CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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Enumerating relator consequences does not decide nontriviality

Statement refuted

Enumerating all relator consequences of a recursive presentation decides nontriviality of words in that presentation.

Facts & Assumptions

Given: The recursive presentation $⟨ a ∣ \emptyset ⟩$ and the word $a$ .

[L1]

The trivial words of a recursively presented group form a recursively enumerable language. (The trivial words of a recursively presented group form a recursively enumerable language)

[L2]

A reduced nonempty word in a free group is nonidentity. (The word problem for a finitely generated free group is solvable by free reduction)

Counterexample

technique · direct

1.1

The presentation $⟨ a ∣ \emptyset ⟩$ presents the free group on $a$ . The word $a$ is reduced and nonempty, so [L2] says that it is nontrivial.

L2given

2.1

By [L1], enumerating relator consequences lists precisely the trivial words. Since $a$ is nontrivial by step 1.1, it never appears, and waiting for it gives no halting certificate of nontriviality.

L1step 1.1

3.1

Hence the asserted decision procedure is false, and the claimed principle is refuted.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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)