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It Is Easy to Determine Whether a Given Integer Is Prime
, 2004
"... The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be super ..."
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Cited by 12 (1 self)
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The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
Normal Forms for SecondOrder Logic over Finite Structures, and Classification of NP Optimization Problems
 Annals of Pure and Applied Logic
, 1996
"... We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rstorder part is a boolean combination ..."
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Cited by 9 (5 self)
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We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rstorder part is a boolean combination of existential formulas, and (ii) over nite successor structures, the KolaitisThakur hierarchy of minimization problems collapses completely and the KolaitisThakur hierarchy of maximization problems collapses partially. The normal form theorem for 1 2 fails if 1 2 is replaced with 1 1 or if innite structures are allowed. 1 Introduction We consider secondorder logic with equality (unless otherwise stated explicitly) and without function symbols of positive arity. Predicates are denoted by capitals and individual variables by lower case letters; a bold face version of a letter denotes a tuple of corresponding symbols. For brevity, we say that a formula reduces t...
It Is Easy to Determine Whether a Given Integer Is
, 2005
"... Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wis ..."
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Cited by 6 (0 self)
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Dedicated to the memory of W. ‘Red ’ Alford, friend and colleague Abstract. “The problem of distinguishing prime numbers from composite numbers, and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. Nevertheless we must confess that all methods that have been proposed thus far are either restricted to very special cases or are so laborious and difficult that even for numbers that do not exceed the limits of tables constructed by estimable men, they try the patience of even the practiced calculator. And these methods do not apply at all to larger numbers... It frequently happens that the trained calculator will be sufficiently rewarded by reducing large numbers to their factors so that it will compensate for the time spent. Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated... It is in the nature of the problem
On Valiant’s holographic algorithms
"... Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primit ..."
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Cited by 2 (2 self)
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Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speedups for certain computational problems compared to naive algorithms for the same problems. The methodology uses Pfaffians and (planar) perfect matchings as basic computational primitives, and attempts to create exponential cancellations in computation. In this article we survey this new theory of matchgate computations and holographic algorithms.
Higman's Embedding Theorem. An Elementary Proof
, 1995
"... In 1961 G. Higman proved a remarkable theorem establishing a deep connection between the logical notion of recursiveness and questions about finitely presented groups. The basic aim of the present paper is to provide the reader with a rigorous and detailed proof of Higman's Theorem. All the necessar ..."
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In 1961 G. Higman proved a remarkable theorem establishing a deep connection between the logical notion of recursiveness and questions about finitely presented groups. The basic aim of the present paper is to provide the reader with a rigorous and detailed proof of Higman's Theorem. All the necessary preliminary material, including elements of group theory and recursive functions theory, is systematically presented and with complete proofs. The aquainted reader may skip the first sections and proceed immediately to the last. 1 Subgroups We assume familiarity with the concept of subgroup. We shall use the standard notation, i.e. H # G means that H is a subgroup of G. Fact 1.1. If H is a subgroup of a group G and K is a subset of H, then K is a subgroup of H i# K is a subgroup of G. Fact 1.2. For any family {H i } i#I of subgroups of a group G the intersection \ i#I H i is also a subgroup of G. Proof. First, we have \ i#I H i #= #. Indeed, if H i # G, for all i # I, th...
Normal Forms for SecondOrder Logic over Finite Structures, and Classification of NP Optimization Problems
, 1996
"... We start with a simple proof of Leivant's normal form theorem for \Sigma 1 1 formulas over finite successor structures. Then we use that normal form to prove the following: (i) over all finite structures, every \Sigma 1 2 formula is equivalent to a \Sigma 1 2 formula whose firstorder part ..."
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We start with a simple proof of Leivant's normal form theorem for \Sigma 1 1 formulas over finite successor structures. Then we use that normal form to prove the following: (i) over all finite structures, every \Sigma 1 2 formula is equivalent to a \Sigma 1 2 formula whose firstorder part is a boolean combination of existential formulas, and (ii) over finite successor structures, the KolaitisThakur hierarchy of minimization problems collapses completely and the KolaitisThakur hierarchy of maximization problems collapses partially. The normal form theorem for \Sigma 1 2 fails if \Sigma 1 2 is replaced with \Sigma 1 1 or if infinite structures are allowed. 1 Introduction We consider secondorder logic with equality (unless otherwise stated explicitly) and without function symbols of positive arity. Predicates are denoted by capitals and individual variables by lower case letters; a bold face version of a letter denotes a tuple of corresponding symbols. For brevity...
Annals of Pure and AppliedLogic, 78 (1996), 111125.
, 1997
"... We start with a simple proof of Leivant's normal form theorem for \Sigma 1 formulas over finite successor structures. Then we use that normal form to prove the following: (i) over all finite structures, every \Sigma 2 formula is equivalenttoa\Sigma 2 formula whose firstorder part is a boolean ..."
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We start with a simple proof of Leivant's normal form theorem for \Sigma 1 formulas over finite successor structures. Then we use that normal form to prove the following: (i) over all finite structures, every \Sigma 2 formula is equivalenttoa\Sigma 2 formula whose firstorder part is a boolean combination of existential formulas, and (ii) over finite successor structures, the KolaitisThakur hierarchy of minimization problems collapses completely and the KolaitisThakur hierarchy of maximization problems collapses partially. The normal form theorem for \Sigma 2 fails if \Sigma 2 is replaced with \Sigma 1 or if infinite structures are allowed.