Results 11  20
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25
On the Robustness of Functional Equations
 SIAM Journal on Computing
, 1994
"... In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for ..."
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In this paper, we study the general question of how characteristics of functional equations influence whether or not they are robust. We isolate examples of properties which are necessary for the functional equations to be robust. On the other hand, we show other properties which are sufficient for robustness. We then study a general class of functional equations, which are of the form 8x; y F [f(x \Gamma y); f(x + y); f(x); f(y)] = 0, where F is an algebraic function. We give conditions on such functional equations that imply robustness. Our results have applications to the area of selftesting/correcting programs. We show that selftesters and selfcorrectors can be found for many functions satisfying robust functional equations, including algebraic functions of trigonometric functions such as tan x; 1 1+cotx ; Ax 1\GammaAx ; cosh x. 1 Introduction The mathematical field of functional equations is concerned with the following prototypical problem: Given a set of properties (fun...
On the Hardness of Permanent
, 1999
"... . We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm w ..."
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Cited by 22 (3 self)
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. We prove that if there is a polynomial time algorithm which computes the permanent of a matrix of order n for any inverse polynomial fraction of all inputs, then there is a BPP algorithm computing the permanent for every matrix. It follows that this hypothesis implies P #P = BPP. Our algorithm works over any sufficiently large finite field (polynomially larger than the inverse of the assumed success ratio), or any interval of integers of similar range. The assumed algorithm can also be a probabilistic polynomial time algorithm. Our result is essentially the best possible based on any black box assumption of permanent solvers, and is a simultaneous improvement of the results of Gemmell and Sudan [GS92], Feige and Lund [FL92] as well as Cai and Hemachandra [CH91], and Toda (see [ABG90]). 1 Introduction The permanent of an n \Theta n matrix A is defined as per(A) = X oe2Sn n Y i=1 A i;oe(i) ; where Sn is the symmetric group on n letters, i.e., the set of all permutations of f1;...
Testing membership in languages that have small width branching programs
 SIAM Journal on Computing
"... Abstract. Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and ..."
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Cited by 22 (5 self)
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Abstract. Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and
Transparent Proofs and Limits to Approximation
, 1994
"... We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researc ..."
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Cited by 16 (0 self)
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We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spotchecks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the PNP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...
A Combinatorial Consistency Lemma with application to proving the PCP Theorem
 PRELIMINARY VERSION IN RANDOM
, 1997
"... The current proof of the PCP Theorem (i.e., NP = PCP(log; O(1))) is very complicated. One source of difficulty is the technically involved analysis of lowdegree tests. Here, we refer to the difficulty of obtaining strong results regarding lowdegree tests; namely, results of the type obtained and u ..."
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Cited by 14 (3 self)
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The current proof of the PCP Theorem (i.e., NP = PCP(log; O(1))) is very complicated. One source of difficulty is the technically involved analysis of lowdegree tests. Here, we refer to the difficulty of obtaining strong results regarding lowdegree tests; namely, results of the type obtained and used by Arora and Safra and Arora et. al. In this paper, we eliminate the need to obtain such strong results on lowdegree tests when proving the PCP Theorem. Although we do not get rid of lowdegree tests altogether, using our results it is now possible to prove the PCP Theorem using a simpler analysis of lowdegree tests (which yields weaker bounds). In other words, we replace the strong algebraic analysis of lowdegree tests presented by Arora and Safra and Arora et. al. by a combinatorial lemma (which does not refer to lowdegree tests or polynomials).
Approximate Checking of Polynomials and Functional Equations
 PROC. 37TH FOUNDATIONS OF COMPUTER SCIENCE
, 1997
"... In this paper, we show how to check programs that compute polynomials and functions defined by addition theorems  in the realistic setting where the output of the program is approximate instead of exact. We present results showing how to perform approximate checking, selftesting, and selfcorrec ..."
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Cited by 14 (3 self)
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In this paper, we show how to check programs that compute polynomials and functions defined by addition theorems  in the realistic setting where the output of the program is approximate instead of exact. We present results showing how to perform approximate checking, selftesting, and selfcorrecting of polynomials, settling in the affirmative a question raised by [GLR + 91, RS92, RS96]. We then show how to perform approximate checking, selftesting, and selfcorrecting for those functions that satisfy addition theorems, settling a question raised by [Rub94]. In both cases, we show that the properties used to test programs for these functions are both robust (in the approximate sense) and stable. Finally, we explore the use of reductions between functional equations in the context of approximate selftesting. Our results have implications for the stability theory of functional equations.
Spreading Rumors Cheaply, Quickly, And Reliably
, 2002
"... Gossip protocols have been shown to be a useful tool in the development of simple, robust, and efficient distributed systems. This thesis addresses a number of problems associated with gossip protocols, including dealing with the failure of a large fraction of the hosts in a system, accommodating th ..."
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Cited by 10 (0 self)
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Gossip protocols have been shown to be a useful tool in the development of simple, robust, and efficient distributed systems. This thesis addresses a number of problems associated with gossip protocols, including dealing with the failure of a large fraction of the hosts in a system, accommodating the topology of the underlying network, improving the efficiency of information exchange between hosts, and tolerating Byzantine failures.
Testing Multivariate Linear Functions: Overcoming the Generator Bottleneck
 Proc. 27th STOC
, 1994
"... The problem of testing program correctness has received considerable attention in computer science. One approach to this problem is the notion of selftesting programs [BLR90]. Selftesting usually becomes more costly in the case of testing multivariate functions. In this paper we present efficien ..."
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Cited by 9 (1 self)
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The problem of testing program correctness has received considerable attention in computer science. One approach to this problem is the notion of selftesting programs [BLR90]. Selftesting usually becomes more costly in the case of testing multivariate functions. In this paper we present efficient methods for selftesting multivariate linear functions. We then apply these methods to several multivariate linear problems to construct efficient selftesters. Cornell University. email: ergun@cs.cornell.edu. This work is supported by ONR Young Investigator Award N000149310590 1 1 Introduction Selftesting/correcting programs, which were introduced in [BLR90], are a powerful tool for attacking the problem of program correctness. Various problems have been shown to have selftesters and selfcorrectors[BLR90][BF90][Lip91][CL90][GLRSW91][RS92][RS93]. In this paper we investigate the problem of selftesting multivariate linear functions, i.e., given a multivariate linear function f a...
SelfTesting Without The Generator Bottleneck
 SIAM J. on Computing
, 1995
"... Suppose P is a program designed to compute a function f defined on a group G. The task of selftesting P , that is, testing if P computes f correctly on most inputs, usually involves testing explicitly if P computes f correctly on every generator of G. In the case of multivariate functions, the numb ..."
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Cited by 7 (2 self)
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Suppose P is a program designed to compute a function f defined on a group G. The task of selftesting P , that is, testing if P computes f correctly on most inputs, usually involves testing explicitly if P computes f correctly on every generator of G. In the case of multivariate functions, the number of generators, and hence the number of such tests, becomes prohibitively large. We refer to this problem as the generator bottleneck . We develop a technique that can be used to overcome the generator bottleneck for functions that have a certain nice structure, specifically if the relationship between the values of the function on the set of generators is easily checkable. Using our technique, we build the first efficient selftesters for many linear, multilinear, and some nonlinear functions. This includes the FFT, and various polynomial functions. All of the selftesters we present make only O(1) calls to the program that is being tested. As a consequence of our techniques, we also obtain efficient program resultcheckers for all these problems.
Probabilistic Proof Systems  A Survey
 IN SYMPOSIUM ON THEORETICAL ASPECTS OF COMPUTER SCIENCE
, 1996
"... Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems  interactive proofs, zeroknowledge proofs, and probabilistic checkable proofs  stressing the essen ..."
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Cited by 5 (0 self)
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Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems  interactive proofs, zeroknowledge proofs, and probabilistic checkable proofs  stressing the essential role of randomness in each of them.