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The statement $A = B \rightarrow \forall \; O \quad ; A^O = B^O$ is incorrect. For example, it is known that $\mathrm{IP} = \mathrm{PSPACE}$, but we know that there exists an oracle $A$ such that $\mathrm{IP}^A \ne \mathrm{PSPACE}^A$. Direct substituion does not work here, because different characterisations of one particular class can behave different when we relativize them. |
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The statement $A = B \rightarrow \forall \; O \; quad A^O = B^O$ is incorrect. For example, it is known that $IP \mathrm{IP} = PSPACE$, \mathrm{PSPACE}$, but we know that there exists an oracle $A$ such that $IP^A \mathrm{IP}^A \ne PSPACE^A$.\mathrm{PSPACE}^A$. Direct substituion does not work here, because different characterisations of one particular class can behave different when we relativize them. |
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The statement $A = B \rightarrow \forall \; O \; A^O = B^O$ is incorrect. For example, it is known that $IP = PSPACE$, but we know that there exists an oracle $A$ such that $IP^A \ne PSPACE^A$. Direct substituion does not work here, because different characterisations of one particular class can behave different when we relativize them. |
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