S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science, pages 2-13, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....had unexpected pay offs in other areas as well. Some of the more significant pursuits along these lines include the work on interactive proof systems [GMR89, Bab85, BM88, LFKN92] multiple prover 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 ....

S. Arora and S. Safra. Probabilistic checking of proofs. In Proc. 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 2--13, 1992.

.... in the other (cf. the surveys [J] B] Two major results in the rst area assert that (i) every NP language has transparent proofs veri able with con dence 1 using r = O(log n j log( j) random bits via O(j log( j) bit queries (Arora, Lund, Motwani, Sudan, Szegedy [ALMSS] Arora, Safra [AS]) ii) 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. ....

....of each of these polynomials in deciding whether to accept or to reject. It is important here that the veri er knows which values it will use before it receives the answers. This makes it possible to replace this prover by two provers providing a quasi encoding scheme for this answer. This is an [AS] type recursion, increasing the number of provers to 3. Lemma 7. BGLR] Let k = O(log n) and h = max(k; log log n) Then there are parameters r = O(k log n) a = O(k log n2 h ) and q = O(k log n) such that NP MIP 1 (r; 2; a; q; 2 k ) We shall need further details from the protocol proving ....

S. Arora and S. Safra. Probabilistic checking of proofs. In: Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, 1992, 2-13.

....the central question in complexity theory, namely, whether or not P is equal to NP. The technical mechanisms that allow to prove such strong results come from work on probabilistically checkable proofs and, most notably, from the probabilistic characterization of NP, the well known PCP theorem [3, 4]. It is natural to ask whether or not the approximability of non NP hard problems can be determined using similar techniques. Certain natural problems, such as Tournament Dominating Set and computing the VapnisChervonenkis (V C) Dimension, and many restricted versions of NP complete problems have ....

....Dominating Set can be approximated to the ratio log n in time O(2 n ) for any 1. It is still an open question whether the NP hard optimization problems Dominating Set or Clique can be approximated to some constant ratio in time O(2 n ) for some 1. However, it has been shown that [3, 4, 5, 15] that they do not adopt polynomial time approximation algorithms achieving any constant ratio unless P=NP. These results (even with worse ratios) have been obtained through 5 the connection between recently developed probabilistic characterizations of NP languages and the inapproximability of ....

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S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 2-13. IEEE Computer Society Press, October 1992.

....an approximate solution should be easier than finding the exact solution. While this assumption holds for the usual list of approximable problems (Bin Packing, Euclidean Travelling Salesman Problem, Set Cover, Subset Sum, etc. recent results on probabilistically checkable proofs [FGL 91, AS92, ALM 92] have shown that for the CLIQUE problem, even coarse approximations are not achievable in polynomial time unless P = NP. Furthermore, these non approximability results extend to other NP complete problems; most notably to Graph Coloring, Set Cover, MAX3SAT and all MAXSNP complete ....

S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of the IEEE Symposium on Foundations of Computer Science, pages 2--13, 1992.

....can exist unless some standard intractability assumption is violated (e.g. P = NP or the polynomial hierarchy (PH) collapses) Recently, Arora et al. 3] showed that the problem of finding the largest clique in a graph is in the latter category. Following a series of breakthrough results [4, 5, 18, 26, 29], they showed that there exists a constant # such that no deterministic polynomial time algorithm can approximate the maximum clique size #(G) of a graph G with n vertices within a factor of n # unless P = NP. While this result strongly suggests that no e#cient algorithm can find good ....

S. Arora and S. Safra, Probabilistic checking of proofs, in Proc. IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1992, pp. 2--13.

.... Department of Computer Science and Engineering, FR 35 University of Washington Seattle, WA 98195 Time Space Tradeoffs for Graph s t Connectivity by Gregory Barnes A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 1992 Approved by (Chairperson of Supervisory Committee) Program Authorized to Offer Degree Date In presenting this dissertation in partial fulfillment of the requirements for the Doctoral degree at the University of Washington, I agree that the Library shall make its copies freely available for ....

.... ustcon algorithm be made to estimate the path length from s to t Is estimating the length of an undirected shortest path to a certain accuracy equivalent to knowing the exact path length The recent work on approximation algorithms for NP complete problems by Arora and Safra, and Arora et al. AS92, ALM 92] gives a new characterization of NP. Perhaps work on approximation problem for NL could yield a similar new understanding of NL. Finally, it is worthwhile to consider the apparent differences between undirected and directed s t connectivity, as outlined in Section 4.3. While it does ....

S. Arora and S. Safra. Probabilistic checking of proofs. In 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, PA, October 1992. IEEE.

....applications, and to isolate unique elements, was observed by Naor and Naor [NN90] see also [ABN 92] It was also known to Tarui [Tar92] in a non uniform context. Large error correcting codes of constant density are integral to the recent work on transparent (holographic) proofs [BFLS91, AS92, ALM 92, Sud92, Bab93, BGLR93] Indeed, the code in the next section is a known simplification of the code originally used by Babai et.al. BFLS91] see Sudan [Sud92] and the Acknowledgments below. Ours is the first formulation we know of the coding idea in the context of improving the bounds ....

S. Arora and S. Safra. Probabilistic checking of proofs. In The Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 2--13, 1992.

....the central question in complexity theory, namely, whether or not P is equal to NP. The technical mechanisms that allow to prove such strong results come from work on probabilistically checkable proofs and, most notably, from the probabilistic characterization of NP, the well known PCP theorem [3, 4]. It is natural to ask whether or not the approximability of non NP hard problems can be determined using similar techniques. Certain natural problems, such as Tournament Dominating Set and computing the VapnisChervonenkis (V C) Dimension, and many restricted versions of NP complete problems have ....

....the ratio fi log n in time O(2 n ffi ) for any fi 1. It is still an open question whether the NP hard optimization problems Dominating Set, Hypergraph Vertex Cover, or Clique can be approximated to some constant ratio in time O(2 n ffi ) for some ffi 1. However, it has been shown that [3, 4, 5, 15] that they do not adopt polynomial time approximation algorithms achieving any constant ratio unless P=NP. These results (even with worse ratios) have been obtained through the connection between recently developed probabilistic characterizations of NP languages and the inapproximability of these ....

[Article contains additional citation context not shown here]

S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 2--13. IEEE Computer Society Press, October 1992.

....algorithms for the maintenance of approximate solutions for optimization problems whose decision version is NP complete. Generally speaking, such a topic has not been sufficiently studied, in spite of the fact that approximation algorithms are acquiring more and more relevance [19] 18] 1] [2]. In contrast to this general situation, the Min Bin Packing problem is an example of an NP complete problem previously considered in terms of the maintenance of approximate solutions [10] 11] 9] 24] 25] 16] Another important computationally intractable problem that has been ....

S. Arora, S. Safra. Probabilistic checking of proofs. Proceedings 33rd IEEE Symposium on Foundations of Computer Science (1992),2--132--13

....of this quantity for a maximization problem. 4 to P = NP is true. Early results on non approximability [53, 161] were sporadic and used very specialized techniques. The recent connection drawn between results from the study of interactive proof systems from communication complexity theory [5, 6, 41, 110] and proving non approximability results has provided a unified way to prove such results. Another tool that has been useful in applying non approximability results is a strengthening of the notion of an NP hardness reduction called an L reduction [122] An L reduction reduces an NP hard problem ....

S. Arora, and S. Safra, "Probabilistic checking of proofs," Proc. of the 33rd Annual IEEE Symp. on Foundations of Computer Science, (1992) pp. 2-13.

....level is chosen to be faulty by an adversary. This bound is within a constant factor of the complexity of arbitrary symmetric functions (that depend on all n inputs) in the fault free case [96] 6 There is a connection between our model and recently discovered constructions for proof encodings [10, 11] (also see [13] for a survey) We show that from certain constructions of fault tolerant circuits the theorem of Arora and Safra [10] NP = PCP (log n; log n) follows. Our results verify the existence of such fault tolerant circuits. However, combining these results we do not get a simpler proof ....

.... (that depend on all n inputs) in the fault free case [96] 6 There is a connection between our model and recently discovered constructions for proof encodings [10, 11] also see [13] for a survey) We show that from certain constructions of fault tolerant circuits the theorem of Arora and Safra [10], NP = PCP (log n; log n) follows. Our results verify the existence of such fault tolerant circuits. However, combining these results we do not get a simpler proof to the characterization of NP in [10] our construction uses an even harder result stating that NP = PCP (log n; 1) 11] The above ....

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S. Arora and S. Safra, "Probabilistic checking of proofs," In Proc. of "33-rd IEEE Symposium on the Foundations of Computer Science", 1992, pp. 2-13. 64 65

....and NISZK come from relating the corresponding complete problems. This general theme of using completeness to simplify the study of a class, rather than as evidence for computational intractability (as is the traditional use of NP completeness) has been evidenced in a number of recent works (cf. [18, 27, 34, 1, 2]) and has been particularly useful in understanding statistical zero knowledge (cf. 32, 33, 11, 21] 1.1 The non interactive model Let us recall the definition of a non interactive statistical zero knowledge proof system from [6] 2 We will adapt the definition to promise problems. Note ....

Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs. In Proceedings of the Thirty Third Annual Symposium on Foundations of Computer Science, pages 2--13, 1992.

.... hash functions and limited independence of random variables, and shows that poly(S) random bits do not add any power to space(S) bounded machines Another complexity theoretic pursuit that has made significant use of quasirandom constructions is the area of probabilistically checkable proofs [AS92b, ALM 92] The breakthrough results of Arora et al. ALM 92] uses a combination of efficient quasirandom constructions some of them studied earlier in coding theory with limited independence. The essence of their result can be summarized by the following example. If j(x 1 ; x n ....

S. Arora and S. Safra. Probabilistic checking of proofs. In Proc. 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 2--13, 1992.

....probabilistic games, proof systems, PSPACE AMS subject classification. 68Q15 PII. S0097539793260738 1. Introduction. Recently, there has been great progress in understanding the precision with which one can approximate solutions to NP hard problems e#ciently. Feige et al. 13] Arora et al. [2, 3], and others proved strong negative results for several fundamental problems such as Clique and Satisfiability. This progress has led to renewed study of approximation algorithms for PSPACE hard problems. Not surprisingly, PSPACE hard problems also display a wide variation in the precision with ....

....that these same problems have polynomial time approximation schemes if and only if NP = P. In [10] we considered optimization versions of several other problems in PSPACE, including Quantified Satisfiability, Generalized Geography, and Finite Automata Intersection. Building on the techniques of [2, 3, 13], we showed that it is in fact PSPACE hard to approximate these problems closely (where closely depends on the problem) Using direct reduction arguments, Hunt et al. 14] showed that some generalized quantified satisfiability problems (for example, satisfiability of quantified formulas in which ....

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S. Arora and M. Safra, Probabilistic checking of proofs, in Proc. 33rd Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1992, pp. 2--13; final version, J. Assoc. Comput. Mach., to appear.

....[Imp88, For94] By definition, oracles tell us the limits of what can be proved by relativizable techniques. This is useful to researchers, because it tells us when not to give up on relativizable techniques (e.g. Imm88, Sze88, BRS95] and when an utterly novel technique will be required (e.g. [LFKN92, Sha92, BFL91, AS92]) We really want to know what happens in the real world, and oracles don t answer this question. People have studied special classes of oracles such as random [BG81] and sparse [BBS86] but neither do these reflect what happens in the real world in general [Kur83, BBS86] Despite all this, ....

Sanjeev Arora and Shmuel Safra, Probabilistic checking of proofs, Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science, 1992, pp. 2--13.

....Our contribution is in providing a more e cient execution environment for detecting misbehavior by hosts participating in Metacomputing infrastructures. A promising approach to improving the guarantees provided by our system involves the application of techniques such as those of Arora and Safra [6] for devising probabilistically checkable proofs (PCP) to the veri cation of the transitions between checkable units. We plan to pursue this approach by creating redundancy in the traces so that checking a reasonably small number of the augmented pieces of the proof of execution presented by a ....

S. Arora and S. Safra. Probabilistic Checking of Proofs. In Proceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 2-13. IEEE Computer Society Press, October 1992.

....membership of OE in 3 SAT. Furthermore, since 3 SAT is NP complete and since Cook s reduction [7] is witness preserving, this can be used as a procedure to obtain robust proofs for membership in any NP language. Note the difference between these robust proofs and PCP proofs such as the ones in [3, 4]. In our case an adversary may choose the bits that we get to see and still we must be able to reconstruct the full witness, whereas the verifier of a PCP proof chooses the bits of the proof that he reads (usually randomly) and only has to verify the validity of the proof. 1 1.1 Summary of ....

S. Arora and S. Safra, Probabilistic checking of proofs, In Proc. of "33-rd IEEE Symposium on the Foundations of Computer Science", 1992, pp. 2-13.

....complete problems. This general theme of using completeness to simplify the study of a class, rather than as evidence for computational intractability (as is the traditional use of NP completeness) has been evidenced in a number of recent works (cf. GMW91, LFKN90, Sha90, ALM 92, AS92] and has been particularly useful in understanding statistical zero knowledge (cf. SV97, SV99, DDPY98, GV99] 1.1 The non interactive model Let us recall the definition of a non interactive statistical zero knowledge proof system from [BDMP91] 3 We will adapt the definition to promise ....

Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs. In Proceedings of the Thirty Third Annual Symposium on Foundations of Computer Science, pages 2--13, 1992.

....intractability assumption is violated (e.g. P = NP or the Polynomial Hierarchy collapses) Recently, Arora, Lund, Motwani, Sudan and Szegedy [ALM 92] showed that the problem of finding the largest clique in a graph is in the latter category. Following a series of breakthrough results [AS92, BFL91, FGL 91, LFKN90, Sha90] they showed that there exists a constant ffl such that no deterministic polynomial time algorithm can approximate the maximum clique size (G) of a graph G with n vertices within a factor of n ffl , unless P = NP. While this result strongly suggests that no ....

S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of the IEEE Symposium on Foundations of Computer Science, pages 2--13, 1992.

.... that IP = PSPACE was shown by giving an interactive proof for QUANTIFIED BOOLEAN FORMULA, which is complete for PSPACE [25, 30] More recently, the celebrated PCP theorem characterizing NP was proven by designing efficient probabilistically checkable proofs for a specific NP complete language [3, 4]. In this paper, we present a complete promise problem 1 for SZK, the class of languages possessing statistical zeroknowledge proofs against an honest verifier. For traditional complexity classes, such as NP and PSPACE, the construction of natural complete problems has become a routine task. ....

S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of the Thirty Third Annual Symposium on Foundations of Computer Science, pages 2--13, 1992.

....for a full treatment. 1 ) 2.2. Honest martingales To connect our concept to standard notions of honesty in complexity theory, we redefine the above formalism so that bounds are expressed in terms of n rather than N . The new formalism is essentially the same as that for holographic proofs in [7, 5, 4, 30, 6], with w playing the role of the proof. Namely, define a query machine M to have a standard TM input tape, any number of standard worktapes, and a query tape that provides random access to bits of a string w given as an auxiliary input. M is given as input the length N of w in standard dyadic ....

S. Arora and S. Safra. Probabilistic checking of proofs. In Proc. 33rd FOCS, pages 2--13, 1992.

....is to develop local algorithms with globally optimum performance guarantees. The problems considered are related to fractional versions of maximum independent set and minimum coloring in hyper graphs. While integer versions of these problem appear to be hard to approximate, BGLR93, FGL 91, AS92, ALM 92] the versions, that happen to be the ones that matter in practice, do not fall into this class. Thus, there is no excuse for substituting local maximality for global maximum , since the gap between the two often grows linearly in the size of the problem. This is in fact the ....

.... routing) that capacity of each resource exceeds size of each job by a logarithmic factor, i.e. min e2P c(e) Omega Gamma127 n) Delta max s2X d s (1) We comment that the general integer problem, without making an assumption of such type, is provably un approximable [BGLR93, FGL 91, AS92, ALM 92] unless P = NP. Indeed, the maximumthroughput (minimum time) problems, are in fact, generalization of maximum independent set (mini mum coloring, resp. problem on hyper graphs. 2.2 Max throughput full access algorithm The algorithm executed by each job s (see Figure 1) works ....

Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs. In Proc. 33rd IEEE Symp. on Found. of Comp. Science, pages 2-- 13, October 1992.

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S. Arora and S. Safra. Probabilistic checking of proofs. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science, pages 2-13, 1992.

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S. Arora and S. Safra, "Probabilistic checking of proofs," In Proc. of "33-rd IEEE Symposium on the Foundations of Computer Science", 1992, pp. 2-13.

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S. Arora and S. Safra. Probabilistic checking of proofs. In Proc. 33rd Annual IEEE Symposium on Foundations of Computer Science, pages 2--13, 1992.

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