Identifiability and Order-Dimension Limits of In-Context Learning on Partial Orders

Abstract: In-context learning is commonly formalized as inference from examples of a function. Partial orders instead combine transitivity, antisymmetry, and incomparability, so a finite prompt may not determine a queried comparison. We develop a theory of in-context learning on partial orders that separates logical identifiability, prompt teaching cost, structural complexity, and the exact capacity of a formal coordinate-decoder class. A version-space semantics makes background knowledge and open- versus closed-world assumptions explicit. For finite open-world prompts with positive and negative comparisons, we prove an exact completion trichotomy: after taking the reflexive transitive closure of the positive demonstrations, a query is forced true, forced false because every true completion creates a cycle or violates a negative demonstration, or remains genuinely ambiguous. For a known $n$-element universe, we characterize the open-world teaching number as the number of covers plus a blocker-set hitting number, prove that its maximum over all $n$-element posets is $n(n-1)$ and is uniquely attained by the antichain, and identify the blocker term as the exact cost of open-world rather than complete-Hasse semantics. We formalize prompt-dependent $s$-coordinate decoders and use the classical coordinate-order equivalence to obtain an exact representation boundary: dimension at most $s$ is necessary and sufficient, while width at most $s$ is a convenient sufficient condition.

• The paper establishes an exact open-world trichotomy for query answers—true, false, or unidentifiable—and provides an O(|U|³) preprocessing method with bitset-based query checks.
• The paper proves that the minimum open-world teaching set is |Cov(P)| + β(P), with the antichain requiring n(n−1) comparisons and β(P) forming a generally NP-hard hitting-set problem.
• The paper shows that exact conjunctive coordinate decoders succeed precisely when every target poset has Dushnik–Miller dimension at most s, while certificate sizes expose limits for proving positive reachability and negative nonreachability.

Overview

The paper studies in-context learning of partial orders from a logical identifiability standpoint. Given a finite known universe $U$, a prompt $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$ consists of positive and negative pairwise comparisons, and a background theory $\mathcal{B}$ is a class of posets on $U$. The version space $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$ collects all posets consistent with the prompt, and a binary answer to a query $(a,b)$ is universally sound exactly when all posets in the version space agree on its truth value (2608.14004). The paper's central contributions are: an exact trichotomy for query identifiability under open-world semantics; an exact characterization of teaching numbers via covers and blocker hitting sets; an exact capability boundary for coordinate-based decoders expressed through Dushnik–Miller order dimension; and certificate-size bounds for positive and negative answers.

Sound answers and the open-world trichotomy

A deterministic or randomized answer rule can be universally correct on a satisfiable prompt if and only if the truth value of the query is constant across the version space; otherwise every realized output errs on some consistent poset (2608.14004). Under the least restrictive background—all posets on $U$—the paper takes $R=\mathrm{TC}(\mathcal{D}^+)$, the reflexive transitive closure of positive demonstrations, and proves that $R$ is contained in every consistent poset. The key technical device is a one-edge closure formula: adding $(a,b)$ to a reflexive transitive relation $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$0 yields closure equal to $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$1, which remains antisymmetric precisely when $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$2. This yields a complete trichotomy: a query is identified true if $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$3; identified false if $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$4 or the predecessor–successor rectangle intersects $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$5; and unidentifiable otherwise, where both a consistent true completion $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$6 and false completion $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$7 exist. Algorithmically, Floyd–Warshall preprocessing costs $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$8, after which each query is decidable in $\mathcal{D}=(\mathcal{D}^+,\mathcal{D}^-)$9 word operations using bitsets—an exact logical oracle rather than a learned predictor.

Exact ambiguity enumeration

To quantify how prevalent unidentifiability is, the paper exhaustively enumerates all 219 labeled posets on a four-element universe over all $\mathcal{B}$0 observed-pair subsets, under a uniform scheme over targets, prompt subsets, and unobserved queries. The resulting ambiguity rate $\mathcal{B}$1 declines smoothly but remains substantial even at near-complete observation:

$\mathcal{B}$2	0	1	2	3	4	5	6	7	8	9	10	11
$\mathcal{B}$3	1.0000	.9726	.9306	.8772	.8177	.7566	.6968	.6398	.5863	.5365	.4902	.4475

Even when eleven of twelve non-reflexive pairs are observed, roughly 45% of remaining queries remain ambiguous. This is an exact integer computation—no sampling error, completed in about 1.6 seconds—and the authors are explicit that it illustrates one fully specified averaging scheme on four elements and does not approximate any natural distribution over larger posets.

Teaching numbers and the open-world surcharge

Under open-world semantics, where the prompt must isolate the target among all posets on $\mathcal{B}$4, the paper proves an exact formula: the minimum teaching set has size $\mathcal{B}$5, where $\mathcal{B}$6 is the cover set and $\mathcal{B}$7 is the minimum hitting-set size for the blocker sets $\mathcal{B}$8 over ordered incomparable pairs. Necessity of all covers follows from a deletion argument showing that removing any single cover yields another poset consistent with the prompt. For extremal families, the chain satisfies $\mathcal{B}$9 while the antichain requires $U$0, and the antichain uniquely attains the class maximum of $U$1 among all $U$2-element posets. Under closed-world semantics, where the prompt is declared to be the complete Hasse diagram, only the $U$3 covers are needed, so the open-world surcharge is exactly $U$4. The paper notes that computing $U$5 is a minimum hitting-set instance, NP-hard in general, though it leaves undetermined the complexity of the structured instances induced by posets—a stated open question.

Order dimension as a decoder capacity limit

The paper connects prompt-conditioned decoding to classical dimension theory via the standard equivalence between realizers and coordinate representations: a poset admits an $U$6-coordinate representation $U$7 for all $U$8 if and only if its Dushnik–Miller dimension is at most $U$9. The consequence is an exact capability boundary: a fixed $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$0-coordinate decoder with conjunctive coordinatewise decisions can be exact on a task family if and only if every target has dimension at most $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$1. Width at most $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$2 suffices by Dilworth's inequality, while a single high-dimension target makes zero-error decoding impossible. Standard families illustrate unboundedness: chains have dimension one, the Boolean lattice $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$3 has dimension exactly $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$4, divisor lattices of $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$5 have dimension $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$6, and the divisibility poset on $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$7 has dimension growing without bound since primorials embed $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$8 whenever $\mathcal{V}_{\mathcal{B}}(\mathcal{D})$9. The authors stress that this is an exact boundary for the specified decoder form only; the sufficiency direction is not an efficient learning algorithm, and approximate decoders and unrestricted neural representations fall outside the claim.

Certificate bounds

For positive queries, any certificate composed solely of demonstrated cover edges must contain at least $(a,b)$0 edges—the shortest directed cover path—so any procedure restricted to witnesses of length at most $(a,b)$1 cannot certify all true queries. For negative queries, nonreachability in the Hasse DAG is witnessed by forward-closed separator sets, and the reachable set $(a,b)$2 is shown to be the unique inclusion-minimal such separator containing $(a,b)$3 and excluding $(a,b)$4, hence also of minimum cardinality. This yields a canonical witness statistic $(a,b)$5 independent of the particular nonreachable target. Structural profiles for chains, Boolean lattices, divisor lattices, and divisibility posets instantiate these statistics—for example, height $(a,b)$6 and maximum negative witness size $(a,b)$7 for $(a,b)$8.

Assumptions and limitations

The completion and teaching results rest on the one-edge closure lemma and assume a fixed finite universe with a satisfiable mixed-label prompt; the fixed universe is essential, since with fresh elements no finite prompt isolates an antichain. The enumeration figure concerns only the four-element uniform scheme and supports no theorem. The decoder boundary applies to exact, prompt-dependent monotone-coordinate decoders and says nothing about approximate or neural decoding. The complexity of computing $(a,b)$9 on poset-induced blocker structures is left open, as is whether efficient global reachability procedures can circumvent the path-length certificate bound.

Conclusion

The paper provides exact, assumption-explicit characterizations of what can be inferred, taught, certified, and decoded about partial orders from comparison prompts. Its main quantitative findings—the trichotomy for query identifiability, the teaching formula $U$0 with antichain worst case $U$1, and the dimension-$U$2 capability ceiling for coordinate decoders—frame in-context learning on orders as a problem where logical identifiability, not statistical estimation, is the binding constraint.