159,549 research outputs found
The Algebraic View of Computation
We argue that computation is an abstract algebraic concept, and a computer is
a result of a morphism (a structure preserving map) from a finite universal
semigroup.Comment: 13 pages, final version will be published elsewher
Algorithms in algebraic number theory
In this paper we discuss the basic problems of algorithmic algebraic number
theory. The emphasis is on aspects that are of interest from a purely
mathematical point of view, and practical issues are largely disregarded. We
describe what has been done and, more importantly, what remains to be done in
the area. We hope to show that the study of algorithms not only increases our
understanding of algebraic number fields but also stimulates our curiosity
about them. The discussion is concentrated of three topics: the determination
of Galois groups, the determination of the ring of integers of an algebraic
number field, and the computation of the group of units and the class group of
that ring of integers.Comment: 34 page
Algebraic extensions in free groups
The aim of this paper is to unify the points of view of three recent and
independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich
and Miasnikov 2002), where similar modern versions of a 1951 theorem of
Takahasi were given. We develop a theory of algebraic extensions for free
groups, highlighting the analogies and differences with respect to the
corresponding classical field-theoretic notions, and we discuss in detail the
notion of algebraic closure. We apply that theory to the study and the
computation of certain algebraic properties of subgroups (e.g. being malnormal,
pure, inert or compressed, being closed in certain profinite topologies) and
the corresponding closure operators. We also analyze the closure of a subgroup
under the addition of solutions of certain sets of equations.Comment: 35 page
Logic Programming and Logarithmic Space
We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation
Linear-algebraic lambda-calculus
With a view towards models of quantum computation and/or the interpretation
of linear logic, we define a functional language where all functions are linear
operators by construction. A small step operational semantic (and hence an
interpreter/simulator) is provided for this language in the form of a term
rewrite system. The linear-algebraic lambda-calculus hereby constructed is
linear in a different (yet related) sense to that, say, of the linear
lambda-calculus. These various notions of linearity are discussed in the
context of quantum programming languages. KEYWORDS: quantum lambda-calculus,
linear lambda-calculus, -calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language
interpreter/simulator file (see "other formats"). See the more recent
arXiv:quant-ph/061219
Computing Multidimensional Persistence
The theory of multidimensional persistence captures the topology of a
multifiltration -- a multiparameter family of increasing spaces.
Multifiltrations arise naturally in the topological analysis of scientific
data. In this paper, we give a polynomial time algorithm for computing
multidimensional persistence. We recast this computation as a problem within
computational algebraic geometry and utilize algorithms from this area to solve
it. While the resulting problem is Expspace-complete and the standard
algorithms take doubly-exponential time, we exploit the structure inherent
withing multifiltrations to yield practical algorithms. We implement all
algorithms in the paper and provide statistical experiments to demonstrate
their feasibility.Comment: This paper has been withdrawn by the authors. Journal of
Computational Geometry, 1(1) 2010, pages 72-100.
http://jocg.org/index.php/jocg/article/view/1
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