Equivalence-Based Planning Techniques

• Equivalence-based planning is a family of techniques that abstract states, actions, or policies by grouping functionally indistinguishable elements to reduce complexity.
• Methodologies such as CEGAR, aggregation, and policy-space compression improve scalability, regret reduction, and computational efficiency.
• The approach enables robust solutions in uncertain and adversarial environments while offering soundness, completeness, and performance guarantees.

Equivalence-based planning encompasses a family of algorithmic and representational techniques in automated planning, verification, and reinforcement learning that identify, leverage, or construct equivalence relations over states, actions, policies, or models. The central objective is to reduce the effective size and complexity of the planning space by abstracting or aggregating those elements that are functionally indistinguishable with respect to planning objectives (such as reachability, strategy optimality, or value functions). Modern equivalence-based methods have enabled advances in the scalability, robustness, and interpretability of planning algorithms under uncertainty and adversariality.

1. Formal Definitions of Equivalence in Planning

Equivalence in planning is formalized by defining relations that group states, actions, transitions, or policies exhibiting indistinguishable behavior for the purposes of strategy synthesis or analysis. The specific definitions are domain-dependent:

• State–Action Equivalence (RL, MDPs): State–action pairs are considered equivalent if their transition probability distributions and expected immediate rewards (possibly after sorting by probability profiles) are indistinguishable under an $L_1$ metric (Asadi et al., 2019). This grouping is realized via profile mappings $\sigma_{s,a}$ and equivalence relations:

$$(s,a) \sim (s',a') \iff \|p(\sigma_{s,a}(\cdot)|s,a) - p(\sigma_{s',a'}(\cdot)|s',a')\|_1 = 0 \quad \text{and} \quad r(s,a) = r(s',a')$$

• Functional Domain Model Equivalence: Two planning domains $D_1$ and $D_2$ are functionally equivalent if there exists a bijective mapping of predicates such that their reach sets over all objects coincide:

$$\Gamma(F_p(D_1), \text{Obj}) = \Gamma(D_2, \text{Obj}) \quad \forall\, \text{Obj}$$

(where $\Gamma$ is the set of all induced state transitions) (Shrinah et al., 2021).

• Value Equivalence in MBRL: Models $m$ and $m^*$ are value equivalent with respect to policy set $\Pi$ and function set $V$ if their BeLLMan operators produce matching updates:

$$T_\pi[v] = \tilde T_\pi[v] \quad \forall\, \pi \in \Pi, v \in V$$

(Grimm et al., 2020, Grimm et al., 2021, Arumugam et al., 2022).

These definitions enable the construction (or detection) of aggregate structures that preserve essential planning properties while discarding redundant distinctions, a foundation for abstraction and compression.

2. Methodologies for Equivalence-Based Planning

The implementation of equivalence-based planning involves diverse methodologies, depending on the formalism and task:

Abstraction and Refinement

The Counterexample-Guided Abstraction Refinement (CEGAR) paradigm generates coarse abstractions by merging states, refining them iteratively post-counterexample analysis until spurious distinctions are eliminated (Chatterjee et al., 2012). Given an abstract game structure $G^a$, abstraction operators such as $\text{Focus}$ and $\text{ValueFocus}$ adaptively refine abstractions only where strategy synthesis or counterexample analysis fails, guided by:

• $r^a(v^a) = \min\{r(v): v \in [v^a]\}$
• $$\text{Focus}(v^a, f_2, val) = \{v \in [v^a]: \exists w \text{ s.t. } (v, w) \in E \land w \in [f_2(v^a)]\}$$
• $$\text{ValueFocus}(v^a) = \{v \in [v^a]: r(v) = r^a(v^a)\}$$

Aggregation via Equivalence-Aware Confidence Sets

In RL, equivalence-aware confidence sets pool observations from all state–action pairs in an equivalence class, leading to tighter statistical bounds and improved exploration–exploitation trade-offs (Asadi et al., 2019). The clustering of equivalence classes can be performed via ApproxEquivalence, which merges clusters as long as their empirical profiles are indistinguishable within confidence intervals.

Policy-Space Equivalence and Compression

In FOND planning, multiple equivalence relations on policies are used to prune redundant explorations: identity equivalence, lanes equivalence (based on “escape” sets), and domain–frontier equivalence (set of mapped plus frontier states) (Messa et al., 28 Mar 2024). Structural state-space symmetries are computed using group-theoretic techniques, canonically representing states in equivalence classes. Policy compression reduces explicit policies over complete states into minimal coverings by partial states using integer programming.

Model Learning and Value Equivalence

Value equivalence can be enforced via loss functions penalizing discrepancies in BeLLMan updates between learned models and the environment (Grimm et al., 2020, Grimm et al., 2021, Arumugam et al., 2022). Proper VE (PVE) provides sufficiency for planning even when models ignore irrelevant environmental details. Rate-distortion theory quantifies information–fidelity trade-offs for approximate value equivalence (Arumugam et al., 2022).

Symbolic Structure and View-Based Planning

In complex robotic domains, splitting the attribute space into shared elementary attributes and decomposing the planning problem into sequentially refined “views” supports both generalization and innovative solution finding while managing intractable combinatorial growth (Hasler et al., 6 May 2024).

3. Theoretical Properties and Guarantees

Equivalence-based planning frameworks offer several critical theoretical guarantees:

• Soundness: If a plan exists for an abstracted or equivalently grouped problem, a corresponding plan exists for the concrete problem; abstraction techniques based on conservative operators ensure no spurious plan is synthesized (Chatterjee et al., 2012).
• Completeness: In model/domain equivalence, soundness and completeness are established for bijective predicate mappings in domains with identical operators and effects (Shrinah et al., 2021).
• Regret Reduction: Exploiting equivalence in RL reduces regret bounds by factors related to the class reduction achieved, e.g., $\sqrt{SA/C}$ improvement with equivalence-aware C-UCRL (Asadi et al., 2019).
• Asymptotic Optimality: Lifting optimal planning to state–cost spaces and employing iterative feasible planners maintains convergence to optimum as iteration number or computation effort increases (Hauser et al., 2015).
• Compression Guarantees: Integer-programmable solution compressors provide minimal partial-state policy representations, maintaining unambiguous execution semantics (Messa et al., 28 Mar 2024).
• Approximate Loss Quantification: The rate-distortion framework precisely connects BeLLMan error (distortion) with model information content, permitting rigorous control over approximation–performance trade-offs (Arumugam et al., 2022).

4. Practical Applications and Impact

Equivalence-based methods have substantially improved the tractability and coverage of planning algorithms across multiple domains:

• Adversarial/Uncertain Environments: CEGAR enables scalable synthesis of robust strategies in stochastic perfect-information games and adversarial settings (Chatterjee et al., 2012).
• Symbolic and SAT-Based Planning: SAS+ formulations and equivalence isomorphism provide more compact and efficient encodings for SAT-based planning (Huang et al., 2014).
• Model-Based RL: VE and PVE principles underlie the empirical success of algorithms such as MuZero, Value Iteration Networks, Predictron, and Value Prediction Networks (Grimm et al., 2020, Grimm et al., 2021).
• Planning Domain Validation: D-VAL offers automatic validation of functional equivalence between planning domain models, facilitating model learning and reformulation (Shrinah et al., 2021).
• FOND Planning: Policy-space equivalence and solution compressors enable AND* and related planners to achieve competitive compactness and coverage with state-of-the-art alternatives (Messa et al., 28 Mar 2024).
• Robotic Planning: Decomposition into elementary attributes supplemented by view-based sequential planning supports generalization and innovative solution synthesis in manipulation and household robotics (Hasler et al., 6 May 2024).

5. Comparison of Methodological Approaches

A structured comparison of key equivalence-based planning methodologies:

Technique	Principle	Primary Benefit
CEGAR	Guided abstraction	Scalability, focused refinement
Equiv.-aware RL clusters	Aggregation	Tighter bounds, lower regret
Policy-space equivalence	Pruning, symmetry	Solution compactness, coverage
Value Equivalence (VE)	BeLLMan preserve	Computation/memory efficiency
View-based Planning	Sequential views	Generalization, flexible solutions

CEGAR and equivalence-aware RL approaches focus on abstraction and aggregation; policy-space equivalence uses group-theoretic symmetry and compression algorithms; value equivalence actively tailors learned models; view-based approaches adapt attribute representation for high-level and robust planning.

6. Limitations and Open Issues

Despite advances, equivalence-based planning faces several unresolved challenges:

• Worst-case Complexity: Iterative abstraction refinement may devolve to concrete level in highly entangled problems (Chatterjee et al., 2012).
• Expressivity Constraints: D-VAL and some mapping techniques are limited to domains expressible in restricted subsets of PDDL/STRIPS (Shrinah et al., 2021).
• Approximation: Value equivalence guarantees hinge on function/policy representational richness; insufficient coverage may lead to suboptimal policies (Grimm et al., 2020, Grimm et al., 2021).
• Interdependency of Attributes: View-based decomposition presumes predicate group independence, which may be violated in some tasks, causing incomplete or suboptimal planning (Hasler et al., 6 May 2024).
• Non-reversibility of Early Decisions: In sequential view-based planning, early subproblem commitments cannot be easily revised, potentially precluding globally optimal solutions (Hasler et al., 6 May 2024).

7. Future Directions

Research into equivalence-based planning continues to evolve along:

• Stronger Guarantees for Approximate Equivalence: Further exploration of rate-distortion theory and its impacts on practical planning fidelity (Arumugam et al., 2022).
• Automated Generation of Symmetry Groups: Efficient computation of structural state-space symmetries for use in planners across diverse domains (Messa et al., 28 Mar 2024).
• Extension to Rich Planning Formulations: Increasing the expressivity of equivalence validation systems to handle temporally extended effects, negative conditions, and dynamic object sets (Shrinah et al., 2021).
• Adaptive and Reversible View-based Planning: Development of replanning strategies and attribute interdependency analysis to overcome current view-based limitations (Hasler et al., 6 May 2024).
• Integration with Learning Systems: Synergy of equivalence-abstraction techniques with deep RL, model compression, and representation learning to improve both learning and planning efficiency (Grimm et al., 2020, Grimm et al., 2021).

Equivalence-based planning remains a foundational concept in bridging theoretical planning principles with scalable, interpretable, and adaptable algorithms for AI, reinforcement learning, and robotics.