Multi-Resolution Approximate Bisimulations

• Multi-resolution approximate bisimulations are defined by a state-dependent resolution function that adapts error tolerances to local system dynamics.
• They generalize both uniform and variable-resolution methods, enabling the creation of compact finite abstractions where traditional methods fall short.
• The synthesis framework employs a scalable CEGIS loop with SMT-based verification, achieving up to 50% smaller abstractions with precise local error guarantees.

Multi-resolution approximate bisimulations generalize classical approximate bisimulation by enabling the specification of state-dependent, locally varying error bounds for behavioral equivalence between continuous-state dynamical systems and their finite abstractions. This approach subsumes both uniform-resolution and variable-resolution abstractions, permits a broader range of systems to be abstracted—including many outside the scope of classical (uniform) ε-approximate bisimulation—and enables sound, efficient, and compact abstraction synthesis tailored to nuanced verification and control requirements (Coppola et al., 22 Sep 2025).

1. Definition and Generalization

Multi-resolution approximate bisimulations replace the traditional uniform error bound with a resolution function ε(x, x̂) that specifies, for each concrete/abstract state pair (x, x̂), the maximum tolerated deviation: $$\overline{R} = \{(x, \hat{x}) \mid d(x, \hat{x}) \leq \varepsilon(x, \hat{x})\}$$ where d(x, x̂) is the metric on state space. This strictly generalizes the classical ε-approximate bisimulation (where ε is uniform), allowing non-uniform, locally adaptive accuracy and coarse-to-fine abstraction granularity across the reachable state space.

This extension is nontrivial: there exist continuous-state systems with unbounded trajectories or locally varying sensitivity that admit multi-resolution bisimulations for appropriately chosen ε(x, x̂), but for which no uniform ε > 0 yields an ε-approximate bisimulation.

2. Resolution Functions and Applicability

A resolution function ε(x, x̂) encodes designer-specified local accuracy—increasing the error tolerance where precise abstraction is unnecessary or the system is less sensitive, and decreasing it near safety-critical or sensitive regions. Example: $$\varepsilon(x, \hat{x}) = u \cdot \min(\|x\|, \|\hat{x}\|) + v$$ for constants u, v ≥ 0. This permits, for example, finer abstraction near the origin (or another region of interest) and coarser abstraction in the tail.

This framework subsumes both variable-resolution (nonconstant ε) and uniform-resolution (constant ε) as special cases—significantly broadening applicability. The main existence result is that for all incrementally uniformly bounded (δ-UB) systems—where trajectories’ pairwise divergence is uniformly bounded by a function α of their initial separation—multi-resolution approximate bisimilar abstractions exist. This class properly contains all incrementally globally asymptotically stable (δ-GAS) systems and extends to non-differentiable dynamics or systems lacking global Lyapunov certificates.

3. Synthesis Framework: The MRASP and CEGIS Loop

The Multi-resolution Abstraction Synthesis Problem (MRASP) is formulated as follows: find a finite-state abstraction and a relation R ⊆ X × X̂ such that

• (a) Coverage: for each initial state in X₀, some abstract state relates to it via R,
• (b) Transition consistency: if (x, x̂) ∈ R and x' = f(x), then there must exist x̂' = f̂(x̂) such that (x', x̂') ∈ R,
• (c) Resolution: all related pairs (x, x̂) satisfy d(x, x̂) ≤ ε(x, x̂).

Synthesis proceeds by a highly scalable Counterexample-Guided Inductive Synthesis (CEGIS) loop:

• Learner: Initializes and refines a mesh/partition of the state space (e.g., via k-means clustering with weights inversely proportional to ε(x̂)), selects anchor points (abstract states), and parameterizes relation templates (e.g., cell radii).
• Verifier: Uses SMT-based reasoning to check the coverage, transition, and resolution properties. Violations yield counterexamples, which trigger mesh or relation refinement.
• Parallelization: Exploits the abstraction's directed acyclic structure, decomposing verification into independent subproblems to scale to high state dimensions.

Mesh refinement and relation parameterization are iteratively improved by localizing splits to parts of the state space that exhibit violation or higher sensitivity, ensuring that the abstraction meets local resolution requirements without unnecessary global refinement.

4. Theoretical Guarantees and Complexity

Existence is established for all δ-UB systems (i.e., systems where

$d(x_k, x_k') \leq \alpha(d(x_0, x_0'))$

for a class K function α and all k ∈ ℕ). As a limit case, δ-GAS systems are trivial.

When specialized to the uniform-resolution case, the approach is equivalent to classical ε-approximate bisimulation. Moreover, there exist systems (e.g., with fast divergence away from specific subsets) that admit no ε-approximate bisimulation but do admit well-structured multi-resolution bisimulations for spatially tuned ε(x, x̂).

5. Experimental Results and Compactness

Extensive benchmarks demonstrate that the new synthesis algorithm yields abstractions up to 50% smaller than uniform gridding based on Lyapunov or δ-GAS assumptions, while enforcing tighter, sharply location-dependent error guarantees. On linear and nonlinear (including non-δ-GAS and non-differentiable) systems, the algorithm:

• Accurately synthesizes compact abstractions even in settings not amenable to classical Lyapunov-based techniques.
• Adapts the mesh granularity, yielding finer abstractions in high-sensitivity regions (e.g., near the origin) and coarser elsewhere.
• Remains computationally tractable as mesh refinement is highly localized and inference/verification subproblems are parallelized.

Concrete experimental setups verify that the achieved error bounds generally improve over the target constraints at critical state regions, with computation time dominated by the SMT-based verification stage but effectively manageable due to mesh/relation decomposition.

6. Mathematical Formulation

The MRASP is formulated as: $\text{minimize}~ |\hat{X}|~ \text{subject to:}$

$$\forall x \in X_0, ~\exists \hat{x} \in \hat{X} : (x, \hat{x}) \in R,$$

$$\forall (x, \hat{x}) \in R, ~ (f(x), f̂(\hat{x})) \in R,$$

$$R \subseteq \{ (x, \hat{x}) : d(x, \hat{x}) \leq \varepsilon(x, \hat{x}) \}$$

where R is the bisimulation relation and ε(x, x̂) is the prescribed, state-dependent tolerance.

7. Implications, Applications, and Future Directions

Multi-resolution approximate bisimulations significantly advance the formal abstraction and verification of continuous-state and hybrid systems. Their key implications include:

• Sound abstraction: Guarantees that the error in property verification is never worse than the specified per-location ε(x, x̂), in contrast to uniform error bounds that can be overly conservative.
• Model compactness: Enables construction of much smaller finite abstractions without sacrificing correctness, particularly relevant for high-dimensional or highly heterogeneous systems.
• Applicability: Extends to any δ-UB system, providing a unified framework for both linear and nonlinear, smooth and nonsmooth, contractive and non-contractive dynamics.
• Scalability: Mesh/local adaptation, decomposition, and parallelization make practical application possible for moderate-dimensional systems.
• Formal methods integration: By bridging continuous dynamics and discrete formal verification, multi-resolution methods support advanced controller synthesis and safety assurance in domains including robotics, autonomy, and networked control.
• Future investigation: Areas for further research include improved solver integration, tighter theoretical bounds for abstraction size versus error, extension to input–output systems, and direct use of the approach in compositional and hierarchical control synthesis frameworks.

Overall, multi-resolution approximate bisimulations enable abstraction-driven verification and synthesis in a principled, locally tunable, and computationally tractable fashion, providing a robust mathematical foundation and practical tools for modern symbolic control and formal verification workflows (Coppola et al., 22 Sep 2025).