Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-12
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FALSE: Rice's theorem applies to having at most a fixed number of states

Statement

Rice's theorem applies to the property "the machine has at most a fixed number of states."

Facts & Assumptions

Given: A fixed state bound k.

[L1]

State-count bounds are syntactic machine properties rather than extensional language properties, by Syntactic machine properties lie outside the scope of Rice's theorem.

[L2]

Rice's theorem applies only to nontrivial extensional properties of recognized languages, by Every nontrivial extensional property of Turing-machine languages is undecidable.

Refutation

technique · direct
1.1

By [L1], the property "has at most k states" can change when one rewrites the machine without changing its recognized language. So it is not an extensional language property.

L1given
2.1

Because the extensionality hypothesis of [L2] fails, Rice's theorem does not apply to this property. Therefore the statement is false.

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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