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139
Quantum Computability
 SIAM Journal of Computation
, 1997
"... Abstract. In this paper some theoretical and (potentially) practical aspects of quantum computing are considered. Using the tools of transcendental number theory it is demonstrated that quantum Turing machines (QTM) with rational amplitudes are sufficient to define the class of bounded error quantum ..."
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Cited by 109 (0 self)
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Abstract. In this paper some theoretical and (potentially) practical aspects of quantum computing are considered. Using the tools of transcendental number theory it is demonstrated that quantum Turing machines (QTM) with rational amplitudes are sufficient to define the class of bounded error quantum polynomial time (BQP) introduced by Bernstein and Vazirani [Proc. 25th ACM Symposium on Theory of Computation, 1993, pp. 11–20, SIAM J. Comput., 26 (1997), pp. 1411–1473]. On the other hand, if quantum Turing machines are allowed unrestricted amplitudes (i.e., arbitrary complex amplitudes), then the corresponding BQP class has uncountable cardinality and contains sets of all Turing degrees. In contrast, allowing unrestricted amplitudes does not increase the power of computation for errorfree quantum polynomial time (EQP). Moreover, with unrestricted amplitudes, BQP is not equal to EQP. The relationship between quantum complexity classes and classical complexity classes is also investigated. It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in P #P and PSPACE. A potentially practical issue of designing “machine independent ” quantum programs is also addressed. A single (“almost universal”) quantum algorithm based on Shor’s method for factoring integers is developed which would run correctly on almost all quantum computers, even if the underlying unitary transformations are unknown to the programmer and the device builder.
Mathematical Problems for the Next Century
 Mathematical Intelligencer
, 1998
"... This report is my response. ..."
Flows on homogeneous spaces and Diophantine approximation on manifolds
, 1998
"... Abstract. We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by ..."
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Cited by 68 (19 self)
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Abstract. We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindˇzuk in 1964. We also prove several related hypotheses of Baker and Sprindˇzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices. 1.
Discrete logarithms in gf(p) using the number field sieve
 SIAM J. Discrete Math
, 1993
"... Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heur ..."
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Cited by 66 (1 self)
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Recently, several algorithms using number field sieves have been given to factor a number n in heuristic expected time Ln[1/3; c], where Ln[v; c] = exp{(c + o(1))(log n) v (log log n) 1−v}, for n → ∞. In this paper we present an algorithm to solve the discrete logarithm problem for GF (p) with heuristic expected running time Lp[1/3; 3 2/3]. For numbers of a special form, there is an asymptotically slower but more practical version of the algorithm.
Metatheory and Reflection in Theorem Proving: A Survey and Critique
, 1995
"... One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an appro ..."
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Cited by 57 (2 self)
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One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fullyexpansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof ...
Floating point verification in HOL Light: the exponential function
 UNIVERSITY OF CAMBRIDGE COMPUTER LABORATORY
, 1997
"... Since they often embody compact but mathematically sophisticated algorithms, operations for computing the common transcendental functions in floating point arithmetic seem good targets for formal verification using a mechanical theorem prover. We discuss some of the general issues that arise in veri ..."
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Cited by 31 (6 self)
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Since they often embody compact but mathematically sophisticated algorithms, operations for computing the common transcendental functions in floating point arithmetic seem good targets for formal verification using a mechanical theorem prover. We discuss some of the general issues that arise in verifications of this class, and then present a machinechecked verification of an algorithm for computing the exponential function in IEEE754 standard binary floating point arithmetic. We confirm (indeed strengthen) the main result of a previously published error analysis, though we uncover a minor error in the hand proof and are forced to confront several subtle issues that might easily be overlooked informally. The development described here includes, apart from the proof itself, a formalization of IEEE arithmetic, a mathematical semantics for the programming language in which the algorithm is expressed, and the body of pure mathematics needed. All this is developed logically from first prin...
Flows on Sarithmetic homogeneous spaces and applications to metric Diophantine approximation
, 2003
"... The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and padic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally ..."
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Cited by 23 (11 self)
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The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and padic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotentinvariant measures on real homogeneous spaces) and KleinbockMargulis (strong extremality of nondegenerate submanifolds of R n) are generalized to the Sarithmetic setting.
Metric Diophantine approximation: The KhintchineGroshev theorem for nondegenerate manifolds
 MOSCOW MATHEMATICAL JOURNAL
, 2002
"... The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on nondegenerate manifolds, which completes earlier results for convergence. ..."
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Cited by 21 (12 self)
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The main objective of this paper is to prove a Khintchine type theorem for divergence for linear Diophantine approximation on nondegenerate manifolds, which completes earlier results for convergence.