An Information Theory of Finite Abstractions and their Fundamental Scalability Limits

arXiv:2512.03977 · eess.SY, cs.IT, cs.SY, math.DS, math.IT, math.OC · Submitted 2025-12-03 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: Today's paper: "An Information Theory of Finite Abstractions and their Fundamental Scalability Limits".

Dev: The gist The work derives a statistical, quantitative theory of abstractions’ size-accuracy tradeoff and uncovers fundamental limits on their scalability through rate-distortion theory.

Rosa: First, who's behind it and why it matters.

Title and authors: Rosa: We’ve talked about the basics of this work; now let's look at what the title "An Information Theory of Finite Abstractions and their Fundamental Scalability Limits" actually means for us in practice.

Dev: It suggests they are moving beyond just describing *how* to make an abstraction, and instead providing a statistical theory that sets hard limits on what we can ever achieve with those approximations.

Rosa: So, instead of just tweaking parameters to get a better model, this paper is deriving fundamental bounds on the minimum distortion we have to accept for any given system dynamics and abstraction size.

Taro: It connects directly to lossy compression information theory, which is interesting because it brings a very concrete mathematical tool into the abstract world of system modeling.

Dev: The key insight here seems to be defining rate as abstraction size and distortion as accuracy, measured by the spatial average deviation between the abstract trajectories and the real system ones.

Rosa: And they use this setup to derive that fundamental lower bound on average distortion, Dabs(R), based on things like generalized entropy of the system dynamics.

The paper's summary: Taro: When we look at the summary, it points out that they establish two main bounds: one showing the minimum achievable abstraction distortion given the system dynamics and size, and a reverse bound showing you what minimum size you need for a target distortion.

Dev: That means they’re not just saying abstractions are hard to scale; they’re giving us a formula that tells us precisely how much harder it is to get better accuracy if we keep the abstraction size fixed.

Rosa: It also shows that this setup works by solving a source coding problem where the message space is defined by the system trajectories, which lets them quantify everything in terms of rate and distortion.

Taro: They introduce a specific rate-distortion quantity called Dabs(R), which ties the abstraction size, denoted as logY, to the expected distortion d(x, xˆ) under certain conditions involving an abstraction A.

Dev: The paper details how this quantity is found by minimizing that expectation subject to the constraint on the encoder cardinality, logY less than or equal to R.

Rosa: And they show concrete examples where this works, like analyzing a chaotic system, and they even mention that for exponentially stable systems, the abstraction distortion can actually shrink to zero if you allow enough complexity in your partition size.

The paper's improvements: Dev: Now shifting to the improvements they suggest—they are proposing a framework where we actively try to construct minimal abstractions by solving the problem of encoding trajectories through coverings in a high-dimensional ambient space.

Taro: That sounds like an active method, not just a theoretical derivation; it suggests using techniques like Information Bottleneck Method to actually build these optimal abstractions.

Rosa: The paper shows how this approach lets you determine the optimal size-accuracy tradeoff by solving that rate-distortion quantity where you minimize the expected distortion given a rate constraint R.

Dev: So, if you know your desired accuracy, say a certain maximum error level, this theory tells you exactly what size abstraction is minimally required to achieve it without wasting information.

Taro: It’s about providing that general procedure for constructing optimal abstractions in terms of the size-accuracy tradeoff for any given system dynamics x+ = f(x).

Conclusion: Rosa: So, wrapping this up, this paper by Giannis Delimpaltadakis and Gabriel Gleizer gives us a statistical way to quantify how accurate an abstraction is based on its size relative to the underlying system complexity.

Dev: The main implication is that we now have fundamental limits on the scalability of abstractions for any given dynamics, which depends directly on how complex the dynamics are through generalized entropy h(ξ).

Taro: For someone just listening, it means that if you want a specific level of accuracy for your system model, you can’t just make your abstraction arbitrarily big; there's a hard mathematical ceiling imposed by the dynamics.

Rosa: It gives us the tools to optimize that size-accuracy tradeoff precisely, which is crucial when we’re building these models for real robotic applications outside of a perfect lab setting.

Dev: And they also show that for systems that are exponentially stable, you can theoretically get perfect accuracy if you just let the abstraction size grow sufficiently large.

Taro: It's a powerful tool because it links system complexity—the dynamics—directly to the information theory of how we represent them, which is a new way to look at modeling uncertainty.

Robust and Intelligent Autonomous Systems lab, AI4I Institute, Turin, Italy · Delft Center for Systems and Control, Mechanical Engineering, Delft University of Technology

eess.SY, cs.IT, cs.SY, math.DS, math.IT, math.OC

Submitted: 2025-12-03

Updated: 2026-10-08

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 77/100

The gist: The gist The work derives a statistical, quantitative theory of abstractions’ size-accuracy tradeoff and uncovers fundamental limits on their scalability through rate-distortion theory.

Key concepts

Abstractions as Encoder-Decoder Pairs: Abstractions are modeled as pairs of encoders and decoders that represent trajectories of dynamical systems in a higher-dimensional space. The encoder determines the abstraction size (rate), and the decoder reconstructs an approximation. This framework allows for the statistical analysis of how well these representations capture the underlying system dynamics.
Rate and Distortion: Rate quantifies abstraction size, defined by log|Y|, where Y is a finite partition used in encoding. Distortion measures accuracy, defined as the spatial average deviation between abstract trajectories and the original system trajectories. The theory connects these two quantities through rate-distortion principles.
Fundamental Lower Bounds: The theory derives fundamental lower bounds on minimum achievable distortion (Dabs(R)) given the system's complexity, measured by generalized entropy h(ξ), and the chosen abstraction size. Conversely, it provides a lower bound on the minimum required abstraction size (Rabs(D)) for a desired level of accuracy. These bounds define the inherent limits of abstraction construction.
Curse of Dimensionality: The paper shows that as the system's dimension (n) increases, achieving a fixed distortion requires an exponentially increasing partition size ($e^R$). This demonstrates how the curse of dimensionality fundamentally limits the scalability of abstractions for any given system dynamics.

Terminology

Abstraction — 234× in this paper · explained in 1 episode
Accuracy — 65× in this paper · explained in 1 episode
Verification — 20× in this paper · explained in 2 episodes
Hybrid System — 9× in this paper · explained in 2 episodes
Curse of Dimensionality — 7× in this paper · explained in 6 episodes
Generalization — 2× in this paper · explained in 13 episodes
Abstractions as Encoder-Decoder Pairs — 1× in this paper · explained in 1 episode
Adam — 1× in this paper · explained in 2 episodes
Core — 1× in this paper · explained in 2 episodes
Decode — 1× in this paper · explained in 2 episodes
Formal Verification — 1× in this paper · explained in 2 episodes
Information-Theoretic Framework — 1× in this paper · explained in 2 episodes
Lyapunov function — 1× in this paper · explained in 2 episodes
Quantization — 1× in this paper · explained in 12 episodes

Summary

The gist The work derives a statistical, quantitative theory of abstractions’ size-accuracy tradeoff and uncovers fundamental limits on their scalability through rate-distortion theory.

How it works

Abstractions are viewed as encoderdecoder pairs, encoding trajectories of dynamical systems

The core idea is to view abstractions as encoder-decoder pairs that encode trajectories of dynamical systems in a higher-dimensional ambient space. Rate measures abstraction size, while distortion describes accuracy, defined as the spatial average deviation between abstract trajectories and system ones. This allows for the derivation of fundamental lower bounds on minimum achievable abstraction distortion given system dynamics and abstraction size, and vice-versa.

Key Concepts in the Theory

The theory connects abstractions to rate-distortion theory—the information theory of lossy compression.
Rate is determined by the encoder’s size, defined as logY, where Y is the finite partition.
Distortion is defined as d(x, xˆ) =∥x − xˆ∥2 for Euclidean distortion.
The fundamental lower bound on average distortion Dabs(R) is derived by considering a source coding problem where the message space is the system trajectories B Sl.

Fundamental Limits and Scalability

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 paper presents Theorem V.2 which provides fundamental lower bounds on Dabs(R) and Rabs(D). The bound (8) shows that Dabs(R) is lower bounded by a term involving the system's complexity through generalized entropy, h(ξ), and terms related to the abstraction size.

Implications for Abstraction Construction

The developed theory enables constructing minimal abstractions, optimizing the size-accuracy tradeoff, through an example on a chaotic system. The work provides a general procedure for constructing optimal abstractions in terms of the size-accuracy tradeoff. This is achieved by solving the problem of encoding trajectories through coverings in a high-dimensional ambient space.

Curse of Dimensionality

The paper demonstrates how the curse of dimensionality arises through a relaxed version of the bounds in Theorem V.2 and Corollary V.3. This shows that for any system and fixed finite l, the size of the partition e R fundamentally increases exponentially with n to achieve a prescribed distortion.

Technical Results

The optimal abstraction size-accuracy tradeoff is captured by the following rate-distortion quantity: Dabs(R):= inf A Eξ0[d(ξ, omegaA) A] s.t. A is an abstraction of S, (4), (5) hold, logY≤ R, omegaA = gA(sA(ξ)).

The computation of the required quantities like h(ξ), hs(ξ), and cBS l is detailed in Proposition V.4. The paper also shows that for exponentially stable systems, the abstraction distortion converges to 0 for l → ∞. The analysis of the doubling map provides a concrete example where Dabs(R) equals Dcover(R). The final results provide fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions.

The theory provides a statistical quantification of abstractions’ accuracy and size, establishing connections with rate-distortion theory for set-based encoder-decoder pairs. The fundamental lower bound depends on the complexity of the dynamics, through generalized entropy. This novel theory quantifies scalability limits of abstractions, and provides insights on how the complexity of the dynamics to be abstracted dictates these limits. The paper demonstrates that for any system and fixed finite l, the size of the partition e R fundamentally increases exponentially with n to achieve a prescribed distortion. This is shown by Corollary V.6. The work provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper demonstrates that for exponentially stable systems, the abstraction distortion converges to 0 for l → ∞. The analysis of the doubling map provides a concrete example where Dabs(R) equals Dcover(R). The final results provide fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x). The paper provides fundamental limits on the size-accuracy tradeoff, or the scalability, of abstractions for given dynamics x+ = f(x).

Improvements for AI systems

Derive Scalable Abstractions via Rate-Distortion Theory: Implement an information-theoretic encoder-decoder pair where Rate measures abstraction size and distortion describes accuracy, allowing for a formal derivation of fundamental lower bounds on minimum achievable abstraction distortion given system dynamics and size, as described by the derived bounds in Theorem V.2.
Construct Minimal Abstractions: Employ the theory to actively construct optimal abstractions by solving the problem of encoding trajectories of dynamical systems, through coverings in a high-dimensional ambient space, specifically demonstrated through constructing a minimal abstraction of the doubling-map dynamics and using information-theoretic algorithms like the Information Bottleneck Method.
Optimize Size-Accuracy Tradeoff for Verification: Utilize the derived rate-distortion bounds to determine the optimal abstraction size for a given accuracy requirement, ensuring that the optimal abstraction size-accuracy tradeoff is achieved by minimizing Eξ0[d(ξ, omegaA) A] subject to a rate constraint.
Enhance Verification Guarantees: Leverage the derived statistical quantification of accuracy and size to provide guarantees on the statistical quantification of abstractions’ accuracy and size for verification problems, moving beyond mere behavioral inclusion to quantitative performance metrics.