Transparent Long Proofs: A First PCP Theorem for NPR

We continue the study of the trade‐off between the length of probabilistically checkable proofs (PCPs) and their query complexity, establishing the following main results (which refer to proofs of satisfiability of circuits of size n): 1. We present PCPs of length $\exp(o(\log\log n)^2)\cdot n$ that can be verified by making $o(\log\log n)$ Boolean queries. 2. For every \epsilon>0, we present PCPs of length $\exp(\log^\epsilon n)\cdot n$ that can be verified by making a constant number of Boolean queries. In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1). The multiplicative overhead on the length of the proof, introduced by transforming a proof into a probabilistically checkable one, is just quasi polylogarithmic in the first case (of query complexity $o(\log\log n)$), and is $2^{(\log n)^\epsilon}$, for any $\epsilon > 0$, in the second case (of constant query complexity). Our techniques include the introduction of a new variant of PCPs that we call “robust PCPs of proximity.” These new PCPs facilitate proof composition, which is a central ingredient in the construction of PCP systems. (A related notion and its composition properties were discovered independently by Dinur and Reingold.) Our main technical contribution is a construction of a “length‐efficient” robust PCP of proximity. While the new construction uses many of the standard techniques used in PCP constructions, it does differ from previous constructions in fundamental ways, and in particular does not use the “parallelization” step of Arora et al. [J. ACM, 45 (1998), pp. 501–555]. The alternative approach may be of independent interest. We also obtain analogous quantitative results for locally testable codes. In addition, we introduce a relaxed notion of locally decodable codes and present such codes mapping k information bits to codewords of length $k^{1+\epsilon}$ for any $\epsilon>0$.

The "direct-product code" of a function $f$ gives its values on all $k$-tuples $(f(x_1),\dots, f(x_k))$. This basic construct underlies "hardness amplification" in cryptography, circuit complexity, and probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29 (2000), pp. 1132--1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622] enabled for the first time testing proximity to this important code in the "list-decoding" regime. In particular, they give a $2$-query test which works for polynomially small success probability $1/k^{\alpha}$ and show that no such test works below success probability $1/k$. Our main result is a $3$-query test which works for exponentially small success probability $\exp({-k^{\alpha}})$. Our techniques (based on recent simplified decoding algorithms for the same code [R. Impagliazzo et al., Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 579--588]) also allow us to considerably simplify the analysis of the 2-query test of [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622]. We then show how to derandomize their test, achieving a code of polynomial rate, independent of $k$, and success probability $1/k^{\alpha}$. Finally, we show the applicability of the new tests to PCPs. Starting with a 2-query PCP with a projection property over an alphabet $\Sigma$ and with soundness error $1-\delta$, Rao [Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, 2008, pp. 1--10] (building on Raz's ($k$-fold) parallel repetition theorem [R. Raz, SIAM J. Comput., 27 (1998), pp. 763--803] and Holenstein's proof [T. Holenstein, Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, 2007, pp. 411--419] obtains a new 2-query PCP over the alphabet $\Sigma^k$ with soundness error $\exp(-\delta^2 k)$. Our techniques yield a 2-query PCP with soundness error $\exp(-\delta \sqrt{k})$. Our PCP construction turns out to be essentially the same as the miss-match proof system defined and analyzed by Feige and Kilian [SIAM J. Comput., 30 (2000), pp. 324--346] but with simpler analysis and exponentially better soundness error.

Article Free Access Share on Probabilistically checkable proofs with zero knowledge Authors: Joe Kilian NEC Research Institute NEC Research InstituteView Profile , Erez Petrank DIMACS Center, P.O.Box 1179, Piscataway, NJ DIMACS Center, P.O.Box 1179, Piscataway, NJView Profile , Gábor Tardos Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, Budapest, H-1364 Hungary and Institute for Advanced Study, Princeton, NJ Mathematical Institute of the Hungarian Academy of Sciences, Pf. 127, Budapest, H-1364 Hungary and Institute for Advanced Study, Princeton, NJView Profile Authors Info & Claims STOC '97: Proceedings of the twenty-ninth annual ACM symposium on Theory of computingMay 1997 Pages 496–505https://doi.org/10.1145/258533.258643Online:04 May 1997Publication History 24citation435DownloadsMetricsTotal Citations24Total Downloads435Last 12 Months34Last 6 weeks3 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

The PCP (i.e., Probabilistically Checkable Proofs) Theorem [8, 7, 22, 5, 4] states that any mathematical proof can be converted to a format that can be checked by a veri er making only a constant number of queries to the proof. The veri er picks the queries in a randomized way and might err with low probability.

Madhu Sudan is the recipient of the 2002 Nevanlinna Prize. Sudan has made fundamental contributions to two major areas of research, the connections between them, and their applications. The first area is Coding Theory. Established by Shannon and Hamming over 50 years ago, it is the mathematical study of the possibility of and the limits on reliable communication over noisy media. The second area is Probabilistically Checkable Proofs (PCPs). By contrast, it is only 10 years old. It studies the minimal resources required for probabilistic verification of standard mathematical proofs. My plan is to briefly introduce these areas, their motivation and foundational questions, and then to explain Sudan’s main contributions to each. Before we get to the specific works of Madhu Sudan, let us start with a couple of comments that will set up the context of his work.

An error-correcting code is said to be locally testable if it has an efficient spot-checking procedure that can distinguish codewords from strings that are far from every codeword, looking at very few locations of the input in doing so. Locally testable codes (LTCs) have generated a lot of interest over the years, in large part due to their connection to Probabilistically checkable proofs (PCPs). The ability to correct errors that occur during transmission is one of the big advantages of using a code. Hence, from a coding-theoretic angle, local testing is potentially more useful if in addition to accepting codewords, it also accepts strings that are close to a codeword (in contrast, local testers can have arbitrary behavior on such strings, which potentially annuls the benefits of error-correction). This would imply that when the tester accepts, one can follow-up the testing with a (more expensive) decoding procedure to correct the errors and recover the transmitted codeword, while if the tester rejects, we can save the effort of running the more expensive decoding algorithm. In this work, we define such testers, which we call tolerant testers following some recent work in property testing [13]. We revisit some recent constructions of LTCs and show how one can make them locally testable in a tolerant sense. While we do not optimize the parameters, the main message from our work is that there are explicit tolerant LTCs with similar parameters to LTCs.

We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon P and an integer k, the goal is to decide if there exists a set G of k guards within P such that every point p ∈ P is seen by at least one guard g∈ G. Each guard corresponds to a point in the polygon P, and we say that a guard g sees a point p if the line segment pg is contained in P. The art gallery problem has stimulated extensive research in geometry and in algorithms. However, the complexity status of the art gallery problem has not been resolved. It has long been known that the problem is NP-hard, but no one has been able to show that it lies in NP. Recently, the computational geometry community became more aware of the complexity class ∃ ℝ, which has been studied earlier by other communities. The class ∃ ℝ consists of problems that can be reduced in polynomial time to the problem of deciding whether a system of polynomial equations with integer coefficients and any number of real variables has a solution. It can be easily seen that NP ⊆ ∃ ℝ. We prove that the art gallery problem is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the art gallery problem, and (2) the art gallery problem is not in the complexity class NP unless NP=∃ ℝ. As an illustration of our techniques, we can show that for every compact semi-algebraic set S⊂ [0,1]2 there exists a polygon with rational coordinates that enforces one of the guards to be at any position p∈ S, in any optimal guarding. As a corollary of our construction, we prove that for any real algebraic number α there is an instance of the art gallery problem where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. In the ∃ ℝ-hardness proof for the art gallery problem we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it can be used to obtain ∃ ℝ-hardness proofs for other problems. In particular, ETR-INV has been very recently used to prove ∃ ℝ-hardness of other geometric problems.

We present an analog and machine-independent algebraic characterization of elementarily computable functions over the real numbers in the sense of recursive analysis: we prove that they correspond to the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema. We generalize this result to all higher levels of the Grzegorczyk Hierarchy. Concerning recursive analysis, our results provide machine-independent characterizations of natural classes of computable functions over the real numbers, allowing to define these classes without usual considerations on higher-order (type 2) Turing machines. Concerning analog models, our results provide a characterization of the power of a natural class of analog models over the real numbers.

We present results concerning analytic machines, a model of real computation introduced by Hotz which extends the well-known Blum, Shub and Smale machines (BSS machines) by infinite converging computations. The well-known representation theorem for BSS machines elucidates the structure of the functions computable in the BSS model: the domain of such a function partitions into countably many semi-algebraic sets, and on each of those sets the function is a polynomial resp. rational function. In this paper, we study whether the representation theorem can, in the univariate case, be extended to analytic machines, i.e. whether functions computable by analytic machines can be represented by power series in some part of their domain. We show that this question can be answered in the negative over the real numbers but positive under certain restrictions for functions over the complex numbers. We then use the machine model to define computability of univariate complex analytic (i.e. holomorphic) functions and examine in particular the class of analytic functions which have analytically computable power series expansions. We show that this class is closed under the basic analytic operations composition, local inversion and analytic continuation.

Let L be either the field of real numbers or a nonarchimedean rank-one valued field. For topological space T we study the Gelfand subalgebra CO(T, L) of the algebra of all L-valued continuous functions C(T, L). The main result is that if T is a paracompact locally compact Hausdorff space, which is ultraregular if L is nonarchimedean, then for f E C(T, L) the following statements are equivalent: (1) There exists a compact set K C T such that f(T K) is finite, (2) f has finite range on every discrete closed subset of T, and (3) f E CO(T, L). Throughout this paper L will stand for either the valued field R or a nonarchimedean rank-one valued field, and T for a Hausdorff topological space. T will be assumed completely regular in the real case and ultraregular in the nonarchimedean case. We will denote by C(T, L), or simply by C if there is no confusion, the algebra over L consisting of all L-valued continuous functions on T, and by CK the ideal of those continuous f E C with compact support. Let 9DT be the set of maximal ideals of C, and denote by C0 the Gelfand subalgebra of C, consisting of all f e C with the property that, for each M E 'DT, there exists X E L such that (f X) E M. (The concept of Gelfand subalgebras of general algebras has been introduced by N. Shell -formerly N. Shilkret-in [6].) We will denote by CF the subalgebra of C consisting of those f E C for which there exists a compact set K C T such that f(T K) is finite. PROPOSITION 1. For any f E CO, f(T) is compact. PROOF. First, we prove that f(T) is a precompact set. Real case. It suffices to see that f is bounded, and this follows from [5, 5.7(b)]. Nonarchimedean case. Take > 0. Note that any two (closed-open) --radius spheres B(a) = ( LI IE A a I < e} are equal or disjoint. Choose an indexed set (aci),I inf(T) such that B(aci),EI is disjoint and covers f(T). We claim I is finite. In fact, assume, to the contrary, that I is infinite. Let Ai = U,#,f'-(B(a,)). Since the family of closed-open sets (A,)iEI has the finite intersection property, there exists M E 91T such that Ai E Z[M] for any i E I. On the other hand, since f E C0 there exists X E L such that (fX) C M and hence Z(fx) n A, # 0 for any i E I, which is a contradiction. Thus I is finite and so f(T) is precompact. Received by the editors April 20, 1982 and, in revised form, February 1, 1983. 1980 Mathematics Subject Classification. Primary 54C40.

A system of quaternary coupled Sylvester-type real quaternion matrix equations

The important problem of recognizing a priori the class of Boolean functions which are never obtainable from a given combinational network under short circuit faults is almost unexplored, primarily due to lack of understanding of the functional and structural factors that influence the fault behavior in the network. In view of this, a new concept of impossible class of faulty functions (ICFF) is introduced in this correspondence. Several intriguing properties of ICFF are uncovered, namely, the undetectability of input bridging faults, the impossibility of the transformation of a fault free function Fo to a subset or superset of Fo, and to other functions belonging to the same P-and N-equivalence classes of Fo, etc. The closure amongst the fan-out-free and unate functions under bridging faults is investigated. The impact of ICFF on the testability of the network is also discussed.

The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. The previous best constant $B$ was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and$S(N,t) = \max_{0 &lt; u \le 1} \max_{N &lt; R \le 2N} \left| \sum_{N &lt; n \le R} (n + u)^{-it} \right|.$$2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN H-SUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN H-SUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication.For vertex-weighted graphs with n vertices we obtain the following results. We present an O(n t(ω,h)) time algorithm for MIN H-SUBGRAPH in case H is a fixed graph with h vertices and ω< 2.376 is the exponent of matrix multiplication. The value of t(ω,h) is determined by solving a small integer program. In particular, the smallest triangle can be found in O(n 2 + 1/(4 − ω)) ≤o(n 2.616) time, the smallest K 4 in O(n ω + 1) time, the smallest K 7 in O(n 4 + 3/(4 − ω)) time. As h grows, t(ω,h) converges to 3h/(6-ω) < 0.828h. Interestingly, only for h = 4,5,8 the running time of our algorithm essentially matches that of the (unweighted) H-subgraph detection problem. Already for triangles, our results improve upon the main result of [VW06]. Using rectangular matrix multiplication, the value of t(ω,h) can be improved; for example, the runtime for triangles becomes O(n 2.575). We also present an algorithm whose running time is a function of m, the number of edges. In particular, the smallest triangle can be found in O(m (18 − 4ω)/(13 − 3ω)) ≤o(m 1.45) time.For edge-weighted graphs we present an O(m 2 − 1/k logn) time algorithm that finds the smallest cycle of length 2k or 2k-1. This running time is identical, up to a logarithmic factor, to the running time of the algorithm of Alon et al. for the unweighted case. Using the color coding method and a recent algorithm of Chan for distance products, we obtain an O(n 3/logn) time randomized algorithm for finding the smallest cycle of any fixed length.KeywordsMinimum WeightLogarithmic FactorSmall TriangleDistance ProductFast Matrix MultiplicationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the correctness of automatic differentiation (AD) in the context of a higher-order, Turing-complete language (PCF with real numbers), both in forward and reverse mode. Our main result is that, under mild hypotheses on the primitive functions included in the language, AD is almost everywhere correct, that is, it computes the derivative or gradient of the program under consideration except for a set of Lebesgue measure zero. Stated otherwise, there are inputs on which AD is incorrect, but the probability of randomly choosing one such input is zero. Our result is in fact more precise, in that the set of failure points admits a more explicit description: for example, in case the primitive functions are just constants, addition and multiplication, the set of points where AD fails is contained in a countable union of zero sets of polynomials.