ExampleConstruction: Literature-sourcedVerification: 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.

Adding an unreachable state changes syntax without changing the recognized language

Example

Start with any Turing machine $M$ . Form a new machine $M^{'}$ by adding one extra state $q_{d e a d}$ and no transitions into it.

Facts & Assumptions

Given: A Turing machine $M$ and the modified machine $M^{'}$ .

[L1]

Syntactic properties can change while the recognized language stays fixed, and those properties therefore fall outside Rice's theorem, by Syntactic machine properties lie outside the scope of Rice's theorem.

Verification

technique · direct

1.1

Because no transition ever enters $q_{d e a d}$ , the computation of $M^{'}$ on every input follows exactly the same states, tape symbols, and head moves as the computation of $M$ . Hence $L (M^{'}) = L (M)$ .

givenconstruct

2.1

Even so, $M^{'}$ has a different state count and visibly has an unreachable state. Step 1.1 therefore gives a concrete instance of [L1]: the code-level property changes while the language does not.

L1step 1.1∎

Depends on

Syntactic machine properties lie outside the scope of Rice's theorem

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

EECS 376 Course Notes, Part 6: Computability (standard reference, not scraped)