Ravi B. Boppana, Johan Hastad, and Stathis Zachos. Does co-np have short interactive proofs? Inf. Process. Lett., 25(2):127--132, 1987. 12  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[Sip83] showed that this implies NP to have small circuits. By Karp and Lipton [KL82] it follows that the polynomial time hierarchy collapses to its second level. If BPP path SBP then we get coNP BPP path SBP AM from the Theorems 2.7 and 3.8. The result of Boppana, Hastad, and Zachos [BHZ87] shows that coNP AM implies a collapse of the polynomial time hierarchy to its second level. 2 Corollary 5.2 There exists an oracle A such that BPP 6= SBP 6= BPP path . Proof: This holds since theorem 5.1 is relativizable and since there exists a relativized world where the ....

....relativized worlds, SBP AM 2 , and a collapse of the polynomial time hierarchy is implied by 2 2 . 2 Remember that AM contains classes like NP;BPP;MA, and it is unlikely that AM is contained in 2 . So in this light AM seems to be quite powerful. However, Boppana, Hastad and Zachos [BHZ87] showed that unless the polynomial time hierarchy collapses AM (and therefore also SBP) is not powerful enough to contain coNP. Together with Yao s oracle this has the following consequence. 17 Theorem 5.11 ( Yao85, BHZ87] There exists an oracle A such that coNP . Proof: Yao [Yao85] ....

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R. B. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127--132, 1987.

....of ac tive research, no polynomial time algorithm for it has yet been found. At the same time, while clearly belonging to NP, no proof has been provided that it is NP complete. Indeed, there is strong evidence that this cannot be the case, for otherwise the polynomial hierarchy would collapse [14, 52]. The current belief is that the problem lies strictly between the P and NPcomplete classes. The subgraph isomorphism problem is more general and in fact more difficult, being NP complete [19] Given two graphs, it is the prob lem of determining whether one is isomorphic to a subgraph of the ....

BOrrANA, R. B., HASTAD, J., AND ZACHOS, S.: 'Does co-NP have short interactive proofs?', Inform. Process. Left. 25 (1987), 127-132.

....Mansour, Sipser and Zachos [FGM 89] showed that one can assume that for positive instances the prover can succeed with no error. Goldreich, Micali and Wigderson [GMW91] show that the set of pairs of nonisomorphic graphs has a bounded round interactive proof system. Boppana, H astad and Zachos [BHZ87] show that if the complement of any NP complete language has bounded round interactive proofs than the polynomial time hierarchy collapses. This remains the best evidence that the graph isomorphism 9 problem is probably not NP complete. In 1990, Lund, Fortnow, Karlo and Nisan [LFKN92] showed ....

R. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127-132, 1987.

....knowledgecomplexity of their interactive proof systems. The main result known (for the above definition) is that languages with logarithmic statistical knowledge complexity are in AM coAM (cf. 79] building on [1] and [60] Thus, unless the polynomial time hierarchy collapses (cf. [20]) NPcomplete set have super logarithmic statistical knowledge complexity. In general, it seems that quantitative notions are harder to handle than qualitative ones. 11 Resettability of a party s random tape (rZK and rsZK) Having gained a reasonable understanding of the security of ....

R. Boppana, J. Hastad, and S. Zachos. Does Co-NP Have Short Interactive Proofs? Information Processing Letters, 25, May 1987, pp. 127-132.

....supports the latter observation might be he graph ismorphism problem, GI . Here, the equivalence problem for graphs, which is in fact an equality problem, is trivially solvable in polynomial time. Therefore GI is in NP. But GI is not NP complete, unless the polynomial hierarchy collapses [GMW91, BHZ87] see also [Sch89] For a restricted class of branching programs, the one time only branching programs , where, on each path, each variable is tested at most once (see next section for precise definitions) the equivalence problem is easier then for general ones: it can be efficiently solved by ....

....programs, we have 1 BPI 2 NP Delta coRP. An obvious question is whether 1 BPI is NP hard. In this paper, we show that the problem to decide whether two one time only branching programs are not isomorphic, 1 BPNI, is in BP Delta NP. Combined with the result of Boppana, Hastad, and Zachos [BHZ87] see also Schoning [Sch89] it follows that 1 BPI cannot be NP hard, unless the polynomial hierarchy collapses to its second level, Sigma p 2 . This result covers also the case of ordered branching programs. Note however that here, the isomorphism problem is in NP. 2.4 Arithmetic Circuits ....

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Boppana, Hastad, and Zachos. Does co-NP have short interactive proofs? IPL: Information Processing Letters, 25:27--32, 1987.

....between two sets of multivariate polynomials equations. In this section, we prove that the Deciding IP problem is not NP complete, under the classical hypothesis that the so called polynomial time hierarchy does not collapse. The proof is based on the following general results (see [9] and [3] for proofs) Theorem 6.1 (Goldwasser, Sipser) If a problem has a constant round interactive proof, then it also has a constant round Arthur Merlin protocol. 8 Theorem 6.2 (Boppana, Hastad, Zachos) If the complement of a problem # has an Arthur Merlin protocol with a constant number of rounds, ....

Ravi B. Boppana, Johan Hastad, Stathis Zachos, Does co-NP have short interactive proofs, Information Proc. Letters, vol. 25, 1987, pp. 127-132.

....A such that relative to A the class AM is not a subset of PP, i.e. AM A 6 PP A . Thus, AM is not a subset of BPP path relative to A. On the other hand, BPP path is not a subset of AM unless the polynomial hierarchy collapses. This follows from the result of Boppana, Hastad, and Zachos [BHZ87] that if coNP AM then the polynomial hierarchy collapses to its second level. Since coNP BPP path , we get the same consequence from the assumption that BPP path is contained in AM. Sipser and G acs ( Sip83] see also [Lau83] showed that BPP R NP . It is an open question whether the ....

R. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25:127--132, 1987. 26

....between the knowledge complexity and the computational complexity of languages. We show that languages with logarithmic knowledge complexity are in AM co AM. This result has a very interesting implication on languages in NP. Recall that if NP coAM then the polynomial time hierarchy collapses [BHZ 87] Assuming that the polynomial time hierarchy does not collapse, we get that NP complete languages do not have logarithmic knowledge complexity. Prior to our result, there was no indication that would contradict all NP languages having knowledge complexity 1. Note that, if a one way function ....

....equivalently, have a constant round interactive proof. There is no restriction on the knowledge complexity of this constant round interactive proof. See [BM 88, GS 89] for de nitions of Arthur Merlin proofs, for some basic properties, and for the equivalence of the de nitions. It was shown in [BHZ 87] that if NP co AM then the polynomial time hierarchy collapses. It is believed that the polynomial time hierarchy does not collapse, and under this assumption, our result implies that NP complete languages do not have logarithmic knowledge complexity. Prior to this result, there was no ....

R. Boppana, J. H astad and S. Zachos. Does co-NP Have Short Interactive Proofs. Information Processing Letters, Vol 25 (1987), No. 2, pp 127-132.

....[BM88] to study the power of randomization in interaction. Goldwasser and Sipser [GS89]proved soon afterwards that these classes are equivalent in power to Interactive Proof Systems introduced by [GMR85] In the last 15 years, this study has proved to be exceedingly successful in complexity theory [ZH86, BHZ87, ZF87, LFKN90, Sh90]. Eventually the study of these proof systems (and multi prover systems) led to perhaps the most spectacular achievement in the Theory of Computing in the last decade [BGKW88, LFKN90, Sh90, BFL91, BFLS91, FGLSM91, AS92, ALMSS92] It is well known that some traditional complexity classes can be ....

R. Boppana, J. Hastad and S. Zachos, Does co-NP have short interactive proofs?, Information Processing Letters, 25:127-132, 1987.

....do not distinguish between them. The graph isomorhism problem is clearly in the class of NP, and it is not known whether it is in P. It is also unknown whether the problem is NP complete, but this seems to be unlikely since it would imply that the polynomial hierarchy collapses to its second level [12]. In fact, evidence that graph isomorphism is not NP complete was given already by Mathon [70] who showed that the decision problem for graph isomorphism and its counting version (i.e. the problem to compute the number of isomorphisms of two given graphs) are polynomial time Turing equivalent. ....

Ravi Boppana, Johan Hastad, and Stathis Zachos. Does co-NP Have Short Interactive Proofs? Information Processing Letters, 25:127--132, May 1987.

....oracle to C. It is easy to see that any problem in NP which is low for some level of the polynomial time hierarchy (PH) is not NP complete unless PH collapses. Thus, lowness of a problem for some level of PH is an evidence that the problem is unlikely to be complete for NP. It is shown in [Sch88, BHZ87] that GI is low for Sigma p 2 , the second level of PH. Since the introduction of this notion the study of lowness of problems to various complexity classes has been of much interest (see [Kob95] for a survey on lowness) 1.1 Counting complexity classes and lowness Among various complexity ....

R. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25:127--132, 1987.

.... then the complement of the language has a bounded round interactive proof, i.e. SZK coIP [2] From this theorem we can deduce that it is unlikely that SZK contains all of NP since if NP SZK then co NP IP [2] which further implies that the polynomial time hierarchy collapses to IP [2] by [BHZ]. This is especially interesting because it implies that membership in SZK can be taken as evidence that a language is not NP complete. A particularly important example of this is Graph Isomorphism, which was shown to be in SZK by [GMW] While Fortnow s result did imply that SZK is probably ....

Boppana R., J. Hastad and S. Zachos "Does co-NP Have Short Interactive Proofs", Information Processing Letters, Vol 25 (1987), No. 2, pp 127--132.

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Ravi B. Boppana, Johan Hastad, and Stathis Zachos. Does co-np have short interactive proofs? Inf. Process. Lett., 25(2):127--132, 1987. 12

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Ravi B. Boppana, Johan Hastad, and Stathis Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127--132, 1987.

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R. Boppana, J. Hastad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25:127-132, 1987.

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Ravi Boppana, Johan Hastad, and Spyros Zachos. Does coNP have short interactive proofs? Information Processing Letters, 25:127--132, 1987.

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R. Boppana, J. Hastad, and S. Zachos. Does coNP have short interactive proofs? Inf. Process. Lett., 25:127--132, 1987.

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R. B. Boppana, J. H astad, and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127--132, 1987.

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Ravi B. Boppana, Johan Hastad, and Stathis Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25:127-132, 1987.

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Ravi Boppana, Johan Hastad, and Stathis Zachos. Does co-NP Have Short Interactive Proofs? Information Processing Letters, 25:127--132, May 1987.

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R. Boppana, J. Hastad, and S. Zachos. Does co-np have short interactive proofs? Information Processing Letters, 25(2):127--132, 1987.

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R. Boppana, J. Hastad, S. Zachos. Does co-NP have short interactive proofs?, Manuscript, 1986.

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R.B. Boppana, J. Hastad and S. Zachos. Does co-NP have short interactive proofs? Information Processing Letters, 25(2):127-132, 1987.

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R. B. Boppana, J. Hastad and S. Zachos. Does coNP have short interactive proofs? Inform. Process. Lett. 25 (1987), 127-132.

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R. Boppana, J. Hastad, and S. Zachos. Does Co-NP Have Short Interactive Proofs? IPL, 25, May 1987, pp. 127-132.

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