DefinitionDefinition: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Boone machine semigroup and augmented configurations

Definition

Let $T$ be the fixed machine of Normalized fixed halting machine for boone simulation, with tape alphabet $S$ , blank $s_{0}$ , states $Q$ , start $q_{s t a r t}$ and stop $q_{0}$ . Take new, mutually distinct symbols $h, q$ . Set $A = S ⊔ Q ⊔ {h, q}$ . Words and concatenation have the convention of Computation alphabets, words, the empty word, and $Σ^{*}$ . On the nonempty words $A^{+}$ use the following equations, oriented as displayed:

Instruction of $T$	Relations (for every $s_{β} \in S$ )
write $s_{k}$ , enter $q_{b}$ , from $(q_{a}, s_{j})$	$q_{a} s_{j} = q_{b} s_{k}$
move right, enter $q_{b}$ , from $(q_{a}, s_{j})$	$q_{a} s_{j} s_{β} = s_{j} q_{b} s_{β}$ ; $q_{a} s_{j} h = s_{j} q_{b} s_{0} h$
move left, enter $q_{b}$ , from $(q_{a}, s_{j})$	$s_{β} q_{a} s_{j} = q_{b} s_{β} s_{j}$ ; $h q_{a} s_{j} = h q_{b} s_{0} s_{j}$

Add $q_{0} s = q_{0}$ and $s q_{0} h = q_{0} h$ for every $s \in S$ , and $h q_{0} h = q$ . Erasure never ranges over $h$ .

An elementary replacement substitutes either side of an equation for the other inside arbitrary contexts $u, v \in A^{*}$ . Write $a \sim b$ when a finite sequence of such symmetric replacements joins them; the sequence of length zero is allowed. Define $Γ = A^{+} / \sim$ with multiplication $[a] [b] = [a b]$ , using the quotient-set convention of Equivalence relation, equivalence class, and the quotient set $A / \sim$ . Its well-definedness is verified below.

The augmented configurations are $q$ and words $h U q_{a} V h$ , where $U, V \in S^{*}$ , $q_{a} \in Q$ , and $V \neq ε$ unless $q_{a} = q_{0}$ . The initial word is $C (w) = h q_{s t a r t} v (w) h$ . The symbol $q$ has no outgoing oriented rule.

For subsequent algebra put $\overset{ˉ}{S} = S ⊔ {h}$ and $\overset{ˉ}{Q} = Q ⊔ {q}$ . Index the finite relation list by $I$ and write each relation as $F_{i} q_{a (i)} G_{i} = H_{i} q_{b (i)} K_{i},$ where $F_{i}, G_{i}, H_{i}, K_{i} \in {\overset{ˉ}{S}}^{*}$ , possibly empty, and both sides have exactly one state letter. This change of notation adds no erasure rule.

Facts & Assumptions

Given: The displayed relation list and contextual replacement on nonempty words.

[F1]

The machine has finite tape and state sets and a finite table. (Normalized fixed halting machine for boone simulation)

[F2]

Concatenation joins finite words, including the empty word. (Computation alphabets, words, the empty word, and $Σ^{*}$ )

[F3]

A reflexive, symmetric, transitive relation defines the quotient set of equivalence classes. (Equivalence relation, equivalence class, and the quotient set $A / \sim$ )

[F4]

A set with an associative binary operation is a semigroup. (Semigroup and monoid)

Verification

1.1

Every table entry contributes at most $∣ S ∣ + 1$ transition equations, and cleanup contributes $2 ∣ S ∣ + 1$ . Every side is nonempty and contains exactly one state letter, including $q$ on the final right side. Thus replacements stay in $A^{+}$ and the list is finite.

F1given

2.1

The length-zero path proves $a \sim a$ . Reversing a path proves symmetry because each replacement permits both directions. Joining two finite paths with common endpoint proves transitivity. Hence $\sim$ is an equivalence relation and its classes form a set.

F3step 1.1

3.1

Prefixing and suffixing every word of a path by fixed words prefixes and suffixes every replacement context. Therefore $a \sim a^{'}$ implies $u a v \sim u a^{'} v$ . If also $b \sim b^{'}$ , first use the path from $a$ to $a^{'}$ with suffix $b$ , then the path from $b$ to $b^{'}$ with prefix $a^{'}$ , obtaining $a b \sim a^{'} b^{'}$ . Thus $[a] [b] = [a b]$ is independent of both representatives.

F2step 2.1

4.1

Word concatenation is associative: both $(a b) c$ and $a (b c)$ list the letters of $a$ , then $b$ , then $c$ at the same indices. Consequently $([a] [b]) [c] = [(a b) c] = [a (b c)] = [a] ([b] [c])$ . Products stay nonempty, so this is the claimed semigroup. Empty contexts and one-letter words cause no exception; an empty word is not adjoined as an identity element.

F2F4step 3.1∎

Source conventions

Rotman, printed pp.426–427, gives the eight relation families. The augmented domain permits empty right tape during stopping cleanup, repairing the nonempty-right-tape assertion in his Lemma 12.3.

Depends on

Used by

Boone group presentation and special word Definition
Boone halting cleanup through an empty right tape Example
Boone sharp is not word inversion Example
Boone hnn tower and auxiliary subgroups Lemma
Boone positive history reconstruction Lemma
Boone semigroup histories detect halting Lemma
Fixed finite semigroup has undecidable terminal equality Theorem
Novikov boone undecidability of the word problem Theorem

Dependency tree · two levels

20 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

Joseph J. Rotman, An Introduction to the Theory of Groups, Chapter 12, pp.426–427, associated semigroup presentation (standard reference, not scraped)