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Safe Language Generation in the Limit

Antonios Anastasopoulos, Giuseppe Ateniese, Evgenios M. Kornaropoulos

TL;DR

The paper addresses the foundational problem of safe language generation in the limit, formalizing safe generation and safe identification under learning from positive and negative examples. It proves that safe language identification is impossible and that safe generation is at least as hard as standard language identification, which is also impossible, using adversarial constructions and reductions. The results reveal a fundamental tension: safety tasks can be strictly harder than their non-safety counterparts in the limit, suggesting that practical safety may require restricted, implementable notions. The work also identifies a tractable regime where infinite set differences between safe and harmful languages enable KM-style generation, while noting broad intractability in general, guiding future theory and real-world safety design.

Abstract

Recent results in learning a language in the limit have shown that, although language identification is impossible, language generation is tractable. As this foundational area expands, we need to consider the implications of language generation in real-world settings. This work offers the first theoretical treatment of safe language generation. Building on the computational paradigm of learning in the limit, we formalize the tasks of safe language identification and generation. We prove that under this model, safe language identification is impossible, and that safe language generation is at least as hard as (vanilla) language identification, which is also impossible. Last, we discuss several intractable and tractable cases.

Safe Language Generation in the Limit

TL;DR

The paper addresses the foundational problem of safe language generation in the limit, formalizing safe generation and safe identification under learning from positive and negative examples. It proves that safe language identification is impossible and that safe generation is at least as hard as standard language identification, which is also impossible, using adversarial constructions and reductions. The results reveal a fundamental tension: safety tasks can be strictly harder than their non-safety counterparts in the limit, suggesting that practical safety may require restricted, implementable notions. The work also identifies a tractable regime where infinite set differences between safe and harmful languages enable KM-style generation, while noting broad intractability in general, guiding future theory and real-world safety design.

Abstract

Recent results in learning a language in the limit have shown that, although language identification is impossible, language generation is tractable. As this foundational area expands, we need to consider the implications of language generation in real-world settings. This work offers the first theoretical treatment of safe language generation. Building on the computational paradigm of learning in the limit, we formalize the tasks of safe language identification and generation. We prove that under this model, safe language identification is impossible, and that safe language generation is at least as hard as (vanilla) language identification, which is also impossible. Last, we discuss several intractable and tractable cases.
Paper Structure (33 sections, 6 theorems, 13 equations, 2 figures, 1 table, 2 algorithms)

This paper contains 33 sections, 6 theorems, 13 equations, 2 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Related Work
  4. Language Identification in the Limit.
  5. Language Generation in the Limit.
  6. On Refined Notions of Success.
  7. Why Is This Work Different.
  8. Safe Language Identification is Impossible
  9. The Languages.
  10. The Language Collections.
  11. Phase 1 of Construction of Enumeration $E$.
  12. Phase 2 of Construction of $E$.
  13. Inductive Argument.
  14. The Safe Generation Problem
  15. On the Emptiness of Set Difference.
  16. ...and 18 more sections

Key Result

Theorem 3.1

For any safe language identification algorithm $A^\mathcal{SI}$, there exists a countable ${\cal X}$, a pair of language collections ${\cal L}_{{K}}$ and ${\cal L}_{{H}}$, and a choice of ${K}$ and ${H}$, such that $A^\mathcal{SI}$ can not safely identify from ${K}$ given a harmful ${H}$ in the limi

Figures (2)

  • Figure 1: An illustration of the collection of ${\cal L}_{{K}}$ and ${\cal L}_{{H}}$ for the constructed instantiation for which there is no timestep in which an algorithm identifies the correct language ${K} \setminus {H}$ consistently. Let $I$ denote the set of integers, $E$ the set of even integers, and $O$ the set of odd integers.
  • Figure 2: An illustration of collections in which each phase seemingly has infinite safe strings, but the enumerated languages $(L^{{K}}_{i^{{K}}},L^{{H}}_{i^{{H}}})$ have no safe strings.

Theorems & Definitions (15)

  • Definition 1: Safe Identification $\mathcal{SI}$
  • Theorem 3.1
  • proof
  • Definition 2: Safe Generation $\mathcal{SG}$
  • Theorem 5.1
  • Definition 3: Safe Generation - Relaxed $\mathcal{SG}^R$
  • proof
  • Theorem 6.1
  • proof
  • Corollary 6.1
  • ...and 5 more