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Notes on finite group theory
2 Preface Group theory is a central part of modern mathematics. Its origins lie in geometry (where groups describe in a very detailed way the symmetries of geometric objects) and in the theory of polynomial equations (developed by Galois, who showed how to associate a finite group with any polynomial equation in such a way that the structure of the group encodes information about the process of...
متن کاملOn the Complexity of Finding Paths in a Two-Dimensional Domain II: Piecewise Straight-Line Paths
The problem of finding a piecewise straight-line path, with a constant number of line segments, in a two-dimensional domain is studied in the Turing machine-based computational model and in the discrete complexity theory. It is proved that, for polynomial-time recognizable domains associated with polynomial-time computable distance functions, the complexity of this problem is equivalent to a di...
متن کاملNotes on Complexity Theory Last updated : November , 2011 Lecture 25
Randomization provides unconditional benefits in many settings; examples include cryptography (where random keys are used to provide protection against an adversary) and distributed computing (where randomness can be used as a means to break symmetry between parties). Randomness also appears to help in algorithm design. But is it possible that, from a complexity-theoretic perspective, randomnes...
متن کاملCharacterizing polynomial time computable functions using theories with weak set existence principles
In this paper we define a sequence of second order theories (with one sort of variables ranging over natural numbers and another sort ranging over sets of natural numbers) whose provably recursive functions from the domain of sets into sets (with a naturally restricted complexity of the graphs) are exactly functions of the corresponding levels of polynomial hierarchy. In particular, the collect...
متن کاملEfficient algorithms for highly compressed data: The Word Problem in Higman's group is in P
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag group is is decidable in polynomial time. Before that the best known upper bound was non-elementary. In the present paper we provide new results for power cir...
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تاریخ انتشار 1992