H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In C. Pueach and R. Reischuk, editors, 13th Annual Symposium on Theoretical Aspects of Computer Science, number 1046 in Lecture Notes in Computer Science, pages 75-86. Springer, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... interactive proof systems [GMW91, CCL94, BFL91] probabilistically checkable proof systems [BFLS91, AS92b, ALM 92] proof systems with verifiers of restricted complexity [Con93, AAI 97] as well as a considerable body of work on various structural aspects of proof systems [BKT94, JT95, BT96, FFNR96] see [Sel96, JT97] for recent surveys) 0.1 Partially publishable proof systems In this paper, we study proof systems for NP from the following viewpoint. Let L be a language in NP, and let RL (x; y) be a polynomial time decidable binary predicate such that for all x, x 2 L iff IBM ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In Proc. 13th Annual Symposium on Theoretical Aspects of Computer Science, volume 1046 of Lecture Notes in Computer Science, pages 75--86. Springer-Verlag, 1996. 11

....output different assignments for different random bits and therefore does not strictly compute a function. The existence of such a randomized machine computing a function in FSAT is still an open problem. In spite of the fact that Theorem 5. 1 holds for almost every oracle, Buhrman and Thierauf [BT96] have obtained a relativization in the opposite direction. Using results about exponential time, they have constructed an oracle A such that FP NP A tt FSAT = From Theorems 5.1 and 5.2, one might get the idea that if some solution for an NP complete problem like SAT can be computed in FP ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. Proceedings of the 13th Annual Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science 1046, pp. 75--86, Springer-Verlag, Berlin, 1996.

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H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In C. Pueach and R. Reischuk, editors, 13th Annual Symposium on Theoretical Aspects of Computer Science, number 1046 in Lecture Notes in Computer Science, pages 75-86. Springer, 1996.

....a relativized world where P = UP 6= NP but the proof heavily uses the fact that the UP machines must have one accepting path for all inputs. An easy application of Lemma 3.1 allows one to randomly find a satisfying assignment of a formula making nonadaptive queries to SAT. Buhrman and Thierauf [BT96] give a relativized world where this fails deterministically. Theorem 1.4 gives the first relativized counterexample to Hypothesis 1.2. In fact Theorem 1.4 shows a stronger result. Buss and Hay [BH91] and Wagner [Wag90] show that languages computable with a polynomial number of nonadaptive ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science, volume 1046 of Lecture Notes in Computer Science, pages 75--86. Springer, Berlin, 1996.

....a formula can be enumerated in output polynomial time. We also give straightforward proofs that BPP is in Sigma p 2 (first proven by G acs (see [Sip83] and create relativized worlds where assignments to SAT cannot be found with non adaptive queries to SAT (first proven by Buhrman and Thierauf [BT96]) and where EXP = NEXP but there exists a NEXP machine whose accepting paths cannot be found in exponential time (first proven by Impagliazzo and Tardos [IT89] These results in their original form require a great deal of time to fully understand the proof because either the ideas and or ....

....version of SAT A [GJ93] by adding a series of extra predicates A 0 ; A 1 ; A 2 ; such that A n (x 1 ; x n ) is true if x 1 : x n is in A. Toda and Watanabe [WT93] showed that relative to a random oracle F sat T FP NP tt 6= On the other hand Buhrman and Thierauf [BT96] showed that there exists an oracle where F sat T FP NP tt = Their result also holds relative to the set constructed in Theorem 21. Theorem 36 Relative to the set A constructed in Theorem 21, F sat T FP NP tt = Proof: For some n, let OE be the formula on n variables such that OE(x) ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In C. Pueach and R. Reischuk, editors, 13th Annual Symposium on Theoretical Aspects of Computer Science, number 1046 in Lecture Notes in Computer Science, pages 75--86. Springer, 1996.

....one can compute the size of the largest clique of a given graph. Then, with one more query, one can find out whether there is more than one clique of that size. Papadimitriou and Zachos [PZ83] asked whether UOCLIQUE is complete for P NP[log] This is still an open problem. Buhrman and Thierauf [BT96] provide strong evidence that UOCLIQUE is not complete for P NP[log] UOCLIQUE can be disjunctively reduced to USAT [BT96] To see this let UCLIQUE be the unique CLIQUE version. That is, given a graph G and a integer k, decide whether G has a unique clique of size k. Now, observe that G 2 ....

....more than one clique of that size. Papadimitriou and Zachos [PZ83] asked whether UOCLIQUE is complete for P NP[log] This is still an open problem. Buhrman and Thierauf [BT96] provide strong evidence that UOCLIQUE is not complete for P NP[log] UOCLIQUE can be disjunctively reduced to USAT [BT96]. To see this let UCLIQUE be the unique CLIQUE version. That is, given a graph G and a integer k, decide whether G has a unique clique of size k. Now, observe that G 2 UOCLIQUE ( 9k : G; k) 2 UCLIQUE. Since the Cook reduction is parsimonious, this provides a disjunctive reduction of UOCLIQUE to ....

H. Buhrman, T. Thierauf. The complexity of generating and checking proofs of membership. In Symposium on Theoretical Aspects of Computer Sience (STACS) '96 Springer Verlag, Lecture Notes in Computer Sience 1046, 75-87,1996.

....formula can be enumerated in output polynomial time. We also give straightforward proofs that BPP is in Sigma p 2 (first proven by G acs (see [Sip83] and create relativized worlds where assignments to SAT cannot be found with non adaptive queries to SAT (first proven by Buhrman and Thierauf [BT96]) and where EXP = NEXP but there exists a NEXP machine whose accepting paths cannot be found in polynomial time (first proven by Impagliazzo and Tardos [IT89] These results in their original form require a great deal of time to fully understand the proof because either the ideas and or ....

....tt 6= if and only if for all OE 2 SAT there exists a satisfying assignment a of OE such that CND p (a j OE) c log(jOEj) for some polynomial p and constant c. Toda and Watanabe [WT93] showed that relative to a random oracle F sat T FP NP tt 6= On the other hand Buhrman and Thierauf [BT96] showed that there exists an oracle where F sat T FP NP tt = Their result also holds relative to the set constructed in Theorem 4.2. Theorem 6.4 Relative to the set A constructed in Theorem 4.2, F sat T FP NP tt = Proof: For some n, let OE be the formula on n variables such that ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In C. Pueach and R. Reischuk, editors, 13th Annual Symposium on Theoretical Aspects of Computer Science, number 1046 in Lecture Notes in Computer Science, pages 75--86. Springer, 1996.

....proof of Lemma 2.9 and find the assignment in FP. 2 Corollary 5.7 If NP = coNP and FP NP jj = FP NP[log] then P = NP. This argument shows that it is actually sufficient to prove that FP NP jj = FP NP[log] implies that some satisfying assignment can be found in FP NP jj . See [WT93, BT96a] for this question. Watanabe and Toda show that relative to a random oracle it is the case that some satisfying assignment can be found in FP NP jj . However relative to a random oracle all of the six hypotheses fail (see Section 7) 6 Unique SAT is in P The hypotheses Unique SAT is in P is ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In C. Puech and R. Reischuk, editors, 13th Annual Symposium on Theoretical Aspects of Computer Science, volume 1046 of Lecture Notes in Computer Science, pages 75--86. Springer, 1996.

....a relativized world where P = UP 6= NP but the proof heavily uses the fact that the UP machines must have one accepting path for all inputs. An easy application of Lemma 3.1 allows one to randomly find a satisfying assignment of a formula making nonadaptive queries to SAT. Buhrman and Thierauf [BT96] give a relativized world where this fails deterministically. Theorem 1.4 gives the first relativized counterexample to Hypothesis 1.2. In fact Theorem 1.4 shows a stronger result. Buss and Hay [BH91] and Wagner [Wag90] show that languages computable with a polynomial number of nonadaptive ....

H. Buhrman and T. Thierauf. The complexity of generating and checking proofs of membership. In Proceedings of the 13th Symposium on Theoretical Aspects of Computer Science, volume 1046 of Lecture Notes in Computer Science, pages 75--86. Springer, Berlin, 1996.

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