Diversified Planning Initialization

• Diversified planning initialization is a method that employs explicit grid-based feature extraction to generate diverse, cost-bounded plan portfolios across various domains.
• It iteratively invokes SAT/SMT solvers with behavioral exclusion constraints, ensuring each plan occupies a unique cell in a defined n-dimensional grid.
• Practical applications span numerical, over-subscription, and multi-agent planning, with empirical studies demonstrating improved diversity and computational tractability.

Diversified planning initialization is a foundational technique to ensure plan portfolios cover a broad, user-relevant range of behaviors or solutions in combinatorial, continuous, and mixed-symbolic planning domains. Rather than outputting redundant or trivially distinct plans, a diversified initializer seeks systematic coverage according to precisely defined diversity criteria. Recent approaches operationalize this via explicit behavior-space models, grid-based feature selection, iterative solver invocation denying prior solution behaviors, and integration with simulation, evolutionary, or stochastic algorithms. These frameworks now support classical, numerical, over-subscription, and multi-agent planning, as well as planning initialization for generative modeling, multi-modal perception, and RL-guided search.

1. Mathematical Foundations of Diversified Planning Initialization

At its core, diversified initialization is formalized over a planning problem $\Pi=(P,A,I,G)$ with constraints: a cost bound $c$ and target pool cardinality $k$ (Abdelwahed et al., 2024). Central to behavior-based diversification is the definition of an $n$-dimensional grid model. The feature set $F=\{f_1, ..., f_n\}$ comprises discrete or domain-specific plan descriptors, each mapped by a DimensionConstructor to a finite set of bins $\text{vals}_i \subset \mathbb{Z}$, a binning function $\otimes_i: \mathbb{Z}\to \text{vals}_i$, and a plan feature extractor $\odot_i(\pi)\in\mathbb{Z}$.

The behavior space is defined:

$$\text{B-Space}(\Xi) = \text{vals}_1 \times \text{vals}_2 \times \dots \times \text{vals}_n$$

For plan $\pi$, the behavior vector is $$f(\pi) = \langle \otimes_1(\odot_1(\pi)), ..., \otimes_n(\odot_n(\pi))\rangle \in \text{B-Space}$$.

The objective is to select $k$ plans $\Psi = \{\pi_1, ..., \pi_k\}$, each of cost at most $c$, maximizing behavior cell coverage:

$$\text{BC}(\Xi,\Psi) = \sum_{b\in \text{B-Space}} I_b(\Psi)$$

where $I_b(\Psi)=1$ iff $\exists\pi\in\Psi$ such that $f(\pi)=b$, and $0$ otherwise.

For initialization, one often imposes a hard constraint that $f(\pi_i)\ne f(\pi_j)\ \forall i<j$, ensuring pairwise behavioral distinctness until $k$ cells or exhaustion of the grid.

2. Iterative Initialization Algorithms and Complexity

The primary initialization algorithm, Forbid-Behaviour-Iterative, operates by successively invoking a planning-as-Satisfiability (SAT/SMT) solver, blocking previously covered behavior cells at each iteration (Abdelwahed et al., 2024). Each call encodes (a) existence of a cost-bounded plan, (b) blocking clauses asserting $\neg(f(\pi)=b)$ for each $b$ previously covered.

Pseudocode structure:

1. Initialize $\Psi=\emptyset$, Covered=$\emptyset$.
2. Repeat until $|\Psi|=k$ or no uncovered cell remains: a. Build SAT/SMT formula encoding planning + all forbidden behaviors. b. If UNSAT, terminate. c. Extract solution, compute behavior, append to $\Psi$, add cell to Covered.
3. Return $\Psi$.

Worst-case complexity is $O(\prod_{i=1}^n |\text{vals}_i|)$ SMT calls (one per valid cell), each costing roughly a single cost-bounded query. Practically, one uses small $n$ (2–4), and coarse discretization. If $k$ exceeds reachable cells, a "fill-up" phase can use plan-forbid enumeration.

3. Feature Selection, Grid Resolution, and Domain Integration

Grid-based diversification is highly tunable. Increasing $n$ (features) or the granularity of discretization enlarges $\text{B-Space}$, directly trading off between computational effort and expressiveness of behavioral coverage. Common features include plan length (e.g., $\{optimal, suboptimal\}$), resource use (e.g., $\{0,1,\geq2\}$), and goal predicate orderings (e.g., all $3!$ permutations). Each feature's bins should be chosen to balance search tractability and semantic relevance; excessive fineness can render search infeasible.

Domain integration supports not only classical planning, but also over-subscription, numerical, and multi-agent domains. Example: in a multi-rover scenario, one could use sample acquisition order and number of rovers as axes, yielding $3\times2=6$ cells; initialization iteratively blocks previously generated order-count pairs to spread solutions over all combinations (Abdelwahed et al., 2024).

4. Behavioral Semantics and Explanation

Grid models enable expressive, composable semantics: each cell is interpretable as an $n$-vector of feature bins, allowing explanations for why plans differ by referencing distinct feature assignments. Unlike distance-based diversity criteria, which measure only syntactic dissimilarity, behavior-grid approaches can encode domain-relevant user requirements (e.g., resource allocations, temporal orderings, scheduling constraints).

Additionally, this model facilitates post-hoc analysis and visualization (e.g., frequency histograms over grid cells, coverage heatmaps), crucial for evaluating coverage quality and diagnosing diversity bottlenecks (Abdelwahed et al., 2024).

5. Case Studies and Empirical Evidence

Empirical validation demonstrates that grid-based initialization outperforms prior distance-based approaches in behavior cell coverage and total diversity (Abdelwahed et al., 2024). In rover domain studies, initializing over discrete order-count grids guaranteed all first $k$ plans occupied distinct cells, maximizing diversity with bounded cost.

The framework generalizes to multiple other domains:

• In numerical planning: custom feature bins can include resource budgets or continuous metric thresholds, discretized appropriately.
• Over-subscription planning: features can model selected priority goals or resource assignments, with initialization forced over their Cartesian product.

Robustness to parameter tuning is reported: small numbers of features and coarse bins yield fast enumeration and widespread coverage; increasing grid resolution proportionally raises computational effort.

6. Relationship to Other Diversified Planning Paradigms

Behaviour Planning initialization with grid-based coverage is distinct from, but complementary to:

• Monte Carlo tree search with diversity post-processing (Benke et al., 2023)
• Genetic algorithm pools with explicit diversity-compatibility objectives (Poddar et al., 24 Jun 2025)
• Occupancy-driven search in multi-robot coordination (Han et al., 2019)
• Domain randomization for planner curriculum construction (Khandelwal et al., 2024)
• Stochastic policy libraries optimized via occupancy-measure constraints (Ghasemi et al., 2020)
• LTL-guided semantic diversity in simulation-based planners (Abdelwahed et al., 20 Oct 2025)

The essential differentiator is the explicit mapping of plans to interpretable, combinatorial grid cells, directly aligning diversity with the end-user's behavioral desiderata.

7. Practical Guidance and Future Directions

For practitioners, the principal steps are:

1. Specifying a set of discrete, salient plan features.
2. Constructing grid bins and ensuring efficient plan-behavior extraction.
3. Implementing iterative solver invocation enforcing behavioral exclusion.
4. Optionally, entering secondary enumeration phases (plan-forbid, relaxed constraints) when cell exhaustion or $k$ oversaturation occurs.

Key open directions include scaling grid-based initialization to high-dimensional behavioral spaces, integrating with learning-based domain feature extractors, and expanding expressivity to continuous and hierarchical plan attributes.

In summary, diversified planning initialization using explicit $n$-dimensional grid models provides a tractable, interpretable, and broadly applicable basis for maximizing plan behavioral coverage in state-of-the-art automated reasoning frameworks (Abdelwahed et al., 2024).