1,874 research outputs found
The Frobenius Problem in a Free Monoid
Given positive integers c1,c2,...,ck with gcd(c1,c2,...,ck) = 1, the Frobenius problem (FP) is to compute the largest integer g(c1,c2,...,ck) that cannot be written as a non-negative integer linear combination of c1,c2,...,ck. The Frobenius problem in a free monoid (FPFM) is a non-commutative generalization of the Frobenius problem. Given words x1,x2,...,xk such that there are only finitely many words that cannot be written as concatenations of words in {x1,x2,...,xk}, the FPFM is to find the longest such words. Unlike the FP, where the upper bound g(c1,c2,...,ck)ā¤max 1ā¤iā¤k ci2 is quadratic, the upper bound on the length of the longest words in the FPFM can be exponential in certain measures and some of the exponential upper bounds are tight. For the 2FPFM, where the given words over Ī£ are of only two distinct lengths m and n with 1<m<n, the length of the longest omitted words is ā¤g(m, m|Ī£|n-m + n - m).
In Chapter 1, I give the definition of the FP in integers and summarize some of the interesting properties of the FP. In Chapter 2, I give the definition of the FPFM and discuss some general properties of the FPFM. Then I mainly focus on the 2FPFM. I discuss the 2FPFM from different points of view and present two equivalent problems, one of which is about combinatorics on words and the other is about the word graph. In Chapter 3, I discuss some variations on the FPFM and related problems, including input in other forms, bases with constant size, the case of infinite words, the case of concatenation with overlap, and the generalization of the local postage-stamp problem in a free monoid. In Chapter 4, I present the construction of some essential examples to complement the theory of the 2FPFM discussed in Chapter 2. The theory and examples of the 2FPFM are the main contribution of the thesis. In Chapter 5, I discuss the algorithms for and computational complexity of the FPFM and related problems. In the last chapter, I summarize the main results and list some open problems.
Part of my work in the thesis has appeared in the papers
Abelian networks II. Halting on all inputs
Abelian networks are systems of communicating automata satisfying a local
commutativity condition. We show that a finite irreducible abelian network
halts on all inputs if and only if all eigenvalues of its production matrix lie
in the open unit disk.Comment: Supersedes sections 5 and 6 of arXiv:1309.3445v1. To appear in
Selecta Mathematic
Geometry of abstraction in quantum computation
Quantum algorithms are sequences of abstract operations, performed on
non-existent computers. They are in obvious need of categorical semantics. We
present some steps in this direction, following earlier contributions of
Abramsky, Coecke and Selinger. In particular, we analyze function abstraction
in quantum computation, which turns out to characterize its classical
interfaces. Some quantum algorithms provide feasible solutions of important
hard problems, such as factoring and discrete log (which are the building
blocks of modern cryptography). It is of a great practical interest to
precisely characterize the computational resources needed to execute such
quantum algorithms. There are many ideas how to build a quantum computer. Can
we prove some necessary conditions? Categorical semantics help with such
questions. We show how to implement an important family of quantum algorithms
using just abelian groups and relations.Comment: 29 pages, 42 figures; Clifford Lectures 2008 (main speaker Samson
Abramsky); this version fixes a pstricks problem in a diagra
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