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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

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 rst-order part is a boolean combination of existential formulas, and (ii) over nite successor structures, the Kolaitis-Thakur hierarchy of minimization problems collapses completely and the Kolaitis-Thakur 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 second-order 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...

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

Leslie Valiant recently proposed a theory of holographic algorithms. These novel algorithms achieve exponential speed-ups 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.

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...

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 first-order part is a boolean combination of existential formulas, and (ii) over finite successor structures, the Kolaitis-Thakur hierarchy of minimization problems collapses completely and the Kolaitis-Thakur 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 second-order 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 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 first-order part is a boolean combination of existential formulas, and (ii) over finite successor structures, the Kolaitis-Thakur hierarchy of minimization problems collapses completely and the Kolaitis-Thakur 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.

Abstract not found