False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-12

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FALSE: nondeterministic acceptance requires every branch to halt

Statement

False claim: a nondeterministic Turing machine accepts an input only when every computation branch on that input halts.

Facts & Assumptions

Given: A nondeterministic machine with states $q_{0}, q_{a c c}, q_{r e j}, q_{L}$ , input alphabet ${1}$ , tape alphabet ${1, ⊔}$ , and allowed transitions $Δ (q_{0}, 1) = {(q_{a c c}, 1, R), (q_{L}, 1, L)}, Δ (q_{L}, 1) = {(q_{L}, 1, L)} .$ Take input word $w = 1$ .

[A1]

The statement refuted is: nondeterministic acceptance requires every branch on the input to halt.

[L1]

A nondeterministic machine accepts an input when there exists an accepting computation on that input, by Accepting computations of a nondeterministic machine.

Refutation

technique · direct

1.1

One allowed first move from the initial configuration is $(q_{a c c}, 1, R)$ . So there is a one-step computation branch that ends immediately in the accept state.

given

1.2

Another allowed first move is $(q_{L}, 1, L)$ . From then on the only allowed move is again $(q_{L}, 1, L)$ , and the left-boundary rule keeps the head at cell $0$ , so this branch never leaves the nonhalting state $q_{L}$ and therefore does not halt.

given

2.1

By [L1] and step 1.1, the machine accepts $w$ because an accepting computation exists. But step 1.2 gives a branch that does not halt. Therefore [A1] is false.

A1L1step 1.1step 1.2∎

Depends on

Accepting computations of a nondeterministic machine

Used by

A branching configuration graph with one accepting path and one looping path Example

Dependency tree · two levels

6 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

John E. Savage, Models of Computation: Exploring the Power of Computing, Chapter 5 (standard reference, not scraped)