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43
On Boolean Decision Trees with Faulty Nodes
 In Random Structures and Algorithms
, 1994
"... We consider the problem of computing with faulty components in the context of the Boolean decision tree model, in which cost is measured by the number of input bits queried and the responses to queries are faulty with a fixed probability. We show that if f can be represented in kDNF form and in jC ..."
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We consider the problem of computing with faulty components in the context of the Boolean decision tree model, in which cost is measured by the number of input bits queried and the responses to queries are faulty with a fixed probability. We show that if f can be represented in kDNF form and in jCNF form, then O(n log(min(k; j)=q)) queries suffice to compute f with error probability less than q, where n is the number of input bits. 1 Introduction In this paper, we describe a method for performing reliable computation despite the presence of faulty components. This problem has been well studied in various contexts. The model we consider here is the noisy Boolean decision tree. In a Boolean decision tree, the value of a function on n bits is computed as follows: Each step consists of a query of an input bit, where the choice of the query may depend on the outcome of the previous queries. The cost Current address: LIP ENSLyon, 46 All'ee d'Italie, 69364 Lyon Cedex 07, France. Part...
The boolean functions computed by random boolean formulas or how to grow the right function. Random Structures and Algorithms
, 2005
"... Among their many uses, growth processes (probabilistic amplification), were used for constructing reliable networks from unreliable components, and deriving complexity bounds of various classes of functions. Hence, determining the initial conditions for such processes is an important and challenging ..."
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Among their many uses, growth processes (probabilistic amplification), were used for constructing reliable networks from unreliable components, and deriving complexity bounds of various classes of functions. Hence, determining the initial conditions for such processes is an important and challenging problem. In this paper we characterize growth processes by their initial conditions and derive conditions under which results such as Valiant’s[Val84] hold. First, we completely characterize growth processes that use linear connectives. Second, by extending Savick´y’s [Sav90] analysis, via “Restriction Lemmas”, we characterize growth processes that use monotone connectives, and show that our technique is applicable to growth processes that use other connectives as well. Additionally, we obtain explicit bounds on the convergence rates of several growth processes, including the growth process studied by Savick´y (1990).
How Do ReadOnce Formulae Shrink?
 Math. Syst. Theory
, 1994
"... Let f be a de Morgan readonce function of n variables. Let f " be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 \Gamma ")=2, the value 1 with the same probability, and leaving it unassigned with probability ". We show that f " de ..."
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Cited by 5 (2 self)
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Let f be a de Morgan readonce function of n variables. Let f " be the random restriction obtained by independently assigning to each variable of f , the value 0 with probability (1 \Gamma ")=2, the value 1 with the same probability, and leaving it unassigned with probability ". We show that f " depends, on the average, on only O(" ff n + "n 1=ff ) variables, where ff = log p 5\Gamma1 2 ' 3:27. This result is asymptotically the tightest possible. It improves a similar result obtained recently by Hastad, Razborov and Yao. 1 Introduction Obtaining nontrivial lower bounds on the complexity of Boolean functions is currently a very difficult task. Only a handful of methods yielding such lower bounds are currently known and even they work only in suitably restricted models. The current state of affairs in this respect is summerized in the books of Dunne [5] and Wegener [15] and the survey paper of Boppana and Sipser [2]. Many of the currently known methods for obtaining complexity l...
Fault Tolerance in Wireless Sensor Networks,” Book chapter
 in Handbook of Sensor Networks, I. Mahgoub and M. Ilyas
"... Abstract: In this Chapter, we address fault tolerance in wireless sensor networks. In order to make the presentation selfcontained, we start by providing a short summary of sensor networks and classical fault tolerance techniques. After that, we discuss the three phases of fault tolerance (fault mo ..."
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Abstract: In this Chapter, we address fault tolerance in wireless sensor networks. In order to make the presentation selfcontained, we start by providing a short summary of sensor networks and classical fault tolerance techniques. After that, we discuss the three phases of fault tolerance (fault models, fault detection and identification and resiliency mechanisms) at four levels of abstractions (hardware, system software, middleware, and applications) and four scopes (components of individual node, individual node, network, and the distributed system). The technical cores of the chapter are two casestudies on heterogeneous fault tolerance and discrepancy minimizationbased fault detection and correction. We conclude the chapter with a brief survey of the future directions for fault tolerance research in wireless sensor networks.
Directed monotone contact networks for threshold functions
 Inform. Process. Lett
, 1994
"... In this note we consider the problem of computing threshold functions using directed monotone contact networks. We give constructions of monotone contact networks of size (k − 1)(n − k + 2) ⌈log(n − k + 2) ⌉ computing T n k, for 2 ≤ k ≤ n − 1. Our upper bound is close to the Ω(kn log(n/(k − 1))) low ..."
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In this note we consider the problem of computing threshold functions using directed monotone contact networks. We give constructions of monotone contact networks of size (k − 1)(n − k + 2) ⌈log(n − k + 2) ⌉ computing T n k, for 2 ≤ k ≤ n − 1. Our upper bound is close to the Ω(kn log(n/(k − 1))) lower bound for small thresholds and the k(n − k + 1) lower bound for large thresholds. Our networks are described explicitly; we do not use probabilistic existence arguments.
Amplification by ReadOnce Formulae
, 1995
"... Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone readonce formulae of size O(n ff+2 ) (where ff = log p 5\Gamma1 2 ' 3:27) that amplify (/ \Gamma 1 n ; / + ..."
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Moore and Shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability. Valiant used similar methods to construct monotone readonce formulae of size O(n ff+2 ) (where ff = log p 5\Gamma1 2 ' 3:27) that amplify (/ \Gamma 1 n ; / + 1 n ) (where / = p 5\Gamma1 2 ' 0:62) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) and deduced as a consequence the existence of monotone formulae of the same size that compute the majority of n bits. Boppana has shown that any monotone readonce formula that amplifies (p \Gamma 1 n ; p + 1 n ) to ( 1 4 ; 3 4 ) (where 0 ! p ! 1 is constant) has size\Omega\Gamma n ff ) and that any monotone, not necessarily readonce, contact network (and in particular any monotone formula) that amplifies ( 1 4 ; 3 4 ) to (2 \Gamman ; 1 \Gamma 2 \Gamman ) has size\Omega\Gamma n 2 ). We extend Boppana's results in two ways. We first show that his two lower bounds hold for general readonc...
A MULTILEVEL VIEW OF DEPENDABLE COMPUTING
, 1994
"... This paper serves a dual purpose. It presents a unified framework and terminology for the study of computer system dependability. It also surveys the field of dependable computing in light of the proposed framework. Specifically, impairments to dependability are viewed from six levels, each being m ..."
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This paper serves a dual purpose. It presents a unified framework and terminology for the study of computer system dependability. It also surveys the field of dependable computing in light of the proposed framework. Specifically, impairments to dependability are viewed from six levels, each being more abstract than the previous one. It is argued that all of these levels are useful, in the sense that proven dependability assurance techniques can be applied at each level, and that it is beneficial to have distinct, precisely defined terminology for describing impairments to, and procurement strategies for, computer system dependability at these levels. The six levels are: (I) Defect level or component level, dealing with deviant atomic parts. (2) Fault level or logic level, dealing with deviant signal values or path selections. (3) Error level or information level, dealing with deviant data or internal states. (4) Malfunction level or system level, dealing with deviant functional behavior. (5) Degradation level or service level, dealing with deviant performance. (6) Failure level or result level, dealing with deviant outputs or actions. Briefly, a hardware or software component may be defective (hardware may also become defective due to wear and aging). Certain system states will expose the defect, resulting in the development of faults
Network Reliability and Fault Tolerance
, 1999
"... this article, we will use the term network reliability in a broad sense and cover several subtopics. We will start with network availability and performability, and then discuss survivable network design, followed by fault detection, isolation, and restoration as well as preplanning. We will conclud ..."
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this article, we will use the term network reliability in a broad sense and cover several subtopics. We will start with network availability and performability, and then discuss survivable network design, followed by fault detection, isolation, and restoration as well as preplanning. We will conclude with a short discussion on recent issues and literature.
Monomial ideals and the Scarf complex for coherent systems in reliability theory
 ISSN 00905364. doi: 10.1214/009053604000000373. URL http://dx.doi.org/10
, 2004
"... A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α = (α1,...,αd) is an integer vector representing the state of a system, then the corresponding algebrai ..."
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A certain type of integer grid, called here an echelon grid, is an object found both in coherent systems whose components have a finite or countable number of levels and in algebraic geometry. If α = (α1,...,αd) is an integer vector representing the state of a system, then the corresponding algebraic object is a monomial x α1 1 · · ·xα d d in the indeterminates x1,...,xd. The idea is to relate a coherent system to monomial ideals, so that the socalled Scarf complex of the monomial ideal yields an inclusion–exclusion identity for the probability of failure, which uses many fewer terms than the classical identity. Moreover in the “general position ” case we obtain via the Scarf complex the tube bounds given by Naiman and Wynn [J. Inequal. Pure Appl. Math. (2001) 2 1–16]. Examples are given for the binary case but the full utility is for general multistate coherent systems and a comprehensive example is given. 1. Introduction. The