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The most fundamental unsolved problem in complexity theory is undoubtedly the P=NP problem, which asks (roughly speaking) whether a problem which can be solved by a non-deterministic polynomial-time (NP) algorithm, can also be solved by a deterministic polynomial-time (P) algorithm. The general belief is that , i.e. there exist problems which can be solved by non-deterministic polynomial-time algorithms but not by deterministic polynomial-time algorithms.
One reason why the question is so difficult to resolve is that a certain generalisation of this question has an affirmative answer in some cases, and a negative answer in other cases. More precisely, if we give all the algorithms access to an oracle, then for one choice of this oracle, all the problems that are solvable by non-deterministic polynomial-time algorithms that calls (), can also be solved by a deterministic polynomial-time algorithm algorithm that calls (), thus ; but for another choice of this oracle, there exist problems solvable by non-deterministic polynomial-time algorithms that call , which cannot be solved by a deterministic polynomial-time algorithm that calls , thus . One particular consequence of this result (which is due to Baker, Gill, and Solovay) is that there cannot be any relativisable proof of either or , where “relativisable” means that the proof would also work without any changes in the presence of an oracle.
The Baker-Gill-Solovay result was quite surprising, but the idea of the proof turns out to be rather simple. To get an oracle such that , one basically sets to be a powerful simulator that can simulate non-deterministic machines (and, furthermore, can also simulate itself); it turns out that any PSPACE-complete oracle would suffice for this task. To get an oracle for which , one has to be a bit sneakier, setting to be a query device for a sparse set of random (or high-complexity) strings, which are too complex to be guessed at by any deterministic polynomial-time algorithm.
Unfortunately, the simple idea of the proof can be obscured by various technical details (e.g. using Turing machines to define and precisely), which require a certain amount of time to properly absorb. To help myself try to understand this result better, I have decided to give a sort of “allegory” of the proof, based around a (rather contrived) story about various students trying to pass a multiple choice test, which avoids all the technical details but still conveys the basic ideas of the argument. This allegory was primarily for my own benefit, but I thought it might also be of interest to some readers here (and also has some tangential relation to the proto-polymath project of determinstically finding primes), so I reproduce it below.
The Distinguished Lecture Series at UCLA for this winter quarter is given by Avi Wigderson, who is lecturing on “some topics in computational complexity“. In his first lecture on Wednesday, Avi gave a wonderful talk (in his inimitably entertaining style) on “The power and weakness of randomness in computation“. The talk was based on these slides. He also gave a sort of “encore” on zero-knowledge proofs in more informal discussions after the main talk.
As always, any errors here are due to my transcription and interpretation.