An Order-Theoretic Characterization of Consistent Inductive Inference

2026-09-23

AI Summary

This study addresses the theoretical challenge posed by Lu (2024) regarding the characterization of consistent inductive inference for arbitrary binary hypothesis classes without requiring uniform error bounds. Working within the ZFC framework, we characterize consistency through well-foundedness analysis and linear orderings—specifically the Kleene-Brouwer ordering—over finite executable trajectories. A learner is constructed via conflict-trajectory selection differentials, complemented by an error-driven evidence-decay mechanism that facilitates both its construction and reverse derivation. This work provides the first representation of consistent prediction grounded in finite evidence, thereby fully resolving the aforementioned open problem. Furthermore, it establishes a novel order-theoretic analytical paradigm for computational learning theory.

Abstract

When can a learner make only finitely many prediction errors along every infinite sequence labeled by a fixed, unknown hypothesis? We characterize this form of consistency for arbitrary binary hypothesis classes in ZFC, without requiring a uniform mistake bound. The characterization uses a single linear order on finite realizable traces. Each trace selects its least subtrace, and the order must satisfy two conditions: conflicting traces select different subtraces, and the order is well-founded on the traces of each fixed target. These conditions induce a learner whose selected evidence decreases on every mistake. Conversely, a consistent learner yields such an order through canonical mistake transcripts and the Kleene--Brouwer ordering. The result provides a representation of consistent prediction by finite evidence, answering a question of Lu (2024).

Problem

• Inductive Inference
• Consistent Prediction
• Binary Hypothesis Classes
• Order-Theoretic Characterization
• Prediction Errors

Innovation

• Inductive Inference
• Order-Theoretic Characterization
• Well-Founded Order
• Consistent Prediction
• Kleene-Brouwer Ordering