Open Problems in Natural Abstractions Research
A working list of open problems in the natural abstractions program, building on the natural latents framework developed over the day.
The overarching goal
0. A constructive natural-latent-model algorithm. Produce an algorithm that, given an arbitrary distribution, returns a natural latent model: a collection of latents that jointly account for all dependence among observables, each with the mediation and redundancy properties. Such an algorithm would give a canonical, near-optimal way to compress observations into predictive summaries, let unlike agents (e.g. humans and machines) verify that their concepts refer to the same structure, and identify which concepts are objective. Everything below is, in effect, a sub-problem of building this.
1. The structural target: definitions and their strength
1.1 Is “explain all the structure” the right target? Full explanation demands that every finite family of observables become conditionally independent given the latent, uniformly across the whole observable universe. This is intentionally very strong — so strong that it forces the latent to retain all of \(D\): explaining all the structure means nothing about \(D\) may be forgotten. Since abstraction is precisely about throwing detail away, full explanation may be a non-starter as the target for that reason. The hard question it raises is how to justify leaving certain observable families unexplained — restricting the target to a predictive task or a structured subclass of families — without that choice being arbitrary.
1.2 Which observable-family restrictions preserve the target? The framework quantifies over all observables of \(D\) to avoid privileging a basis by hand, but no agent can search that space. The hope is that the unrestricted theory identifies a stable target also recovered from many smaller, well-chosen families. Which restrictions preserve the same target, and why?
1.3 When is strong redundancy the right nontriviality condition? The coarsening characterization shows redundancy is equivalent to being a minimal mediator over every coarsening, but it remains unclear exactly when this stronger condition is the right one to use versus plain minimal mediation. Characterize the regimes.
1.4 Weaker redundancy notions for distributed abstractions. A genuine abstraction may be distributed — partially present in each observable and recoverable only jointly — in which case strong redundancy excludes real structure. Candidate weakenings include stochastic redundancy and “weak redundancy” (small conditional entropy of the latent given all-but-one observable). Is there a principled family of admissible nontriviality constraints, and which is most useful in practice?
1.5 Constraining the extension space. Latents are taken to live on an extension of the probability space, rather than being readable off \(D\) directly. The open problem is to understand what constraints that extension should obey — and, given some sense data, what counts as a minimal, justified extension space to posit in the first place, rather than an arbitrary one.
2. Existence and obstructions
2.1 Nonexistence of natural latents. Under strong redundancy, natural latents do not always exist for a given observable family. We want clean, canonical examples of nonexistence and an understanding of what nonexistence indicates about the family — when it signals genuinely incompatible observations versus a poorly chosen representation.
2.2 Synergy. Some dependence is synergistic: present jointly, invisible in any individual observable. Strongly redundant latents are built precisely so as not to see it. Can the framework be extended to accommodate synergistic structure, or does synergy demand a fundamentally different nontriviality condition?
2.3 Bound saturation and sharpness. The Mediator-Determines-Redund bound (\(H(Z_{\mathrm{red}} \mid Z_{\mathrm{med}}) \le \varepsilon_{\mathrm{med}} + 2\varepsilon_{\mathrm{red}}\)) is tight in a worked example. Characterize when the structural inequalities are (near-)tight in general; saturation should illuminate the geometry of the mediator/redund latent classes.
2.4 The approximate regime: tolerances, frontiers, underdetermination. Away from exact existence and exact redundancy, the theory is pervasively approximate. This brings Pareto frontiers in \((\varepsilon_{\mathrm{med}}, \varepsilon_{\mathrm{red}})\) space and underdetermination among many near-optimal candidate latents. How should we select among, or canonicalize, near-optimal candidates?
2.5 Easy versus hard distributions. Some laws admit crisp, low-dimensional explanatory variables; others are intrinsically distributed and resist any compact latent. By analogy with macroscopic order parameters near a critical temperature in statistical physics, characterize the class of systems for which clean abstraction is plausible at all.
3. Spurious versus substantive structure
3.1 Distinguishing real structure from partition artifacts. Observables can be dependent for purely combinatorial reasons (e.g. “even” vs. “multiple of 4”), with no underlying latent feature. Two attitudes: directly control for dependence arising from trivial partition relations (careful but hard), or model everything and tolerate some meaningless mediated relations (brute force). A fuller theory should say how to draw the line.
3.2 When is apparent structure rare by chance? In a large state space, how often do independent observables look dependent purely by coincidence? A theory of natural abstraction should be able to say when observed dependence is safe to trust as substantive rather than an artifact of how the space was carved.
4. Discovery, learning, and computation
4.1 Joint learning of distribution, prediction, and abstraction. The framework assumes the law on \(D\) is known before the search begins. In practice an agent learns which abstractions are natural while building an approximate predictive model, and would not try to predict the entire input space. Give an account of prediction, representation learning, and abstraction discovery proceeding in parallel.
4.2 Inference from finite data. The definitions describe the target if the law were known. Build a theory of inference/learning on top: how can full explanation, mediation, and redundancy be estimated, approximated, or searched for from finite samples — perhaps by repurposing existing representation-learning methods?
4.3 Recoverability versus existence (computational cost). A latent may be perfectly real — organizing dependence in the world — yet be computationally or inferentially intractable to extract (the hash-then-perturb example). A theory of natural abstraction should say not only which latents exist in principle but also something about the cost of recovering them.
4.4 Constructive discovery mechanics. Even granting an oracle that produces candidate natural latents when they exist, open questions remain: which observable families to examine, how to merge or prune overlapping latents, which candidates count as canonical, and how cross-agent translation should be operationalized.
5. Modeling assumptions and setup
5.1 Adequacy of the base input space. Whether \(D\) is fine enough to capture the structure of interest is a substantive modeling question. A coarse \(D\) can make a phenomenon look noisy or independent when a richer \(\widetilde D\) (hidden state, history, extra variables) would reveal rigid structure. The framework can return a correct negative answer to the wrong question: no informative latent for the law on \(D\), even though one exists for the phenomenon. How should an agent diagnose and repair an inadequate starting representation?
5.2 Treatment of time. The setup is deliberately noncommittal about time. Options include adding a time-like index as a component of the input, or letting \(\mathcal{D}\) range over trajectories rather than snapshots. Some such move is likely needed for natural latents that mediate among observations of the same system at different times.
6. Conceptual and philosophical
6.1 What type of thing is a concept? The mathematics treats a latent as a random variable (something that takes values), but ordinary concept-talk conflates feature dimension, particular value, and predicate/rule. “The velocity of this object” looks like a natural latent; “velocity in general” is a different type. Work out the more general type and its relation to particular latent variables.
6.2 From conditional translatability to genuine objectivity. The framework establishes a conditional result: among observers predicting efficiently, certain latents are forced by dependence structure. It does not establish objectivity in the pure form — that agreement is explained by the world rather than by anything shared between observers. Closing this gap (or precisely bounding it) is open.
6.3 Coverage: how much of the world do natural latents reach? Examples show natural latents exist, but how many of the concepts we actually use are natural latents in this sense, and how far a complete account of a domain can be built from them, is unresolved — the questions a complete theory would have to answer next.
7. Representation and translation
7.1 Canonical representation of a latent. The framework identifies latents only up to bijection, but some relabelings preserve far more usable structure than others (e.g. 00,01,10,11 exposes a product structure that 1,2,3,4 hides; a collision-free hash preserves distinctions while destroying usability). Define what makes one representation of a latent objectively better, and how to find good ones.
7.2 Constructing the translation maps. Guaranteed Translatability shows two natural latents over shared observables determine each other up to small error, but finding the actual bijection between differently labeled versions of the same latent is left for later work.
7.3 Representation matters for computation, even though the theory forgets it. Information-theoretic quantities are deliberately representation-invariant: they see only the partition a variable induces, never the basis or the labels. That invariance is exactly what buys objectivity — but representation is also what makes computation tractable, and the basis an agent ends up using is shaped by its sensors and by what is cheap to compute. Is there a story in which certain representations get converged on because they enable efficient prediction, thereby justifying the bases they privilege? Building a principled account of representation back on top of the representation-free theory is open.