ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-12

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.

Checking a short accepting history for a unary increment machine

Example

Consider the deterministic machine with states $q_{0}, q_{a c c}, q_{r e j}$ , input alphabet ${I}$ , tape alphabet ${I, ⊔}$ , and transition rules $δ (q_{0}, I) = (q_{0}, I, R), δ (q_{0}, ⊔) = (q_{a c c}, I, R) .$ On input $I I I$ it scans right across the unary block, writes one extra I in the first blank cell, and halts.

Using the configuration-word convention with B for the blank symbol, a candidate accepting history is $q_{0} I I I, I q_{0} I I, I I q_{0} I, I I I q_{0} B, I I I I q_{a c c} B .$

Facts & Assumptions

Given: The machine and input above, together with the displayed candidate history.

[L1]

For a fixed machine and input, encoded accepting computation histories form a decidable language by local checks on the start configuration, successive one-step moves, and the final accepting state, by For a fixed machine and input, valid accepting computation histories form a decidable language.

[L2]

A halting computation history is a finite list beginning at the initial configuration and following the one-step relation at each stage, by Finite computation histories, halting computations, and divergence.

Verification

technique · direct

1.1

The first configuration word $q_{0} I I I$ is the canonical initial configuration on input $I I I$ : the head is at the leftmost I, the state is $q_{0}$ , and the last displayed cell is the last nonblank input cell.

given

1.2

Each of the first three transitions just moves the head one cell to the right while leaving the scanned I unchanged, so $q_{0} I I I \to I q_{0} I I \to I I q_{0} I \to I I I q_{0} B$ is legal. The last transition uses $δ (q_{0}, ⊔) = (q_{a c c}, I, R)$ , so the machine writes I in the scanned blank cell and then moves right onto the next blank cell. Therefore $I I I q_{0} B \to I I I I q_{a c c} B$ is legal as well.

given

2.1

The final configuration word has state $q_{a c c}$ , so it is accepting. Therefore the displayed list is an accepting computation history in the sense of [L2], exactly the sort of object whose validity [L1] decides.

L1L2step 1.1step 1.2∎

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

Michael Sipser, MIT 18.404J Theory of Computation, Lecture 10: The Computation History Method (standard reference, not scraped)