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.
Recursive presentations and finite presentations of groups
Definition
Let be a finite generating alphabet and let be a set of relator words on .
- The presentation is recursive when the language of words representing the members of is recursively enumerable.
- It is finite when itself is finite, equivalently when it is one of the finite presentations of Relators and relations; finitely generated, finitely related, and finite presentations.
The group presented is the quotient from Group presentation by generators and relations.
Depends on
Used by
- The isomorphism problem for a class of finite presentations Definition
- The word problem for a fixed finite presentation Definition
- FALSE: recursively enumerable trivial words already give a decision algorithm False statement
- The trivial words of a recursively presented group form a recursively enumerable language Lemma
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
- Charles F. Miller III, Decision Problems for Groups - Survey and Reflections (standard reference, not scraped)
- Dexter Chua after H. Wilton, Topics in Geometric Group Theory (standard reference, not scraped)