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Abstract:

It has been shown that the class of languages with interactive proofs, IP, is exactly the class PSPACE. This surprising result elegantly places IP in the standard classification of feasible computations. Furthermore, the IP = PSPACE result reveals some very interesting and unsuspected properties of mathematical proofs. In this column we define the width of a proof in a formal system F and show that it is an intuitively satisfying and robust definition. Then, using the IP = PSPACE result, it is seen that the width of a proof (as opposed to the length) determines how quickly one can give overwhelming evidence that a theorem is provable without showing the full proof. 1 On Proofs and Interactive Proofs A mathematician has the most confidence in the truth of a theorem when he/she is given a complete proof of the theorem in a trusted formal system. Let F be such a formal system in which the correctness of a proof can be checked by a verifier in polynomial time. The class NP clearly captures all the theorems which have polynomially long proofs. The NP =? P question is the question about the quantitative computational difference between finding a proof of a theorem and checking the correctness of a given proof. Some years ago, theoretical computer scientists asked whether it is possible to give convincing evidence that a theorem is provable in F without showing a complete

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