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

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.

APFEx: Adaptive Pareto Front Explorer for Intersectional Fairness

Ensuring fairness in machine learning models is critical, especially when biases compound across intersecting protected attributes like race, gender, and age. While existing methods address fairness for single attributes, they fail to capture the nuanced, multiplicative biases faced by intersectional subgroups. We introduce Adaptive Pareto Front Explorer (APFEx), the first framework to explicitly model intersectional fairness as a joint optimization problem over the Cartesian product of sensitive attributes. APFEx combines three key innovations- (1) an adaptive multi-objective optimizer that dynamically switches between Pareto cone projection, gradient weighting, and exploration strategies to navigate fairness-accuracy trade-offs, (2) differentiable intersectional fairness metrics enabling gradient-based optimization of non-smooth subgroup disparities, and (3) theoretical guarantees of convergence to Pareto-optimal solutions. Experiments on four real-world datasets demonstrate APFEx's superiority, reducing fairness violations while maintaining competitive accuracy. Our work bridges a critical gap in fair ML, providing a scalable, model-agnostic solution for intersectional fairness.

Assessing the Limits of In-Context Learning beyond Functions using Partially Ordered Relation

Generating rational and generally accurate responses to tasks, often accompanied by example demonstrations, highlights Large Language Model's (LLM's) remarkable In-Context Learning (ICL) capabilities without requiring updates to the model's parameter space. Despite having an ongoing exploration focused on the inference from a document-level concept, its behavior in learning well-defined functions or relations in context needs a careful investigation. In this article, we present the performance of ICL on partially ordered relation by introducing the notion of inductively increasing complexity in prompts. In most cases, the saturated performance of the chosen metric indicates that while ICL offers some benefits, its effectiveness remains constrained as we increase the complexity in the prompts even in presence of sufficient demonstrative examples. The behavior is evident from our empirical findings and has further been theoretically justified in term of its implicit optimization process. The code is available \href{https://anonymous.4open.science/r/ICLonPartiallyOrderSet}{here}.