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Oracles are used in complexity theory for lower and upper bounds in the decision tree model. Using oracles you can tell if the proof for certain problems (like P vs NP) relativizes or not. If it relativizes then you cannot use a proof based solely on diagonalization (like the halting problem) to solve it. Can also tell you about bad approaches to the problem. A good book to read about this is chapter 3 of Computational Complexity: A Modern Approach (here is the draft). A beautiful paper about oracles in complexity theory was written by Fortnow.

Some people like oracles and some don't. I particularly like them, because lower bounds are very hard to proof in a general Turing machine. Lower bounds in at least a restricted model is a step towards a real lower bound. I believe the best lower bounds we truly understand are AC0 circuits.

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Oracles are used in complexity theory for lower and upper bounds in the decision tree model. Using oracles you can tell if the proof for certain problems (like P vs NP) relativizes or not. If it relativizes then you cannot use a proof based on diagonalization (like the halting problem) to solve it. I Can also tell you about bad approaches to the problem. A good book to read about this is chapter 3 of Computational Complexity: A Modern Approach (here is the draft). A beautiful paper about oracles in complexity theory was written by Fortnow.

Some people like oracles and some don't. I particularly like them, because lower bounds are very hard to proof in a general Turing machine. Lower bounds in at least a restricted model is a step towards a real lower bound. I believe the best lower bounds we truly understand are AC0 circuits.

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Oracles are used in complexity theory for lower and upper bounds in the decision tree model. Using oracles you can tell if the proof for certain problems (like P vs NP) relativizes or not. If it relativizes then you cannot use a proof based on diagonalization (like the halting problem) to solve it. I good book to read about this is chapter 3 of Computational Complexity: A Modern Approach (here is the draft).

Some people like oracles and some don't. I particularly like them, because lower bounds are very hard to proof in a general Turing machine. Lower bounds in at least a restricted model is a step towards a real lower bound. I believe the best lower bounds we truly understand are AC0 circuits.