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

Faizanuddin Ansari, Debanjan Dutta, Swagatam Das

arXiv:2608.14004v1cs.LG

TL;DR

Finite prompts over partial orders may leave queried comparisons logically undetermined, while teaching cost and representational complexity depend on distinct structural properties. The paper formalizes these issues with version spaces and coordinate decoders, proving an exact true/false/unknown classification, open-world teaching bounds, and a dimension-based representation boundary.

  • Problem

    Prior in-context-learning abstractions do not distinguish missing evidence from incomparability or specify what finite relational prompts logically determine.

  • Method

    The framework uses version-space semantics, closure and constraint reasoning, teaching-number decompositions, and prompt-dependent coordinate-order decoders.

  • Results

    The theory gives a three-way forced-true/forced-false/unknown classifier, identifies open-world teaching cost as covers plus blockers, and makes dimension at most s necessary and sufficient for exact s-coordinate decoding.

  • Takeaways & Limitations

    Prompt ambiguity, teaching cost, certificate length, and representation dimension are distinct sources of difficulty rather than one overall notion of poset difficulty.

  • Takeaways & Limitations

    The complexity of the structured blocker-set instances induced by posets remains undetermined.

Abstract

from arXiv · show

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.

1 Introduction · 2 Related Work and Positioning · 3 Formal Model

The paper develops a formal theory of in-context learning for partial orders, separating logical identifiability, teaching cost, and representation capacity. It models prompts as version spaces over a fixed universe, distinguishes open- from closed-world semantics, and connects exact coordinate decoding to order dimension.

  • 1 Introduction: Partial orders expose that missing evidence does not imply incomparability, while their structure can limit exact decoding.A partial order is reflexive, antisymmetric, and transitive, with comparability given by reachability in the reflexive transitive closure of its cover graph.
  • 1 Introduction: The paper asks what relational prompts logically determine, how many labels teach a finite poset, and which posets admit exact coordinate-order decoding.These questions extend prior empirical work on linear-order and divisibility prompts, which reported performance saturation in current language models.
  • 1 Introduction: The contributions formalize poset version spaces, prove a true/false/unknown completion theorem, characterize open-world teaching prompts, and establish a representation boundary.The framework includes background knowledge, cover labels, blocker-set hitting, exact chain and antichain values, and formal coordinate-order representations.
  • 2 Related Work and Positioning: Related ICL theories include function-learning, optimization, Bayesian, task-representation, mechanistic, and concept-subspace accounts.The paper positions its contribution as relational ICL theory rather than another general mechanism for task learning.
  • 2 Related Work and Positioning: Unlike database prediction and graph-reasoning settings, this work studies logical completion of mathematical partial orders using order dimension as the representation invariant.It connects classical Dushnik–Miller dimension results to a precisely restricted ICL representation class.
  • 3 Formal Model: The formal model fixes a finite known universe U shared by every target poset and every hypothesis in the background theory B.A prompt contains positive demonstrations asserting x ⪯P y and negative demonstrations asserting x not⪯P y, while the query is separate.
  • 3 Formal Model: Identifiability is defined through a version space of posets consistent with demonstrations and background knowledge, with universal soundness requiring correctness for every remaining hypothesis.The background theory may encode relation semantics or completeness assumptions, and satisfiability is evaluated within the resulting version space.
  • 3 Formal Model: Open-world semantics permit additional comparisons consistent with the axioms and negative demonstrations, whereas closed-world Hasse semantics treat the displayed DAG as complete.This distinction makes background assumptions explicit when determining whether a query is identified.

4 Open-World Identifiability

Open-world identifiability is exactly the existence of a universally sound answer: a query is forced true by positive closure, forced false by a cycle or negative-label obstruction, or otherwise ambiguous. The completion trichotomy is constructive, supports efficient query testing, and clarifies how background semantics and closed-world assumptions change identifiability.

  • Sound-answer criterion: A universally sound binary answer exists if and only if the query is identified.If consistent posets disagree, neither deterministic nor randomized binary output can guarantee correctness for every completion.
  • Open-world completion trichotomy: The open-world completion trichotomy classifies each query as forced true, forced false, or genuinely unidentifiable after reflexive transitive closure of positive demonstrations.Forced falsity arises when adding the query creates a cycle or forces a pair contradicting a negative demonstration.
  • Open-world completion trichotomy: A three-element example realizes all three outcomes: a ⪯b is true, b ⪯c is false from a transitive negative conflict, and c ⪯b remains ambiguous.The ambiguity persists because both the least closure and its extension by (c, b) satisfy the prompt.
  • Algorithmic consequence: O(|U|^3) preprocessing computes positive closure, after which each query costs O(|D−|) by scanning negative demonstrations.Bitset representations reduce the rectangle test to O(|U|^2/w) word operations per query.
  • Background assumptions: Identifiability depends on background knowledge: stronger semantics can identify queries that demonstrations alone cannot, so benchmarks must distinguish prompt-only induction from pretrained semantic knowledge.Closed-world reachability additionally requires declaring a DAG to be the complete Hasse diagram.

5 Prompt Teaching Complexity

For a known finite universe, open-world teaching of a poset requires positive cover demonstrations and negative labels forming a blocker-set hitting set for every ordered incomparable pair. The maximum teaching cost is n(n−1), uniquely attained by the antichain, and the blocker term is precisely the cost absent complete-Hasse semantics.

  • Exact teaching characterization: Negative labels must hit every blocker set BP(a, b) for ordered incomparable pairs, with the two orientations treated as distinct constraints.If a blocker set is missed, adding a ⪯ b and closing transitively yields a second consistent poset.
  • Exact teaching characterization: Cover demonstrations are necessary because deleting any cover relation preserves a poset consistent with the other labels.A cover has no intermediate element, so removing it cannot create a transitivity violation.
  • Exact teaching characterization: A teaching prompt is obtained by labeling every cover positive and using a minimum blocker-set hitting set as negative labels.The covers generate the poset, while any added ordered pair forces a blocked relation and contradicts a negative label.
  • Extremal values: n(n−1) is the maximum teaching number over n-element posets, uniquely attained by the antichain.For the antichain, each ordered incomparable pair has a singleton blocker set, requiring all n(n−1) negative labels.
  • Price of open-world semantics: Every cover edge is necessary and sufficient under complete-Hasse semantics, so the blocker term β(P) is exactly the additional label cost of open-world semantics.Complete-Hasse prompts require only the displayed cover edges, whereas open-world prompts also need blocker-hitting negative labels.

6 Structural Complexity

Structural complexity is multidimensional: height and positive path length capture chain depth, width captures incomparability, and order dimension captures the number of linear orders needed for realization. Standard families show that dimension and certificate length can vary independently, while finite divisibility posets have unbounded dimension.

  • Standard families: A chain has dimension 1, whereas the Boolean lattice B_m has dimension m and the positive-divisor poset of N = Q_r has dimension r.These families illustrate that order dimension differs across chains, Boolean lattices, and divisor posets.
  • Finite divisibility: If the product of the first r primes is at most n, then dim(D_n) ≥ r; consequently, the dimensions of D_n are unbounded as n → ∞.Squarefree divisors induce a Boolean lattice, and dimension monotonicity under subposets gives the bound.
  • Interpretation: Higher order dimension requires more coordinates for exact uniform representation, but it does not order the behavioral difficulty of every individual query or model.The comparison is explicitly family-level rather than a universal query-level difficulty ordering.

7 Coordinate-Order Representation Limits · 8 Positive and Negative Certificates

The paper establishes an exact dimension boundary for prompt-dependent monotone coordinate decoders and characterizes positive and negative certificates in Hasse DAGs. Representation requires target order dimension at most s, while certificate complexity is governed by shortest positive paths and reachable regions for negative queries.

  • 7 Coordinate-Order Representation Limits: An exact s-coordinate decoder exists for a task family if and only if every target poset has dimension at most s.Coordinates may depend on prompts, but one decoder must work exactly across the whole family.
  • 7 Coordinate-Order Representation Limits: Width at most s for every target is sufficient, whereas one target with dimension greater than s rules out exact decoding.The width condition is convenient rather than necessary; the exact boundary is order dimension.
  • 7 Coordinate-Order Representation Limits: The representation boundary is nonconstructive and does not bound transformer hidden width, attention-update rank, or arbitrary task-vector capacity.The result concerns only the specified coordinatewise monotone decoder class and does not guarantee efficient coordinate recovery.
  • 8 Positive and Negative Certificates: λ+_P(a,b) cover edges are necessary for any positive certificate built only from demonstrated cover edges.Thus, a procedure composing at most T cover edges cannot certify every positive query whenever λ+_P(a,b) > T.
  • 8 Positive and Negative Certificates: A false query is certified exactly by a forward-closed set containing a and excluding b when b is unreachable from a in the Hasse DAG.The canonical witness is Reach_H(a), the set of vertices reachable from a.
  • 8 Positive and Negative Certificates: ν−_P(a,b) = |Reach_H(a)| for unreachable b, and Reach_H(a) is the unique inclusion-minimal and minimum-cardinality negative witness.For fixed H and a, this value is independent of which nonreachable vertex b is queried.
  • 8 Positive and Negative Certificates: Positive certificates may be local paths, whereas canonical negative witnesses may expose large reachable regions rather than merely recording path absence.This distinction does not imply an unrestricted transformer-depth lower bound because models may use global algorithms or encoded summaries.

9 Implications, Scope, and Limitations

The framework separates logical identifiability, teaching cost, certificate complexity, and representation capacity for poset in-context learning. It also clarifies benchmark-design requirements and limits its representation conclusions to a specific monotone coordinate-decoder class.

  • How the framework fits together: The framework treats prompt solvability, teaching cost, structural difficulty, and coordinate-decoder representability as logically distinct questions.Universal soundness requires agreement across every poset in the version space; the framework then separates structural and proof-related difficulty measures from representation capacity.
  • Examples illustrating the separation of difficulty sources: n-1 cover edges teach the chain Cn under both open- and closed-world semantics, although its bottom-to-top positive certificate has length n-1.The chain has order dimension one, illustrating that simple representation can coexist with a long particular proof.
  • Consequences for benchmark design: Poset benchmarks should specify the universe, background theory, demonstration semantics, diagram completeness, and whether unknown is an admissible output.Theorem 3 provides a model-independent logical baseline for mixed positive/negative open-world prompts.
  • Scope of the theoretical conclusions: The results distinguish non-identifiability, prompt-budget, certificate, and representation limitations as four separate sources of failure.High order dimension is a limitation only for the exact monotone coordinate decoders defined in the framework.
  • Scope of the theoretical conclusions: The theory does not claim limitations for unrestricted transformers, softmax attention, general task vectors, approximate decoders, or arbitrary graph algorithms.Transformer universality results remain compatible because the framework concerns a restricted decoder class.

10 Conclusion … 2 Semantic Setup and Sound Answers

The paper develops a version-space theory of partial-order in-context learning in which sound answers require queried comparisons to be identified across all consistent posets. It establishes exact completion, teaching-cost, and coordinate-decoder characterizations, while the supplementary material expands proofs, edge cases, enumerations, and cross-references.

  • 10 Conclusion: The conclusion frames queried-comparison identifiability as the basic requirement for sound in-context learning on partial orders.Finite open-world prompts support a true/false/unknown characterization, while complete Hasse prompts reduce to reachability.
  • 10 Conclusion: Open-world teaching cost decomposes into mandatory covers plus a blocker-set surcharge, whose tight class maximum is uniquely attained by the antichain.The blocker term distinguishes open-world teaching from complete-Hasse semantics.
  • SUPPLEMENTARY MATERIAL FOR “IDENTIFIABILITY AND ORDER-DIMENSION LIMITS OF IN-CONTEXT LEARNING ON PARTIAL ORDERS”: The supplementary material provides expanded proofs, edge-case checks, exact ambiguity enumeration, teaching arguments, decoder boundaries, and structural-profile details.It uses the prefix S and maps main-paper results to corresponding supplementary statements.
  • 1 Cross-Document Reference Map: The cross-document map identifies the supplementary results corresponding to sound answers, completion trichotomies, closed-world reachability, background-theory dependence, and ambiguity enumeration.It links these results to Proposition S1, Lemmas S2–S3, Theorem S4, Observation S5, Proposition S6, and the exact finite-universe ambiguity section.
  • 2 Semantic Setup and Sound Answers: A finite known universe defines prompts as positive and negative comparisons over U, with a background theory B determining the version space of consistent posets.Satisfiable prompts have a universally sound binary answer exactly when every consistent poset assigns the same truth value to the query.
  • 2 Semantic Setup and Sound Answers: If two consistent posets disagree on a query, no deterministic binary answer can be universally correct.The proof also rules out randomized rules that promise correctness, because each possible output fails on one of the disagreeing posets.

3 Expanded Open-World Completion Proof

For finite open-world prompts over all posets, the reflexive transitive closure of positive demonstrations is contained in every consistent completion. Each query is then classified as forced true, forced false by a cycle or negative demonstration, or genuinely unidentifiable.

  • Least positive closure: The positive closure R is a partial order, avoids every negative demonstration, and is contained in every consistent poset.This closure provides the least positive completion under satisfiable demonstrations.
  • One-edge closure: Adding a queried edge aRb preserves antisymmetry exactly when b̸Ra, with the least transitive extension adding the predecessor–successor rectangle PredR(a) × SuccR(b).If bRa, the new edge creates a two-way comparison and violates antisymmetry.
  • Open-world completion trichotomy: Theorem S4 gives exactly three outcomes: aRb forces true; a̸Rb with bRa or a negative rectangle intersection forces false; otherwise the query is unidentifiable.The false case arises because any true extension would violate antisymmetry or a negative demonstration.
  • Open-world completion trichotomy: When neither obstruction occurs, consistent true and false completions coexist, proving the query unidentifiable.The two completions are formed by retaining R alone or adding the least true extension while avoiding negative pairs.
  • Algorithmic consequences: O(|U|3) closure preprocessing supports naive O(|D−|) queries, or O(|U|2/w) rectangle tests with bitsets.These procedures implement a logical oracle rather than a learned predictor.

4 Exact Finite-Universe Ambiguity Enumeration · 5 Expanded Teaching-Number Proof

Section 4 gives an exact finite-universe version-space enumeration of ambiguous comparisons, while Section 5 characterizes open-world teaching cost as covers plus a blocker-set hitting number. The teaching number ranges up to n(n−1), uniquely at the antichain, whereas complete-Hasse semantics costs only the cover edges.

  • 4 Exact Finite-Universe Ambiguity Enumeration: The enumeration is exhaustive over all 219 labeled posets on four elements and all 212 observed-pair masks, with ambiguity defined by nonconstant truth across the version space.It uses equal weighting and no random-poset generator, fitted parameter, Monte Carlo error, or random seed.
  • 4 Exact Finite-Universe Ambiguity Enumeration: Stronger background knowledge shrinks the version space, preserves any identification already obtained, and cannot reduce a target’s minimum teaching-set size.Formally, if B1 ⊆ B2, then VB1(D) ⊆ VB2(D).
  • 5 Expanded Teaching-Number Proof: Open-world teaching on a known finite universe requires every strict cover positively and a negative blocker-set hitting set for ordered incomparable pairs.The fixed universe is essential because allowing fresh elements prevents finite prompts from isolating an antichain.
  • 5 Expanded Teaching-Number Proof: The exact open-world teaching number is |Cov(P)| + β(P), where β(P) is the minimum number of negative labels intersecting every blocker set.The lower bound follows from cover necessity and blocker hitting; the upper bound uses all covers plus a minimum hitting set.
  • 5 Expanded Teaching-Number Proof: For n elements, the teaching number is bounded by n(n−1), with equality uniquely attained by the antichain; the chain requires n − 1 labels.The antichain needs all n(n−1) ordered negative labels, while the chain has n − 1 covers and no incomparable pairs.
  • 5 Expanded Teaching-Number Proof: Under complete-Hasse semantics, the teaching number equals |Cov(P)|, so the open-world overhead is exactly β(P).All covers are sufficient and necessary because their reflexive transitive closure determines the declared poset.
  • 5 Expanded Teaching-Number Proof: β(P) is a minimum hitting-set instance, but the complexity of blocker-set instances induced by posets remains undetermined.Minimum hitting set is NP-hard in general; the paper does not establish hardness or tractability for this structured subclass.

6 Order Dimension and the Conditional Decoder Bound

The section establishes an exact zero-error boundary for fixed s-coordinate decoders: they can represent precisely target posets of dimension at most s, with width at most s as a sufficient condition. It also shows that order dimension is one for chains, equals m for Boolean lattices B_m, and is unbounded in divisor posets as n grows.

  • Exact coordinate-decoder capability: An exact s-coordinate decoder exists if and only if every target poset has dimension at most s.The converse constructs prompt-dependent coordinate realizations and pads shorter realizations by repetition.
  • Exact coordinate-decoder capability: Width at most s is a convenient sufficient condition for exact s-coordinate decoding.This follows from the classical inequality dim(P) ≤ width(P).
  • Scope and limitations: The boundary is exact and zero-error for the specified decoder form, but its sufficiency proof is not an efficient learning algorithm.Approximate decoders and unrestricted neural representations are outside the statement.
  • Standard poset families: r coordinates are necessary and sufficient for the positive-divisor poset of a number with r distinct prime factors, and divisor-poset dimensions are unbounded as n grows.Exponent vectors give the upper bound, while squarefree divisors induce a Boolean lattice; if the first r primes’ product is at most n, then dim(D_n) ≥ r.
  • Standard poset families: m coordinates are necessary and sufficient for the Boolean lattice B_m, so dim(B_m) = m.Membership indicators provide the upper bound, while incomparable singleton-complement pairs force at least m linear extensions.

7 Expanded Certificate Proofs

The section establishes sharp certificate bounds for positive and negative order queries. Cover-path length determines positive-certificate cost, while forward-closed reachability sets provide canonical, minimum-size negative witnesses.

  • Positive chain certificates: λ+_P(a, b) cover edges are necessary for any positive certificate using only demonstrated cover edges.Such a certificate is a directed Hasse-DAG path, so no shorter valid chain exists.
  • Positive chain certificates: A cover-path procedure limited to length T cannot certify every positive query with λ+_P(a, b) > T.This restriction does not exclude alternative global reachability algorithms.
  • Negative witnesses: A nonreachability witness exists exactly when a forward-closed set contains a and excludes b.Forward closure prevents any directed path from leaving the set, while Reach_H(a) supplies such a set whenever b is unreachable.
  • Negative witnesses: Reach_H(a) is the unique inclusion-minimal and minimum-cardinality forward-closed witness excluding any nonreachable b.Therefore ν−_P(a, b) equals the minimum witness size and, for fixed H and a, is independent of b.

8 Assumptions and Dependency Summary

The results rely on a fixed finite universe, a satisfiable mixed-label prompt, and Lemma S3’s one-edge closure formula. The paper also distinguishes its synthesis from classical foundations and treats the deterministic illustration as non-theorem evidence.

  • 8 Assumptions and Dependency Summary: The completion trichotomy assumes all posets share a fixed finite universe and the mixed-label prompt is satisfiable.Its completion and teaching results depend on Lemma S3’s one-edge closure formula.
  • 8 Assumptions and Dependency Summary: The deterministic illustration exhaustively enumerates a specifically defined uniform distribution over four-element targets, prompt subsets, and unobserved queries.No theorem depends on that illustration.
  • 8 Assumptions and Dependency Summary: The synthesis separates prompt entailment, open-world teaching surcharge, structural profile coordinates, certificate size, and specified coordinate-decoder capacity from classical concepts.The cited foundations include version spaces, teaching dimension, Hasse reachability, and Dushnik–Miller dimension.
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