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 word problem for a fixed finite presentation
Definition
Fix a finite presentation . Its word problem is the decision problem whose input is a word on and whose question is whether represents the identity in the group presented by .
Depends on
Used by
- Algebraic relator area and the Dehn function of a finite presentation Definition
- The conjugacy problem for a finitely generated group Definition
- The uniform word problem for finite presentations Definition
- Todd-Coxeter as a partial coset-enumeration procedure Example
- Solvability of the word problem does not depend on the chosen finite generating set Proposition
- Novikov-Boone: some finitely presented group has unsolvable word problem Remark
- Free products and suitable amalgamated free products have solvable word problem Theorem
- The word problem for a finitely generated free group is solvable by free reduction Theorem
- The word problem for finitely generated abelian groups is solvable Theorem
Dependency tree · two levels
4 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)
- Alex Bishop, Minicourse: On Decision Problems in Groups (standard reference, not scraped)