An Information Theory of Finite Abstractions and their Fundamental Scalability Limits

TL;DR

This work establishes an information-theoretic framework for finite abstractions of dynamical systems by modeling abstractions as encoder–decoder pairs with a rate–distortion tradeoff. It derives fundamental lower bounds on the achievable abstraction distortion and minimum abstraction size, showing how trajectory entropy and system dynamics govern scalability. The theory is demonstrated on illustrative examples (e.g., the doubling map and a 3D nonlinear system), revealing both the tightness of the bounds and how minimal abstractions can be constructed via optimal coverings in high-dimensional spaces. Overall, the results provide quantitative limits on abstraction scalability and a path toward principled design of minimal, provably accurate abstractions for complex dynamical systems.

Abstract

Finite abstractions are discrete approximations of dynamical systems, such that the set of abstraction trajectories contains all system trajectories. There is a consensus that abstractions suffer from the curse of dimensionality: for the same ``accuracy" (how closely the abstraction represents the system), the abstraction size scales poorly with system dimensions. And yet, after decades of research on abstractions, there are no formal results on their accuracy-size tradeoff. In this work, we derive a statistical, quantitative theory of abstractions' accuracy-size tradeoff and uncover fundamental limits on their scalability, through rate-distortion theory -- the information theory of lossy compression. Abstractions are viewed as encoder-decoder pairs, encoding trajectories of dynamical systems. Rate measures abstraction size, while distortion describes accuracy, defined as the spatial average deviation between abstract trajectories and system ones. We obtain a fundamental lower bound on the minimum achievable abstraction distortion, given the system dynamics and the abstraction size; and vice-versa a lower bound on the minimum size, for given distortion. The bound depends on the complexity of the dynamics, through trajectory entropy. We demonstrate its tightness on some dynamical systems. Finally, we showcase how this new theory enables constructing minimal abstractions, optimizing the size-accuracy tradeoff, through an example on a chaotic system.

Figures (6)

• Figure 1: The typical source coding setting.
• Figure 2: Abstractions as a source coding scheme. From the left: 1) a sample trajectory $\xi$ of system $S$ with its initial state $\xi_0$ highlighted in blue; 2) the state-space partition $\mathcal{Y}$, and the corresponding abstract initial condition in cyan; 3) the set of abstract trajectories $\Omega_A$, in red; 4) the specific trajectory $\xi'_*\in\Omega_A$ that deviates the most from the true system trajectory $\xi$.
• Figure 3: Consider the dynamical system $x^+ = x^2$ with state-space $\mathcal{X}=[0,1]$. The parabola depicts the set of trajectories ${\mathcal{B}}_2^S$, embedded in $[0,1]^2$. Consider an abstraction $A$ with associated partition sets $Y_i = [0.2(i-1), \,0.2i)$ for $i=1,\dots,4$ and $Y_5=[0.8,1]$. The abstraction transitions, thus, are $\underset{A}{\rightarrow}=\{(Y_1,Y_1),(Y_2,Y_1), (Y_3,Y_1), (Y_3,Y_2), (Y_4,Y_2), (Y_4,Y_3), (Y_4,Y_4), (Y_5, Y_4), (Y_5,Y_5)\}$. The colored rectangles represent the abstraction outputs $\Omega_A$, depending on the initial condition $\xi_0$. For example, if $\xi_0\in Y_4$, then the abstraction output $\Omega_A$ is the green rectangle $\{(x_0,x_1):\, x_0\in Y_4, x_1 \in Y_3\cup Y_4\cup Y_5\}$. Observe how the abstraction's outputs define a $2$-dimensional cover of the curve ${\mathcal{B}}_2^S$. The dots represent the Chebyshev centers for each different abstraction output, and the circles are the corresponding Chebyshev balls. E.g, when $\xi_0\in Y_4$, we have $x_c(\Omega_A) = (.5\, \, .5)$ and $r_c(\Omega_A)=0.3\sqrt2$. The abstraction's expected distortion is lower bounded as per \ref{['eq:prop abstraction vs encoder']}.
• Figure 4: Section \ref{['ssec:chaotic map']}: Comparison between $D_{abs}(R)$ and the fundamental lower bound from Theorem \ref{['thm:slb_abstractions']}, for $l=5$.
• Figure 5: Section \ref{['ssec:chaotic map']}: Optimal cover of ${\mathcal{B}}_l^S, \ l=3$ with $k=2$ (left) and corresponding abstraction trajectories (right, blue boxes). On the right, the lines inside the boxes represent ${\mathcal{B}}_3^S$, with their Chebyshev centers marked in red. The maximal distance between any point in the ${\mathcal{B}}_3^S$ and its corresponding blue box is obtained at one of the edges intersecting with the trajectory.

Theorems & Definitions (32)

• Example II.1: Entropy of trajectories of the doubling map
• Theorem II.1: Generalized Shannon lower bound riegler2023lossy
• proof : Proof Sketch
• Definition II.2: Transition system
• Definition III.1: Measurable Partition
• Definition III.2: Abstraction
• Theorem III.3: Behavioral inclusion tabuada2009verification
• Remark 1: Extension to (approximate) simulations
• Remark 2: Initial-condition distribution
• Remark 3: Statistics of abstractions' accuracy and size