Doubts about Baker-Gill-Solovay

Question

How am I supposed to read the P=?NP relativization proof? I am reading the classical paper Relativization of the P=?NP problem by Baker, Gill and Solovay, in particular the proof that there exist an oracle $B$ such that $\mathsf{P}^B \neq \mathsf{NP}^B$ on page 436. I have some questions and your help will be appreciated.

• The procedure does not seem to be an oracle but a deterministic algorithm.

• Is this construction a counterexample to P = NP?

• What does this mean "Run query machine $P_i$ with oracle $B_i$ on input $x_i = 0^n$?" Does it mean that $P_i$ asks

  1. if $B_i$ accepts $0^n$
  2. if $B_i$ accepts any string of length $n$
  3. ONE BY ONE if $B_i$ accepts a string of length $n$ from the canonical enumeration?

• I assume that the set $B$ or $B_i$ is initially empty. Does it mean that the FIRST string of length $n$ from the canonical enumeration will always be added?

Answer 1

Let me answer your concrete questions:

• An oracle is a language. One way to describe a language is to give a procedure for constructing it.
• The construction is not a counterexample to P=NP. It shows that there exists an oracle relative to which P and NP are different. It also gives another oracle relative to which they are the same. Later, it was shown that P is different from NP relative to a random oracle (almost surely); recently this was extended to the entire polynomial hierarchy. The modern interpretation of such results is that diagonalizing techniques (techniques that would also work relative to an arbitrary oracle) cannot be used to settle the P vs NP question.
• "Run query machine $P_i$ with oracle $B_i$ on input $x_i = 0^n$" has the following meaning. The machine $P_i$ is an oracle machine — it is a Turing machine that has a special mechanism allowing it to access the oracle. We run $P_i$ on the input $0^n$, i.e., the string of length $n$ consisting entirely of zeroes. Whenever $P_i$ makes an oracle access, we answer according to $B_i$.
• Your final question is a bit specific. I suggest trying to read the proof again, and if you fail, check a more modern account. Note that the result can be proved in several ways, so a proof appearing in a different source might be different.