D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXP-time. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, pages 13--18, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....time. They also show that every language accepted by these proof systems lie in NEXP, nondeterministic exponential time. In 1990, Babai, Fortnow and Lund [BFL91] show the surprising converse that every language in NEXP has probabilistically checkable proofs. Babai, Fortnow, Levin and Szegedy [BFLS91] scale this proof down to develop holographic proofs for NP where, with a properly encoded input, the veri er can check the correctness of the proof in very short amount of time. Feige, Goldwasser, Lov asz, Safra and Szegedy [FGL 96] made an amazing connection between probabilistically ....

....problems. Arora [Aro98] after failing to achieve lower bounds for traveling salesman in the plane, has developed a polynomial time approximation algorithm for this and related problems. A series of results due to Cai, Condon, Lipton, Lapidot, Shamir, Feige and Lov asz [CCL92, CCL90, CCL91, Fei91, LS91, FL92] have modi ed the protocol of Babai, Fortnow and Lund [BFL91] 10 to show that every language in NEXP has a two prover, one round proof systems with an exponentially small error. This problem remained so elusive because running these proof systems in parallel does not have the expected ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXP-time. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 13-18. IEEE, New York, 1991.

....proofs over a large alphabet in which number of alphabets that a verifier is allowed to probe is a parameter. This concept is an important ingredient in the recursive construction of probabilistically checkable proofs [AS92, ALMSS92, BGLR93] and is also of independent interest in complexity theory [LS91, FL92a]. The original definition of probabilistically checkable proofs is due to [AS92] based on an implicit notion in [FGLSS91] A very closely related notion that of holographic proofs appears in the work of [BFLS91] The particular choice of parameters made in the following definition is due to ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 13--18, 1991.

....rather than polynomial time ones) The available constant prover proof systems appear in Figure 5 and are discussed below. Throughout this discussion we consider proof systems obtaining an arbitrary small constant error probability. The two prover proofs of Lapidot Shamir and Feige Lov asz [LaSh, FeLo] had poly logarithmic randomness and answer sizes, so [ALMSS] used a modification of these, in the process increasing the Due to Provers Coins Answer size Canonical Can be made canonical [LaSh, FeLo] 2 polylog polylog No Yes [BeSu] ALMSS] poly(ffl ) log polylog No [BGLR] 4 log ....

....small constant error probability. The two prover proofs of Lapidot Shamir and Feige Lov asz [LaSh, FeLo] had poly logarithmic randomness and answer sizes, so [ALMSS] used a modification of these, in the process increasing the Due to Provers Coins Answer size Canonical Can be made canonical [LaSh, FeLo] 2 polylog polylog No Yes [BeSu] ALMSS] poly(ffl ) log polylog No [BGLR] 4 log polyloglog No [Tar] 3 log O(1) No [FeKi1] 2 log O(1) No At cost of one more prover [BeSu] Raz] 2 log O(1) Yes (NA) Figure 5: Constant prover PCPs achieving error which is a fixed, but arbitrarily ....

[Article contains additional citation context not shown here]

D. Lapidot and A. Shamir. Fully parallelized multi-prover protocols for NEXP-time. Proceedings of the 32nd Symposium on Foundations of Computer Science, IEEE, 1991, pp. 13--18.

....system Psi as described above, to an equation system e Psi with the desired properties. We use two techniques iteratively in the construction of e Psi. The arithmetization technique from [BFL91] summarized in lemma 13, and the curve extension technique, which is similar to [ALM 92, LS91, FL92] and is formalized in lemma 16. We also use the linearization technique, described in lemma 17. Aside from having different parameters, these techniques have a similar structure. They receive an equation system as input, and substitute each equation in it with a set of new equations, that ....

....The properties of e Psi are maintained (the weight of the system relative to any assignment may change by at most a factor by this transition, as shown in the proof of proposition 28) 4. 3 The Curve Extension Lemma This lemma uses a technique called curve extension, similar to [ALM 92, LS91, FL92] to reduce the principal degree and dimension of the input system. Lemma 16 (Curve Extension) The curve extension lemma is obtained from the lemma template by setting its C The parameters of the input system should satisfy r yes ; r no ; d; D jFj such that d Delta r yes D r ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 13--18, 1991.

....(2) For I L, for any possible protocol, the verifier accepts with probability smaller than #. For the exact definitions of MIP(2, 1) proof systems and the family of languages MIP(2, 1) see [12, 23] The class of languages MIP(2, 1) turned out to be very powerful. In particular, it follows from [7, 37, 25] that NEXPTIME = MIP(2, 1) with exponentially small probability of error. MIP(2, 1) proof systems have cryptographic applications (see [12, 13, 36, 18] and have also been used as a starting point to prove that certain optimization problems are hard to approximate (see [22, 4, 5, 25, 6, 10, 38, ....

....in analyzing the probability of error of a parallel repetition, not only as a mathematical problem, but also because an e#cient technique to decrease the probability of error was needed. In the literature there are many results that use di#erent techniques to decrease the probability of error [19, 33, 37, 25, 8, 23, 43]. For certain applications, however, these techniques are insu#cient. Parallel repetition was suggested as a technique to decrease the probability of error, because it was believed to be very e#cient, and because it preserves many canonical properties of the proof system (e.g. zero knowledge) ....

D. LAPIDOT AND A. SHAMIR, Fully parallelized multi prover protocols for NEXP-time, in Proc. FOCS 1991, IEEE Computer Society Press, Los Alamitos, CA, pp. 13--18.

....most . Parameters. The parameters of interest are p, r 1 and l = jjCjj l 0 . Our transformation applies to any canonical proof system. The above theorems are obtained by plugging in speci c canonical proof systems as we now describe. The only property missing to make the proof system of [19, 14] canonical is that the second prover s answers are not expressible as a projection of those of the rst. This is simple to x. To see how, recall that the rst prover sends a sequence of polynomials A 1 ; A d : F F where d = O(jF j log 2 n) and F is the underlying eld. These ....

....the analysis a better analysis of the linearity tests, due to [10] which, although known before [7] seems to have been forgotten by them; we even reduce to 13, from the 15 in [7] the number of 3SAT clauses needed to write the quadratic test. A very skimpy description follows. We rst use the [19, 14] proof system to get a canonical two prover proof system as discussed in Section 3.1. We then convert this into a Max 3SAT by expressing the task of the extended veri er by a 3cnf formula: A linearity test requires 4 clauses, a quadraticity test 13, an output test 2 and an input test requires 4 ....

D. Lapidot and A. Shamir. Fully Parallelized Multiprover protocols for NEXP-time. FOCS 91.

....Details of the proof are omitted from this abstract, but they are obvious from reading our proofs (specifically, by noting their algorithmic nature) Past work on constant prover proof systems. The first construction of a nontrivial constant prover 1 round proof system for NP appeared in [23]; others appeared in [16, 10, 34, 14, 27] These systems could not reduce the error probability to below a constant while using O(logn) random bits (the best construction needs O(k log n) random bits to make the error probability 2 Gammak ; see [27] It was also known [15] that some obvious ....

D. LAPIDOT AND A. SHAMIR. Fully Parallelized Multi-prover protocols for NEXP-time. IEEE FOCS 1991.

....every NP language has one round interactive proofs with con dence 1 with a bounded number p of provers, where the veri er uses r = O(log n j log j) random bits and the answer of each prover has length a. The values of the parameters p and a are critical for the applications. Lapidot, Shamir [LS] obtained this result with p = 4 provers and a = poly(log n; log ) answer size; Feige, Lov asz [FL] reduced the number of provers The author was visiting the Automation and Computation Institute of the Hungarian Academy of Sciences, DIMACS Center and the University of Toronto while part of ....

....p 1 and all checkpoints T 2 t P rob R2 r Z(x; R; T ; P 1 ; P p ) 62 fg(R) T ) rejectg The above de nition of the MQS is neither stronger nor weaker than that of the MES. At an MQS the decoding depends on the random string R (this is similar to the de nition of a quasi oracle in [LS]) But the decoding does not depend on the rst p 1 provers. Below all MQS have p = 2 provers unless otherwise stated. 4 Existence of the encoding schemes The main result of this section is a scaled down version of [FL] to achieve a MQS for any polynomial function family F . We start with a ....

[Article contains additional citation context not shown here]

D. Lapidot and A. Shamir. Fully parallelized multi prover protocol for NEXPTIME. In: Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science, 1991, 13-18.

....random line in F m and verify that the restriction of f to this line agrees signi cantly with some univariate degree d polynomial. If this is the case, accept. This test is similar in avor to all other known low degree tests, such as the original tests in [BLR90, BFL91] and later ones in [BFLS91, FGLSS91, GLRSW91] Our interest in the test of [RS96] stems from the fact that it has a potential to yield something meaningful about the function being tested, even when the function passes the test with very low probability. In contrast, several of the other tests above, e.g. those in [BFL91, ....

....FGLSS91, GLRSW91] Our interest in the test of [RS96] stems from the fact that it has a potential to yield something meaningful about the function being tested, even when the function passes the test with very low probability. In contrast, several of the other tests above, e.g. those in [BFL91, BFLS91, FGLSS91] explicitly test the degree of the polynomial in each variable and consequently are unable to say anything useful unless the function passes the test with probability 1 O(1=m) However the best prior analysis of this test was not strong enough to match the potential performance of the ....

[Article contains additional citation context not shown here]

D. Lapidot and A. Shamir. Fully Parallelized Multi-prover protocols for NEXP-time. Journal of Computer and System Sciences, 54(2):215-220, April 1997.

....proofs over a large alphabet in which number of alphabets that a veri er is allowed to probe is a parameter. This concept is an important ingredient in the recursive construction of probabilistically checkable proofs [AS92, ALMSS92, BGLR93] and is also of independent interest in complexity theory [LS91, FL92a]. The original de nition of probabilistically checkable proofs is due to [AS92] based on an implicit notion in [FGLSS91] A very closely related notion that of holographic proofs appears in the work of [BFLS91] The particular choice of parameters made in the following de nition is due to ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 13-18, 1991.

....Sudan [90] The in uence of the former will be apparent in Section 5, while the latter work (together with a lemma of Arora and Safra [6] is used in our analysis of a Low Degree Test described in Section 7.2. Finally, we were in uenced by work on constant prover 1 round interactive proof systems [74, 45]. In fact, our de nition of an outer veri er (De nition 8) may be viewed as a generalization of the de nition of such proof systems, and our Theorem 4 provides the rst known construction of a 2 prover 1 round proof system that uses logarithmic random bits and constant number of communication ....

.... accept] e: In both cases, the probability is over the choice of R in f0; 1g r(jxj) We note that an (r(n) p(n) c(n) a(n) outer veri er with p(n) O(1) is very similar to a constant prover 1 round interactive proof system [46] Fairly ecient constructions of such veri ers are implicit in [74, 45]; for instance, it is shown that NP [c 1RPCP( log) c n; O(1) log) c n; log) c n; 1=n) We also observe that the de nition of RPCP generalizes that of PCP. In particular, RPCP(r(n) p(n) c(n) a(n) 1=2) PCP(r(n) p(n)a(n) since an (r(n) p(n) c(n) a(n) outer veri er examines ....

D. Lapidot and A. Shamir. Fully parallelized multi-prover protocols for NEXP-time. Journal of Computer and System Sciences, 54(2):215-220, April 1997.

....system, the proof consists of p arrays of letters from #. The verifier is only allowed to query 1 letter from each array. Since each letter of # is represented by #log # # bits, the number of bits queried may be viewed as p #log # #. Constructions of such proof systems for NP appeared in [30, 96, 52, 27, 50, 113]. Lund and Yannakakis [108] used these proof systems to prove inapproximability results for SETCOVER and many subgraph maximization problems. The best construction of such proof systems is due to Raz and Safra [114] They show that for each k # # log n, every NP language has a verifier that uses ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 13--18, 1991.

....[90] The influence of the former will be apparent in Section 5, while the latter work (together with a lemma of Arora and Safra [6] is used in our analysis of a Low Degree Test described in Section 7.2. Finally, we were influenced by work on constant prover 1 round interactive proof systems [74, 45]. In fact, our definition of an outer verifier (Definition 8) may be viewed as a generalization of the definition of such proof systems, and our Theorem 4 provides the first known construction of a 2 prover 1 round proof system that uses logarithmic random bits and constant number of communication ....

....e. In both cases, the probability is over the choice of R in 0, 1 r( x ) We note that an (r(n) p(n) c(n) a(n) outer verifier with p(n) O(1) is very similar to a constant prover 1 round interactive proof system [46] Fairly e#cient constructions of such verifiers are implicit in [74, 45]; for instance, it is shown that NP # # c #RPCP( log) c n, O(1) log) c n, log) c n, 1 n) We also observe that the definition of RPCP generalizes that of PCP. In particular, RPCP(r(n) p(n) c(n) a(n) 1 2) # PCP(r(n) p(n)a(n) since an (r(n) p(n) c(n) a(n) outer verifier ....

D. Lapidot and A. Shamir. Fully parallelized multi-prover protocols for NEXPtime. Journal of Computer and System Sciences, 54(2):215-220, April 1997.

....Note that membership proofs for NP have polynomial size, so a straightforward meaning of scaling down would require the running time of the verifier to be polylogarithmic. But this cannot be, since the verifier must take linear time simply to read the input. Babai, Fortnow, Levin, and Szegedy [BFLS91] nevertheless obtained a scale down result (including a scale down in the running time of the verifier) by changing the model of computation: in the new model, the input has to be provided to the verifier in an encoded form using a specific error correcting code. The authors showed that in this ....

....This Paper The notion of efficiently checkable membership proofs is inherently interesting because it represents a new way of looking at classical complexity classes such as NP. The notion becomes even more intriguing in light of the possible trade offs, hinted at in the results of [FGL 91, BFLS91] between the verifier s running time, random bits, and query bits. If these trade offs can be improved, improved nonapproximability results for the clique problem follow, as we will soon see. To facilitate the study of such trade offs, we define below a hierarchy of complexity classes PCP (for ....

[Article contains additional citation context not shown here]

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 13--18, 1991.

....every single paper in the area. Second, the results we get this way are stronger. And finally, we believe that this way the proof is more precise, and easier to verify (although it is still very hard) Our proof starts off with the general scheme of [AS92] as amended (following the technique of [LS91] improved in [FL92] by [ALM 92] This scheme, however, is insufficient when sub constant error probability is sought after. Further alterations are therefore required, in order to allow undisturbed flow of such small error probability between various parts of the proof. The specific ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 13--18, 1991.

....PCP characterization of NP [RS97, DFK ] where each test depends on a constant ( 2) number of variables. Whether or not such a characterization exists remains an open question. Note that it is highly unlikely that this problem is in P since such an interactive proof protocol for NP exists [LS91, FL92, Raz98] with a quasi polynomial blow up. For proving the hardness of approximating the Closest Vector problem, this obstacle was bypassed [DKS98] by showing NP hardness for a different problem called SSAT . SSAT could then be used instead of depend 2 PCP for proving NP hardness for ....

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXPTIME. In Proc. 32nd IEEE Symp. on Foundations of Computer Science, pages 13--18, 1991.

No context found.

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXP-time. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, pages 13--18, 1991.

No context found.

D. Lapidot and A. Shamir. Fully Parallelized Multi-prover protocols for NEXP-time. Proceedings of the Thirty Second Annual Symposium on the Foundations of Computer Science, IEEE, 1991.

No context found.

D. Lapidot and A. Shamir. Fully Parallelized Multi-prover protocols for NEXP-time. Proceedings of the Thirty Second Annual Symposium on the Foundations of Computer Science, IEEE, 1991.

No context found.

D. Lapidot, A. Shamir. "Fully parallelized multi-prover protocols for NEXP-time". JCSS 54(2), 215--220, 1997.

No context found.

Lapidot, D., and Shamir, A. Fully parallelized multi prover protocols for NEXP-time (extended abstract). In Proc. 32nd IEEE Symp. on Foundations of Comp. Science (San Juan, Puerto Rico, 1-4 Oct. 1991), pp. 13-18.

No context found.

Lapidot, D., and Shamir, A. Fully parallelized multi prover protocols for NEXP-time (extended abstract). In Proc. 32nd IEEE Symp. on Foundations of Comp. Science (San Juan, Puerto Rico, 1-4 Oct. 1991), pp. 13-18.

No context found.

D. Lapidot and A. Shamir. Fully Parallelized Multi-prover protocols for NEXP-time. Proceedings of the Thirty Second Annual Symposium on the Foundations of Computer Science, IEEE, 1991.

No context found.

D. Lapidot and A. Shamir. Fully Parallelized Multi-prover protocols for NEXP-time. Proceedings of the Thirty Second Annual Symposium on the Foundations of Computer Science, IEEE, 1991.

No context found.

D. Lapidot and A. Shamir. Fully parallelized multi prover protocols for NEXP-time. In Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, pages 13-18, 1991.

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC