CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Standard semantic properties such as emptiness, finiteness, regularity, and context-freedom are undecidable

Statement

The following decision problems about a Turing machine's recognized language are undecidable:

whether the language is empty,
whether it is finite,
whether it is regular,
and whether it is context-free.

Facts & Assumptions

Given: The four language properties listed in the statement.

[L1]

Rice's theorem says that every nontrivial extensional property of recognizable languages has an undecidable index set, by Every nontrivial extensional property of Turing-machine languages is undecidable.

[L2]

An extensional property depends only on the recognized language, not on the particular machine code, by Index sets and extensional properties of Turing-machine languages.

Proof

technique · direct

1.1L2given

Each listed property is extensional in the sense of [L2]: if two machines recognize the same language, then either both languages are empty, both are finite, both are regular, or both are context-free.

1.2givenconstruct

Each property is nontrivial. For emptiness, $\emptyset$ belongs and ${0}$ does not. For finiteness, ${0}$ belongs and ${0^{n} : n \geq 0}$ does not. For regularity, $\emptyset$ belongs and the standard recognizable language ${0^{n} 1^{n} : n \geq 0}$ does not. For context-freedom, ${0^{n} 1^{n} : n \geq 0}$ belongs and the standard recognizable language ${0^{n} 1^{n} 2^{n} : n \geq 0}$ does not.

2.1L1step 1.1step 1.2∎

Therefore [L1] applies to each of the four properties, so each corresponding index set is undecidable.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

H. G. Rice, Classes of Recursively Enumerable Sets and Their Decision Problems (standard reference, not scraped)
MIT 6.045J / 18.400J, Lecture 9: Mapping Reducibility and Rice's Theorem (standard reference, not scraped)