PropositionStatement: 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.

The Post correspondence problem is recognizable

Statement

The language of PCP instances that have a match is recognizable.

Facts & Assumptions

Given: A PCP instance $(u_{1}, v_{1}), \dots, (u_{n}, v_{n}) .$

[L1]

A yes-instance of PCP is exactly a nonempty finite index sequence $i_{1}, \dots, i_{k}$ with $u_{i_{1}} \dots u_{i_{k}} = v_{i_{1}} \dots v_{i_{k}},$ by The Post correspondence problem.

Proof

technique · direct

1.1

Enumerate all nonempty finite sequences over ${1, \dots, n}$ in length-lexicographic order: $(1), (2), \dots, (n), (1, 1), (1, 2), \dots .$ For each sequence $σ = (i_{1}, \dots, i_{k})$ , compute the two concatenations $u_{i_{1}} \dots u_{i_{k}}$ and $v_{i_{1}} \dots v_{i_{k}}$ and compare them.

L1givenconstruct

2.1

If the instance has a match, [L1] supplies some witnessing sequence $σ$ , and the enumeration from step 1.1 eventually reaches it and accepts. If the instance has no match, every tested sequence fails and the search runs forever.

L1step 1.1

3.1

Therefore there is a Turing machine that accepts exactly the PCP yes-instances, so PCP is recognizable.

step 2.1∎

Depends on

The Post correspondence problem

Used by

Nothing in the library uses this result yet.

Dependency tree · one level

1 result within one dependency step 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

John Watrous, Introduction to the Theory of Computing (standard reference, not scraped)