Sliding Scale Conjectures in PCP

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 $\epsilon>0$.

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 "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.

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 introduce and study the notion of probabilistically checkable proofs (PCP) for real number algorithms. Our starting point is the computational model of Blum, Shub, and Smale (BSS) and the real analogue NPR of NP in that model. We define in a straightforward manner verifiers as well as complexity classes PCPR(r(n),q(n)) for the BSS model. Our main result is, to the best of our knowledge, the first PCP theorem for NPR. It states that each problem in NPR has transparent long proofs, i.e.,NPR \subseteq PCPR(poly,1), where poly denotes the class of univariate polynomial functions. The techniques used extend ideas from [12] for self-testing and self-correcting certain functions over so-called rational domains to more general domains over the real numbers. The latter arise from the particular NPR-complete problem for which we construct a verifier of the required form.

Proof assistants are software tools for formal modeling and verification of software, hardware, design, and mathematical proofs. Due to the growing complexity and scale of formal proofs, compatibility issues frequently arise when using different versions of proof assistants. These issues result in broken proofs, disrupting the maintenance of formalized theories and hindering the broader dissemination of results within the community. Although existing works have proposed techniques to address specific types of compatibility issues, the overall characteristics of these issues remain largely unexplored. To address this gap, we conduct the first extensive empirical study to characterize compatibility issues, using Isabelle as a case study. We develop a regression testing framework to automatically collect compatibility issues from the Archive of Formal Proofs, the largest repository of formal proofs in Isabelle. By analyzing 12,079 collected issues, we identify their types and symptoms and further investigate their root causes. We also extract updated proofs that address these issues to understand the applied resolution strategies. Our study provides an in-depth understanding of compatibility issues in proof assistants, offering insights that support the development of effective techniques to mitigate these issues.

These lectures are intended to give a broad overview of the most important formal verification techniques. These methods are applicable to hardware, software, protocols etc. and even to checking proofs in pure mathematics, though I will emphasize the applications to verification. Since other lecturers will cover some of the topics in more detail (in particular model checking), my lectures will cover mainly the material on deductive theorem proving, which in any case is my special area of interest. The arrangement of material in these notes is roughly in order of logical complexity, starting with methods for propositional logic and leading up to general theorem proving, then finishing with a description of my own theorem proving program ‘HOL Light’ and an extended case study on the verification of a floating-point square root algorithm used by Intel.

Mathematical assistance systems and proof assistance systems in general have traditionally been developed as large, monolithic systems which are often hard to maintain and extend. In this article we propose a component network architecture as a means to design and implement such systems. Under this view a mathematical assistance system is an integrated knowledge-based system composed as a network of individual, specialized components. These components manipulate and mutually exchange different kinds of mathematical knowledge encoded within different document formats. Consequently, several units of mathematical knowledge coexist throughout the system within these components and this knowledge changes non-monotonically over time. Our approach has resulted in a lean and maintainable system code and makes the system open for extensions. Moreover, it naturally decomposes the global and complex reasoning and truth maintenance task into local reasoning and truth maintenance tasks inside the system components. The interplay between neighboring components in the network is thereby realized by nonmonotonic updates over agreed interface representations encoding different kinds of mathematical knowledge.

We prove that for all constants a, NQP = NTIME[n polylog(n)] cannot be (1/2 + 2−log a n )-approximated by 2log a n -size ACC 0 ∘ THR circuits ( ACC 0 circuits with a bottom layer of THR gates). Previously, it was even open whether E NP can be (1/2+1/√n)-approximated by AC 0[⊕] circuits. As a straightforward application, we obtain an infinitely often ( NE ∩ coNE)/1-computable pseudorandom generator for poly-size ACC 0 circuits with seed length 2logє n , for all є > 0. More generally, we establish a connection showing that, for a typical circuit class C, non-trivial nondeterministic algorithms estimating the acceptance probability of a given S-size C circuit with an additive error 1/S (we call it a CAPP algorithm) imply strong (1/2 + 1/n ω(1)) average-case lower bounds for nondeterministic time classes against C circuits. Note that the existence of such (deterministic) algorithms is much weaker than the widely believed conjecture PromiseBPP = PromiseP. We also apply our results to several sub-classes of TC 0 circuits. First, we show that for all k, NP cannot be (1/2 + n −k )-approximated by n k -size Sum∘ THR circuits (exact ℝ-linear combination of threshold gates), improving the corresponding worst-case result in [Williams, CCC 2018]. Second, we establish strong average-case lower bounds and build ( NE ∩ coNE)/1-computable PRGs for Sum ∘ PTF circuits, for various regimes of degrees. Third, we show that non-trivial CAPP algorithms for MAJ ∘ MAJ indeed already imply worst-case lower bounds for TC 3 0 ( MAJ ∘ MAJ ∘ MAJ). Since exponential lower bounds for MAJ ∘ MAJ are already known, this suggests TC 3 0 lower bounds are probably within reach. Our new results build on a line of recent works, including [Murray and Williams, STOC 2018], [Chen and Williams, CCC 2019], and [Chen, FOCS 2019]. In particular, it strengthens the corresponding (1/2 + 1/polylog(n))-inapproximability average-case lower bounds in [Chen, FOCS 2019]. The two important technical ingredients are techniques from Cryptography in NC 0 [Applebaum et al., SICOMP 2006], and Probabilistic Checkable Proofs of Proximity with NC 1-computable proofs.

We investigate the outage performance of a non-orthogonal multiple access (NOMA) network assisted by multiple active simultaneously transmitting and reflecting reconfigurable intelligent surfaces (STAR-ARISs), under hardware impairments (HIS) and imperfect successive interference cancellation (ipSIC). To accurately evaluate system reliability, we employ the moment matching technique to approximate the composite channel gain distribution as a Gamma random variable within a Nakagami-m fading environment. Building on this approximation, closed-form expressions characterizing the outage performance are deduced. To corroborate the theoretical framework, simulation experiments are conducted, demonstrating three major outcomes: i) The outage performance improves as the number of STAR-ARIS or their constituent reflective elements increases; ii) Under the constraint of a constant total number of elements, a single large-scale STAR-ARIS configuration can outperform a distributed setup of multiple smaller surfaces; and iii) The STAR-ARISs-assisted NOMA network achieves a lower outage probability than its orthogonal multiple access (OMA) counterpart.

PROOF CHECKING THE RSA PUBLIC KEY ENCRYPTION ALGORITHM

This paper shows how a dialogue system for information-seeking dialogues can be implemented in a type-theory-based syntax editor, originally developed for editing mathematical proofs. The implementation gives a simple logical metatheory to such dialogue systems and also suggests new functions for them, e.g., a local undo operation. The method developed provides a logically based declarative way of implementing simple dialogue systems that is easy to port to new domains.

Exact solving of systems of linear equations (SLEs) is a fundamental subroutine within number theory, formal verification of mathematical proofs, and exact-precision mathematical programming. Moreover, efficient exact SLE solution methods could be valuable for a growing body of science and engineering applications where current fixed-precision standards have been deemed inadequate. This article contains key derivations relating, and computational tests comparing, two exact direct solution frameworks: roundoff-error-free (REF) LU factorization and rational arithmetic LU factorization. Specifically, both approaches solve the linear system Ax = b by factoring the matrix A into the product of a lower triangular (L) and upper triangular (U) matrix, A = LU . Most significantly, the featured findings reveal that the integer-preserving REF factorization framework solves dense SLEs one order of magnitude faster than the exact rational arithmetic approach while requiring half the memory. Since rational LU is utilized for basic solution validation in exact linear and mixed-integer programming, these results offer preliminary evidence of the potential of the REF factorization framework to be utilized within this specific context. Additionally, this article develops and analyzes an efficient streamlined version of Edmonds’s Q-matrix approach that can be implemented as another basic solution validation approach. Further experiments demonstrate that the REF factorization framework also outperforms this alternative integer-preserving approach in terms of memory requirements and computational effort. General purpose codes to solve dense SLEs exactly via any of the aforementioned methods have been made available to the research and academic communities.

The PCP theorem encapsulates the idea that randomization allows the immediate verification of any mathematical proof. A simple route to this striking result was proposed earlier this year.