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Regular Languages are Testable with a Constant Number of Queries
 SIAM Journal on Computing
, 1999
"... We continue the study of combinatorial property testing, initiated by Goldreich, Goldwasser and Ron in [7]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language L 2 f0; 1g and an integer n there exists a randomized algorithm which always acc ..."
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Cited by 81 (19 self)
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We continue the study of combinatorial property testing, initiated by Goldreich, Goldwasser and Ron in [7]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language L 2 f0; 1g and an integer n there exists a randomized algorithm which always accepts a word w of length n if w 2 L, and rejects it with high probability if w has to be modified in at least n positions to create a word in L. The algorithm queries ~ O(1=) bits of w. This query complexity is shown to be optimal up to a factor polylogarithmic in 1=. We also discuss testability of more complex languages and show, in particular, that the query complexity required for testing contextfree languages cannot be bounded by any function of . The problem of testing regular languages can be viewed as a part of a very general approach, seeking to probe testability of properties defined by logical means. 1
Primality testing with Gaussian periods
, 2003
"... The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new ..."
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Cited by 21 (1 self)
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The problem of quickly determining whether a given large integer is prime or composite has been of interest for centuries, if not longer. The past 30 years has seen a great deal of progress, leading up to the recent deterministic, polynomialtime algorithm of Agrawal, Kayal, and Saxena [2]. This new “AKS test ” for the primality of n involves verifying the
On the Automata Size for Presburger Arithmetic
 In Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science (LICS 2004
, 2004
"... Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Pr ..."
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Cited by 10 (1 self)
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Automata provide an effective mechanization of decision procedures for Presburger arithmetic. However, only crude lower and upper bounds are known on the sizes of the automata produced by this approach. In this paper, we prove that the number of states of the minimal deterministic automaton for a Presburger arithmetic formula is triple exponentially bounded in the length of the formula. This upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that this triple exponential bound is tight (even for nondeterministic automata). Moreover, we provide optimal automata constructions for linear equations and inequations.
The global isoperimetric methodology applied to Kneser’s Theorem
 Preprint August2007
"... We give in the present work a new methodology that allows to give isoperimetric proofs, for Kneser’s Theorem and Kemperman’s structure Theory and most sophisticated results of this type. As an illustration we present a new proof of Kneser’s Theorem. 1 ..."
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Cited by 2 (2 self)
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We give in the present work a new methodology that allows to give isoperimetric proofs, for Kneser’s Theorem and Kemperman’s structure Theory and most sophisticated results of this type. As an illustration we present a new proof of Kneser’s Theorem. 1
Bounds on the Automata Size for Presburger Arithmetic
, 2005
"... Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the number of states of the minimal deterministic automaton for a Pre ..."
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Cited by 1 (0 self)
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Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the number of states of the minimal deterministic automaton for a Presburger arithmetic formula. This bound depends on the length of the formula and the quantifiers occurring in the formula. The upper bound is established by comparing the automata for Presburger arithmetic formulas with the formulas produced by a quantifier elimination method. We also show that our bound is tight, even for nondeterministic automata. Moreover, we provide optimal automata constructions for linear equations and inequations.
Some additive applications of the isoperimetric approach
, 2008
"... Let G be a group and let X be a finite subset. The isoperimetric method investigates the objective function (XB)\X, defined on the subsets X with X  ≥ k and G\(XB)  ≥ k. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and ob ..."
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Let G be a group and let X be a finite subset. The isoperimetric method investigates the objective function (XB)\X, defined on the subsets X with X  ≥ k and G\(XB)  ≥ k. In this paper we present all the basic facts about the isoperimetric method. We improve some of our previous results and obtain generalizations and short proofs for several known results. We also give some new applications. Some of the results obtained here will be used in a coming paper [23] to improve Kempermann structure Theory.
On the Frobenius ’ Problem of three numbers
"... Given k natural numbers {a1,..., ak} ⊂ N with 1 ≤ a1 < a2 <.. < ak and gcd(a1,..., ak) = 1, let be R(a1,..., ak) = {λ1a1 + · · · + λkak  λi ∈ N, i = 1 ÷ k} and R(a1,..., ak) = N \ R(a1,..., ak). It is easy to see that R(a1,..., ak)  < ∞. The Frobenius Problem related to the set ..."
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Given k natural numbers {a1,..., ak} ⊂ N with 1 ≤ a1 < a2 <.. < ak and gcd(a1,..., ak) = 1, let be R(a1,..., ak) = {λ1a1 + · · · + λkak  λi ∈ N, i = 1 ÷ k} and R(a1,..., ak) = N \ R(a1,..., ak). It is easy to see that R(a1,..., ak)  < ∞. The Frobenius Problem related to the set {a1,..., ak} consists on the computation of f(a1,..., ak) = max R(a1,..., ak), also called the Frobenius number, and the cardinal R(a1,..., ak). The solution of the Frobenius Problem is the explicit computation of the set R(a1,..., ak). In some cases it is known a sharp upper bound for the Frobenius number. When k = 3 this bound is known to be F (N) = max f(a, b, N) = 0<a<b<N gcd(a,b,N)=1 This bound is given in [4]. j 2(⌊N/2 ⌋ − 1) 2 − 1 if N ≡ 0 (mod 2), 2 ⌊N/2 ⌋ (⌊N/2 ⌋ − 1) − 1 if N ≡ 1 (mod 2). In this work we give a geometrical proof of this bound which allows us to give the solution of the Frobenius problem for all the sets {α, β, N} such that f(α, β, N) = F (N). Keywords: Frobenius problem, Lshaped tile, Smith normal form, Minimum Distance Diagram 1
WEIGHTED MULTICONNECTED LOOP NETWORKS
"... Given relatively prime integers N,a1,...,ak, a multiconnected loop network is defined as the directed graph with vertex set Z/NZ = {0,1,...,N − 1}, and directed edges i → r ≡ i + aj (mod N). If each edge i → i+aj is given a positive real weight wj for j = 1,...,k, then we have a weighted multiconn ..."
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Given relatively prime integers N,a1,...,ak, a multiconnected loop network is defined as the directed graph with vertex set Z/NZ = {0,1,...,N − 1}, and directed edges i → r ≡ i + aj (mod N). If each edge i → i+aj is given a positive real weight wj for j = 1,...,k, then we have a weighted multiconnected loop network. The weight of a path is the sum of weights on its edges. The distance from a vertex to another is the minimum weight of all paths from the first vertex to the second. The diameter of the network is the maximum distance, and the average diameter is the average distance in the network. In this paper we study the diameter and the average diameter of a weighted multiconnected loop network. We give a unified and generalized presentation of several results in the literature, and also some new results are obtained. 1