Hallucination, abstention, and computable inseparability

TL;DR

This paper explores the limitations of abstention in formal systems, linking it to classical undecidability and inseparability theorems, and analyzes the trade-offs between avoiding hallucination and coverage.

Contribution

It formally characterizes the limitations of abstaining systems using arithmetical hierarchy separation and computability theory, extending classical undecidability results.

Findings

Abstention can trivially avoid hallucination if allowed on all inputs.

The size of guaranteed correct answers is limited by inseparability in the arithmetical hierarchy.

There is a computability-theoretic trade-off between abstention and coverage.

Abstract

The impossibility of eliminating hallucination, understood here as incorrect definite answers, in sufficiently expressive yes-or-no formal domains is an immediate consequence of classical undecidability theorems. This note does not revisit that forced-answer obstruction as its main claim. Instead, it attempts to formally describe the corresponding limitation for abstaining systems. Abstention can trivially avoid hallucination if the system is allowed to abstain on every input; the substantive question is how large the domain of guaranteed correct non-abstaining answers can be. We formulate this question using separation in the arithmetical hierarchy. Given disjoint sets $A$ and $B$, any system that answers Yes on all queries indexed by $A$ and No on all queries indexed by $B$ induces a separator of $A$ from $B$. By combining this observation with the classical existence theorem of $\Delta_{n}^{0}$-inseparable pairs of $\Sigma_{n}^{0}$-sets, we yield a computability-theoretic trade-off between avoiding hallucination by abstention and maintaining a large domain of guaranteed coverage.