Policy-Reachable Model Family

• Policy-reachable model families are sets of models, system states, or control strategies provably attainable under feasible policies defined by system constraints and initial conditions.
• They leverage structured reachability analysis, Lyapunov-based overapproximations, and dependency-preserving parameterizations to reduce computational complexity and certify safe behavior.
• These frameworks integrate techniques from decision-theoretic planning, hybrid systems verification, and control synthesis to enable robust abstraction and controller design.

A policy-reachable model family is an analytically constructed or automatically inferred set of models, system states, or controller strategies that are provably attainable or visited under feasible policies given system constraints, initial states, and domain structure. In contemporary decision-theoretic planning, hybrid systems verification, and control synthesis, the main objective is to compute and exploit compact representations of these families to reduce computational complexity, certify safe or optimal policy behavior, and inform abstraction algorithms. Recent research formalizes policy-reachable model families through structured reachability analysis, overapproximations using Lyapunov methods, dependency-preserving reachable sets, distributional invariant certificates, and compositional strategies for families of Markov decision processes (MDPs).

1. Structured Reachability and Graph-Based Pruning

Structured reachability analysis algorithms, such as the REACHABLEK family (Boutilier et al., 2013), operate on compact representations of MDPs, notably dynamic Bayes nets (DBNs) with conditional action effects and correlated dependencies. These algorithms systematically construct alternating layers—action levels (composed of Conditional Action Effect (CAE) nodes that correspond to branches in DBN conditional probability trees) and propositional levels (variable values with mutual exclusion constraints)—to propagate which combinations of state variables can be reached given known initial conditions. Parameterization by a complexity parameter $K$ dictates the order of exclusion constraints (binary, ternary, ..., k-ary).

A key technical contribution is the use of k-ary mutual exclusion testing: only tuples of variable assignments supported by at least one consistent path through the action-effect graph are retained as reachable. This is particularly effective in pruning infeasible state combinations arising from resource limitations or correlated effects (e.g., in manufacturing domains with conditional dependencies). As a result, unreachable portions of the state space are eliminated, enabling abstraction algorithms to operate over a drastically reduced policy-reachable family, which enhances tractability and policy quality.

2. Lyapunov-Based Overapproximations and Policy Iteration

For piecewise affine systems, overapproximating the set of reachable states hinges on the existence of piecewise quadratic Lyapunov functions (Adjé, 2015). Each cell of the state space partition has an associated quadratic form used to construct an invariant sublevel set $S$ such that

$$x_0 \in S, \qquad x_k \in S \cap X^i \implies f^i(x_k) \in S,$$

for every affine regime $i$. Policy iteration refines these template bounds through fixed-point computation over the template domain, utilizing dualized semidefinite programs for tractability.

This Lyapunov-LMI-policy iteration framework generalizes robust control invariant approaches and enables precise bounding of the system’s policy-reachable set for verification and synthesis. Such analytically obtained invariants define the policy-reachable model family for certified safe operation under any admissible control, forming the basis for controller synthesis, static analysis, and sound overapproximation techniques, even for high-dimensional or nonlinear switching systems.

3. Dependency-Preserving Reachability and Parameterization

A dependency-preserving approach (Kochdumper et al., 2019) represents the reachable set and its subsets parametrically (commonly via polynomial zonotopes), allowing efficient extraction of policy-reachable subsets corresponding to arbitrary initial condition choices or policy parameters: $$\mathcal{PZ} = \{f_{G,E}(\alpha) \oplus \sum_j \beta_j\, G_I(:,j) \mid \alpha_k,\, \beta_j \in [-1,1]\}$$ By evaluating the analytical mapping $f_{G,E}(\alpha)$, one instantly computes the subset of reachable states traceable to any fixed $\alpha$ without re-executing the full reachability computation.

This parameter-indexed mapping directly generalizes to policy reachability: optimizing over $\alpha$ identifies initial sets (or policy configurations) that maximize some safety or reachability measure $J(\cdot)$, efficiently yielding model families suited for falsification, safe maneuver synthesis, or real-time control. The method’s soundness, computational efficiency ($O(n^2)$ extraction), and extensibility underscore its practical value, especially when intersecting with learning-based control architectures.

4. Distributional Certificates and Policy Synthesis

Analysis of MDPs as distribution transformers (Akshay et al., 7 May 2024) yields a formal method for synthesizing policies together with explicit certificates guaranteeing distributional reach-avoid properties. The certificate comprises a convex invariant $I$ (e.g., a polyhedral set of safe distributions over states) and a ranking function $R$ decreasing with each transition: $$\forall\, \mu \in I \setminus T: \qquad R(\mu) \geq R(M^{\pi}(\mu)) + 1,\qquad R(\mu) \geq 0.$$ This formalism characterizes the policy-reachable family by all policies admitting such certificates: every evolution under a synthesized controller remains within $I$ and eventually reaches a target distribution $T$.

Automated synthesis uses SMT solvers and quantifier elimination over template parameters (affine policies, invariant sets, ranking functions) to generate these families efficiently. Applications include robot swarms, chemical networks, and pharmacokinetic verification, signifying that distribution-level reachability generalizes state-wise guarantees and enhances robustness for safety-critical domains.

5. Recursive Abstractions and Policy Trees

The policy tree abstraction (Andriushchenko et al., 17 Jul 2024) provides a hierarchical mapping from a large family of MDPs (each indexed by a system configuration or parameter vector) to a small set of robust memoryless policies and unsatisfiable outcomes. The recursive construction alternates between game-based abstraction and efficient splitting of the MDP family, yielding a tree structure: $\mathcal{T} = (V,\ l,\ r,\ F,\ L)$ where leaves are labeled by robust policies winning for all MDPs in a subfamily, and inner nodes split on distinguishing features.

Empirical evaluations show dramatic scalability improvements relative to naive enumeration; millions of MDPs can be covered by $\ll 1\%$ distinct policies via policy trees. This partitioned structure exposes the underlying policy-reachable model family: subfamilies where a single robust controller suffices (or none exist), and the refinement process operationalizes compositional synthesis in uncertain or configurable systems.

6. Safety-Constrained System Identification and Conformance

Reachset-conformant identification frameworks (Lützow et al., 16 Jul 2024) ensure that a model's reachable set, computed by set-based overapproximation methods (GO models, linearization, zonotopic uncertainty), contains all real system output measurements: $$\forall\, m,k:\quad \text{Reach}_k(S_T, m) \subseteq \text{Reach}_k(S_M, m)$$ Uncertainties are estimated via optimization (LP for white-box, NL/LP integration for gray-box, genetic programming for black-box), adapting the approach to any prior knowledge regime.

This guarantees that any control policy synthesized using $S_M$ is formally safe for the true system $S_T$ provided outputs remain within the precomputed reachable set. Hence, reachset conformance underlies the policy-reachable family for verification and certification in cyber-physical and safety-critical domains.

7. Dynamics-Conditioned Policy Reachability and Transfer

Inverse Constraint Learning (ICL) (Qadri et al., 26 Jan 2025) reveals that constraints inferred from safe demonstrations correspond to the backward reachable tube (BRT) for a given dynamics model, not simply the failure set. The BRT is strictly dynamics-conditioned:

$$S^{safe} = \{s \mid \exists\, \pi:\ \forall\, d,\, t,\ \xi^{\pi,d}(t) \notin \mathcal{F}\},\quad S^{unsafe} = \mathbb{R}^n \setminus S^{safe} = \text{BRT}(\mathcal{F})$$

As such, transferable safe policies must account for underlying dynamic capabilities (e.g., agility, control authority). ICL frameworks thus define policy-reachable families as dynamics-dependent, with direct implications for sample efficiency, cross-domain transfer, and robustness to structural variations.

8. Scalable Reachability via MPC–Deep Learning Integration

Recent hybrid approaches (Feng et al., 4 May 2025) interleave model predictive control (MPC) optimization and deep neural approximation to compute accurate safety value functions (the BRT) in high-dimensional systems. The learning objective combines supervised MPC-generated approximations and PDE residual loss from HJ equations: $$\mathcal{L}_{combined} = \mathcal{L}_{PDE} + \lambda\, \mathcal{L}_{data}$$ This coupling improves accuracy, stability, and safe set recovery (as measured by MSE and verified set volume) versus either method alone. The resulting controllers, derived from the certified network value function, constitute the policy-reachable family for safe operation in otherwise intractable state spaces.

---

Policy-reachable model families represent a unifying concept across decision processes, hybrid systems, and control synthesis, denoting the set of models, policies, or reachable states that can be certified as attainable, safe, or optimal under feasible strategies. Advances in graph-theoretic pruning, Lyapunov invariants, parameterized reachability, certificate synthesis, abstraction trees, and data-driven identification continue to extend the applicability and efficiency of these frameworks for real-world, safety-critical domains.