Alphabeta MathHow 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 undecidable
Statement
The Post correspondence problem is undecidable.
Facts & Assumptions
Given: The language of PCP instances with a match.
Modified PCP many-one reduces to PCP, by Modified PCP many-one reduces to PCP.
Modified PCP is undecidable, by The modified Post correspondence problem is undecidable.
If and is decidable, then is decidable, by Computable many-one reductions transfer decidability and recognizability backward.
Proof
By [L1], there is a total computable map sending each modified-PCP instance to a PCP instance with the same yes-or-no answer.
If PCP were decidable, then [L3] and step 1.1 would imply that modified PCP is decidable. This contradicts [L2].
Therefore PCP is undecidable.
Depends on
Used by
Dependency tree · two levels
9 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
- MIT OpenCourseWare, Lecture 10: Computation History Method (standard reference, not scraped)
- John Watrous, Introduction to the Theory of Computing (standard reference, not scraped)