LOG are labeled with probabilities reflecting their empirical frequency across the training set of instances, i.e., μ(e) = n / nGraph(L₂'), where n is the observed frequency of the directed edge (L₁', L₂'), and nGraph(L₂') is the number of graphs in the dataset containing L₂'.

The p-LOG is denoted as ⟨V_D, E_D, μ(E_D)⟩, where V_D is the set of parameterized landmarks, E_D is the set of directed, weighted edges, and μ(E_D) gives the probability for each edge (Pérez-Corral et al., 21 Sep 2025).

2. Construction Pipeline

The process of creating a p-LOG from a set of solved planning tasks consists of the following steps:

1. Landmark Extraction and Grounded LGG Construction: For each training planning task Πₖ, construct a grounded Landmark Generation Graph (LGG), whose nodes are grounded landmarks, and edges represent ordering constraints established via known extraction methods.
2. Lifting (Parameterization) of Landmarks and Orderings: For each landmark L in an LGG, create a lifted version L' by replacing object constants with variables appropriate for their argument type. Every edge (Lᵢ, L) in the original LGG consequently becomes an edge (Lᵢ', L') in the lifted ordering graph (LOG).
3. Aggregation Across Tasks: For each unique parameterized landmark L', aggregate all LOGs over all tasks to form a unified, weighted LOG (w-LOG), where edge multiplicities n represent frequency counts.
4. Probability Assignment: For every edge e = ⟨(L₁', L₂'), n⟩ in the w-LOG, compute the empirical probability μ(e) = n / nGraph(L₂').

The resulting p-LOG encodes the joint empirical structure of the domain, enabling probabilistic weighting of orderings discovered from the dataset (Pérez-Corral et al., 21 Sep 2025).

3. Instantiation for New Tasks

To utilize a p-LOG for a new planning instance Π^G, a two-phase instantiation is performed:

• Phase 1: Generation of Task-Specific p-LGGs
  • p-LGG_G: Generated from the new task's goal state, instantiating lifted landmarks using objects found in the goal.
  • p-LGG_I: Generated from the initial state, following an analogous procedure while propagating constraints forward.

For each phase, a queue is maintained with the relevant landmarks, and an assignment structure tracks constraints (object vs. variable). Parameterized predicates are instantiated with available domain objects.

• Phase 2: Combination and Unification
  • Merging the two p-LGGs involves iterative matching and "pinning" of variables via equivalence checks—predicates are considered equivalent if their argument types and instantiated values (objects or variables) match or permit feasible substitutions under the current constraints.
  • The process iterates until no further unification is possible, yielding a single, instantiated p-LGG tailored for the new task.

This instantiation mechanism allows the general, domain-learned relationships to be mapped onto specific problem instances, thus extracting instance-specific guidance from the generic p-LOG (Pérez-Corral et al., 21 Sep 2025).

4. Mathematical Formalism

The p-LOG assigns a probability μ(e) to every directed edge e = ⟨(L₁', L₂'), n⟩ as: $μ(e) = \frac{n}{\text{nGraph}(L₂')}$ where n is the frequency of the ordering (L₁', L₂'), and nGraph(L₂') is the number of graphs in which L₂' appears.

For comparing lifted landmarks and edges with their grounded (instance-specific) equivalents, a likelihood function is defined: $$ω(L', L) = \begin{cases} \frac{|\text{objects}(L')|}{|\text{objects}(L)|} & \text{if } |\text{params}(L)| > 0 \ 1 & \text{otherwise} \end{cases}$$ and for edges: $$ω(e₁, e₂) = \frac{1}{2} ( ω(L₁', L₁) + ω(L₂', L₂) )$$ These are used in evaluation metrics for p-LOG precision and recall (Pérez-Corral et al., 21 Sep 2025).

5. Evaluation Protocols and Metrics

Standard set-based precision, recall, and F1-score metrics are adapted to the lifted/probabilistic setting. To account for partial matches and abstraction levels, the paper introduces α-precision and α-recall, which multiply standard values by the likelihood score (ω) reflecting the parameterization match between lifted and grounded landmarks or edges: $$\text{α-Precision} = \text{Precision} + \alpha(1 - \text{Precision})$$ This dual evaluation regime quantifies both the strict (instance-level) and relaxed (domain-level, abstracted) utility of the p-LOG.

Empirical studies over benchmark planning domains show that recall and α-recall values for both vertices and edges are consistently high, implying that the p-LOG recovers most true and even additional ordering constraints beyond those found by classical, task-specific methods (Pérez-Corral et al., 21 Sep 2025).

6. Applications and Implications

Probabilistic lifted ordering graphs present several noteworthy applications and implications in automated planning:

• Instance Transferability: Lifting orderings to the parameterized level enables sharing of discovered domain-level knowledge across instances—removing the need to recompute landmark graphs from scratch.
• Domain-Generalization and Heuristic Guidance: The p-LOG can guide planners in novel tasks by providing probabilistically weighted orderings. Even if not universally necessary, orderings with higher μ(e) can be used to prioritize search directions.
• Improved Generality and Interpretability: Variable parameterization (lifting) ensures that relationships reflect underlying domain regularities rather than artifacts of specific instantiations, leading to improved generalization.
• Computational Efficiency: Instantiating only upon need and leveraging precomputed, lifted relationships reduces the computational overhead compared to approaches relying solely on relaxed planning graphs in each instance.
• Extensions: The approach allows for future expansion to include “reasonable” (non-necessary but helpful) orderings, as well as more complex relationships such as conjunctive or mutex constraints, broadening the utility of the p-LOG formalism.
• Empirical Effectiveness: Precision, recall, and α-based metrics validate the practical performance of p-LOGs, with the graphs recovering additional, otherwise overlooked orderings, supporting enhanced search heuristics and plan extraction (Pérez-Corral et al., 21 Sep 2025).

7. Context within the Literature

The probabilistic lifted ordering graph leverages and advances landmark extraction work by introducing statistical abstraction across instances rather than instance-specific tailoring. Unlike classical landmark approaches, which are detailed and precise for single tasks but brittle and non-reusable, p-LOGs facilitate domain-level learning and transfer—an increasingly important feature in scalable automated planning and generalization-centric AI research.

Furthermore, the explicit identification of orderings not as necessary but probabilistically significant marks a shift towards robust, data-driven planning heuristics that can adapt to exceptions while benefiting from cross-task regularities.

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In summary, the probabilistic lifted ordering graph is a domain-level, parameterized, weighted graph structure for encoding, transferring, and instantiating statistically significant landmark orderings across planning tasks. It improves generalizability, efficiency, and the integration of probabilistic knowledge into planning search heuristics by abstracting away from singleton problem instances to reusable, abstracted domain knowledge (Pérez-Corral et al., 21 Sep 2025).