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As I have read Sipser's TOC, on page 372, there is a $EXPSPACE$-complete language, say, $B$, i.e. $B\in$ $EXPSPACE$ and every $L\in$ $EXPSPACE$ is polynomial time reducible to $B$.

I also have read Aaronson's P=?NP survey. He says, on page 46, that "$P^A\neq EXP^A$ for all oracles $A$."

However, could we infer $EXPSPACE\subseteq P^B\subseteq EXP^B\subseteq EXPSPACE$ so $P^B=EXP^B=EXPSPACE$?

$EXPSPACE\subseteq P^B\subseteq EXP^B$ is trivially true. We then show that $EXP^B\subseteq EXPSPACE$ since we could decide a $EXP^B$ language by a certain $EXPSPACE$ algorithm simulating the $EXP$'s steps and the oracle $B$.

I believe that Aaronson's statement is true but what have I missed in my conclusion?

minh quý lê
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We then show that $EXP^B\subseteq EXPSPACE$ since we could decide a $EXP^B$ language by a certain $EXPSPACE$ algorithm simulating the $EXP$'s steps and the oracle $B$.

Queries to $B$ won't necessarily only require exponential space to simulate, since they might be on strings with length exponential in the original input.

spaceisdarkgreen
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