False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Relativized separations prove unrelativized separations

Statement

False inference schema: a separation $P^{B} \neq N P^{B}$ for one total oracle can be transferred independently of the oracle, yielding $P^{A} \neq N P^{A}$ for every total oracle $A$ (and in particular for the empty oracle).

Facts & Assumptions

Given: the objects and hypotheses in the statement above.

[F1]

There is a total oracle $B$ for which $P^{B} \neq N P^{B}$ . In fact $L_{B} = {1^{n} : B \cap {0, 1}^{n} \neq \emptyset}$ lies in $N P^{B} ∖ P^{B}$ . (An oracle separates p from np).

[F2]

For $A = T Q B F$ , $P^{A} = N P^{A} = P S P A C E$ . (An oracle collapses p and np).

Refutation

1.1

There exists a total $B$ with $P^{B} \neq N P^{B}$ , so the premise of the proposed rule is realized.

2.1

For $A = T Q B F$ , however, $P^{A} = N P^{A}$ . This violates the universally quantified conclusion and refutes the rule. The isolated implication from the separating world to $P \neq N P$ is not here called a known false proposition: determining its truth would require settling the unrelativized question.

F2step 1.1∎

Remark

The refutation supplies total oracles $B$ and $A = T Q B F$ with $P^{B} \neq N P^{B}$ and $P^{A} = N P^{A}$ , respectively. These are counterinstances to the universal transfer rule; no conclusion about the empty oracle follows.

Depends on

Used by

Relativized separations prove unrelativized separations Counterexample

Dependency tree · two levels

4 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

Arora–Barak, Computational Complexity, 2007 draft; §3.5 Theorem3.9 and relativization discussion, pp71–72. (standard reference, not scraped)