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11
PRIMES is in P
 Ann. of Math
, 2002
"... We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1 ..."
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We present an unconditional deterministic polynomialtime algorithm that determines whether an input number is prime or composite. 1
Checking NFA equivalence with bisimulations up to congruence
"... Abstract—We introduce bisimulation up to congruence as a technique for proving language equivalence of nondeterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp [12] that, instead of computing the whole determinised automa ..."
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Cited by 6 (0 self)
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Abstract—We introduce bisimulation up to congruence as a technique for proving language equivalence of nondeterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp [12] that, instead of computing the whole determinised automata, explores only a small portion of it. Although the optimised algorithm remains exponential in worst case (the problem is PSPACEcomplete), experimental results show improvements of several orders of magnitude over the standard algorithm. I.
THE LEAST COMMON MULTIPLE OF CONSECUTIVE TERMS IN ARITHMETIC PROGRESSIONS
, 903
"... Abstract. Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetical function gk,a,b for any positive integer n by gk,a,b(n): = (b+na)(b+(n+1)a)···(b+(n+k)a) lcm(b+na,b+(n+1)a,·· ·,b+(n+k)a). Letting a = 1 and b = 0, then gk,a,b becomes the arithmetical function introduced previously by Farh ..."
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Cited by 3 (2 self)
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Abstract. Let k ≥ 0, a ≥ 1 and b ≥ 0 be integers. We define the arithmetical function gk,a,b for any positive integer n by gk,a,b(n): = (b+na)(b+(n+1)a)···(b+(n+k)a) lcm(b+na,b+(n+1)a,·· ·,b+(n+k)a). Letting a = 1 and b = 0, then gk,a,b becomes the arithmetical function introduced previously by Farhi. Farhi proved gk,1,0 is periodical and k! is a period. Hong and Yang improved Farhi’s period k! to lcm(1, 2,...,k) and conjectured that lcm(1,2,...,k,k+1) k+1 divides the smallest positive period of gk,1,0. Recently, Farhi and Kane proved this conjecture and determined the smallest positive period of gk,1,0. For the general integers a ≥ 1 and b ≥ 0, it is natural to ask the interesting question: Whether gk,a,b is periodical? If the answer is affirmative, then one asks the further question: What is the smallest positive period of gk,a,b? In this paper, we mainly study these questions. We first show that the arithmetical function gk,a,b is periodical. Consequently, we provide detailed padic analysis to the periodical function gk,a,b. Finally, we determine the smallest positive period of gk,a,b. So we answer completely the above two questions. Our result extends the FarhiKane theorem from the set of positive integers to the general arithmetic progression. 1.
Branchingtime Model Checking of Onecounter Processes
 In Proc. of STACS, volume 5 of LIPIcs
, 2010
"... Abstract. Onecounter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from the modal µcalculus for this problem [20]. First, we ana ..."
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Abstract. Onecounter processes (OCPs) are pushdown processes which operate only on a unary stack alphabet. We study the computational complexity of model checking computation tree logic (CTL) over OCPs. A PSPACE upper bound is inherited from the modal µcalculus for this problem [20]. First, we analyze the periodic behaviour of CTL over OCPs and derive a model checking algorithm whose running time is exponential only in the number of control locations and a syntactic notion of the formula that we call leftward until depth. In particular, model checking fixed OCPs against CTL formulas with a fixed leftward until depth is in P. This generalizes a corresponding result from [12] for the expression complexity of CTL’s fragment EF. Second, we prove that already over some fixed OCP, CTL model checking is PSPACEhard, i.e., expression complexity is PSPACEhard. Third, we show that there already exists a fixed CTL formula for which model checking of OCPs is PSPACEhard, i.e., data complexity is PSPACEhard as well. To obtain the latter result, we employ two results from complexity theory: (i) Converting a natural number in Chinese remainder presentation into binary presentation is in logspaceuniform NC 1 [8] and (ii) PSPACE is AC 0serializable [14]. We demonstrate that our approach can be used to obtain further results. We show that modelchecking CTL’s fragment EF over OCPs is hard for P NP, thus establishing a matching lower bound and answering an open question from [12]. We moreover show that the following problem is hard for PSPACE: Given a onecounter Markov decision process, a set of target states with counter value zero each, and an initial state, to decide whether the probability that the initial state will eventually reach one of the target states is arbitrarily close to 1. This improves a previously known lower bound for every level of the Boolean hierarchy shown in [5]. 1
Primality Testing in Polynomial Time
, 2003
"... These notes contain a description and correctness proof of the deterministic polynomialtime primality testing algorithm of Agrawal, Kayal, and Saxena. Some background from number theory and algebra is given in Section 4. 1 A polynomial identity for prime numbers Theorem 1.1 Let n 2 and a 0 be in ..."
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These notes contain a description and correctness proof of the deterministic polynomialtime primality testing algorithm of Agrawal, Kayal, and Saxena. Some background from number theory and algebra is given in Section 4. 1 A polynomial identity for prime numbers Theorem 1.1 Let n 2 and a 0 be integers. 1. If n is a prime number, then in the ring Z n [x].
NAIR’S AND FARHI’S IDENTITIES INVOLVING THE LEAST COMMON MULTIPLE OF BINOMIAL COEFFICIENTS ARE EQUIVALENT
, 2009
"... ∀k ∈ N. Recently, Farhi proved a new identity: lcm ( `k ´ `k ´ `k ´ lcm(1,2,...,k+1) 0 1 k k+1 ∀k ∈ N. In this note, we show that Nair’s and Farhi’s identities are equivalent. Throughout this note, let N denote the set of nonnegative integers. Define N ∗:= N \ {0}. There are lots of known results ab ..."
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∀k ∈ N. Recently, Farhi proved a new identity: lcm ( `k ´ `k ´ `k ´ lcm(1,2,...,k+1) 0 1 k k+1 ∀k ∈ N. In this note, we show that Nair’s and Farhi’s identities are equivalent. Throughout this note, let N denote the set of nonnegative integers. Define N ∗:= N \ {0}. There are lots of known results about the least common multiple of a sequence of positive integers. The most renowned is nothing else than an equivalent of the prime number theory; it says that log lcm(1, 2,..., n) ∼ n as n approaches infinity (see, for instance [6]), where lcm(1, 2, · · · , n) means the least common multiple of 1, 2,..., n. Some authors found effective bounds for lcm(1, 2,..., n). Hanson [5] got the upper bound lcm(1, 2,..., n) ≤ 3 n (∀n ≥ 1). Nair [12] obtained the lower bound lcm(1, 2, · · · , n) ≥ 2 n (∀n ≥ 9). Nair [12] also gave a new nice proof for the wellknown estimate lcm(1, 2, · · · , n) ≥ 2 n−1 (∀n ≥ 1). Hong and Feng [7] extended this inequality to the general arithmetic progression, which confirmed Farhi’s conjecture [2]. Regarding to many other related questions and generalizations of the above results investigated by several authors, we refer the interested reader to [1], [4], [8][10]. By exploiting the integral ∫ 1 0 xm−1 (1 − x) n−m dx, Nair [12] showed the following identity involving the binomial coefficients: Theorem 1. (Nair [12]) For any n ∈ N ∗ , we have n n n lcm ( , 2,..., n) = lcm(1, 2,..., n). 1 2 n Recently, by using Kummer’s theorem on the padic valuation of binomial coefficients ([11]), Farhi [3] provided an elegant padic proof to the following new interesting identity involving the binomial coefficients: Theorem 2. (Farhi [3]) For any n ∈ N, we have n n n lcm(1, 2,..., n + 1) lcm ( , ,..., ) =. 0 1 n n + 1
THE CARDINALITY OF SOME SYMMETRIC DIFFERENCES
"... Abstract. In this paper, we prove that for positive integers k and n, thecardinality of the symmetric differences of {1, 2,...,k}, {2, 4,...,2k}, {3, 6,...,3k},..., {n, 2n,...,kn} is at least k or n, whichever is larger. This solved a problem raised by Pilz in which binary composition codes were stu ..."
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Abstract. In this paper, we prove that for positive integers k and n, thecardinality of the symmetric differences of {1, 2,...,k}, {2, 4,...,2k}, {3, 6,...,3k},..., {n, 2n,...,kn} is at least k or n, whichever is larger. This solved a problem raised by Pilz in which binary composition codes were studied. 1.
EXTENSIONS OF THE HERMITE G.C.D. THEOREMS FOR BINOMIAL COEFFICIENTS
, 1993
"... (n,k) nk + l (ra + U) ..."