303 research outputs found
SgpDec : Cascade (de)compositions of finite transformation semigroups and permutation groups
We describe how the SgpDec computer algebra package can be used for composing and decomposing permutation groups and transformation semigroups hierarchically by directly constructing substructures of wreath products, the so called cascade products.Final Accepted Versio
Isoholonomic Problem and Holonomic Quantum Computation
Geometric phases accompanying adiabatic processes in quantum systems can be
utilized as unitary gates for quantum computation. Optimization of control of
the adiabatic process naturally leads to the isoholonomic problem. The
isoholonomic problem in a homogeneous fiber bundle is formulated and solved
completely.Comment: 7 pages, Proceedings of International Conference on Topology in
Ordered Phases organized by Hokkaido University in March 200
Exploring the concept of interaction computing through the discrete algebraic analysis of the BelousovâZhabotinsky reaction
Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn
Microscopic description of 2d topological phases, duality and 3d state sums
Doubled topological phases introduced by Kitaev, Levin and Wen supported on
two dimensional lattices are Hamiltonian versions of three dimensional
topological quantum field theories described by the Turaev-Viro state sum
models. We introduce the latter with an emphasis on obtaining them from
theories in the continuum. Equivalence of the previous models in the ground
state are shown in case of the honeycomb lattice and the gauge group being a
finite group by means of the well-known duality transformation between the
group algebra and the spin network basis of lattice gauge theory. An analysis
of the ribbon operators describing excitations in both types of models and the
three dimensional geometrical interpretation are given.Comment: 19 pages, typos corrected, style improved, a final paragraph adde
Algebraic hierarchical decomposition of finite state automata : a computational approach
The theory of algebraic hierarchical decomposition of finite state automata
is an important and well developed branch of theoretical computer science
(Krohn-Rhodes Theory). Beyond this it gives a general model for some
important aspects of our cognitive capabilities and also provides possible
means for constructing artificial cognitive systems: a Krohn-Rhodes decomposition
may serve as a formal model of understanding since we comprehend
the world around us in terms of hierarchical representations. In order to
investigate formal models of understanding using this approach, we need
efficient tools but despite the significance of the theory there has been no
computational implementation until this work.
Here the main aim was to open up the vast space of these decompositions
by developing a computational toolkit and to make the initial steps of the
exploration. Two different decomposition methods were implemented: the
VuT and the holonomy decomposition. Since the holonomy method, unlike
the VUT method, gives decompositions of reasonable lengths, it was chosen
for a more detailed study.
In studying the holonomy decomposition our main focus is to develop
techniques which enable us to calculate the decompositions efficiently, since
eventually we would like to apply the decompositions for real-world problems.
As the most crucial part is finding the the group components we
present several different ways for solving this problem. Then we investigate
actual decompositions generated by the holonomy method: automata with
some spatial structure illustrating the core structure of the holonomy decomposition,
cases for showing interesting properties of the decomposition
(length of the decomposition, number of states of a component), and the
decomposition of finite residue class rings of integers modulo n.
Finally we analyse the applicability of the holonomy decompositions as
formal theories of understanding, and delineate the directions for further
research
From quantum circuits to adiabatic algorithms
This paper explores several aspects of the adiabatic quantum computation
model. We first show a way that directly maps any arbitrary circuit in the
standard quantum computing model to an adiabatic algorithm of the same depth.
Specifically, we look for a smooth time-dependent Hamiltonian whose unique
ground state slowly changes from the initial state of the circuit to its final
state. Since this construction requires in general an n-local Hamiltonian, we
will study whether approximation is possible using previous results on ground
state entanglement and perturbation theory. Finally we will point out how the
adiabatic model can be relaxed in various ways to allow for 2-local partially
adiabatic algorithms as well as 2-local holonomic quantum algorithms.Comment: Version accepted by and to appear in Phys. Rev.
A Quantitative Study of Pure Parallel Processes
In this paper, we study the interleaving -- or pure merge -- operator that
most often characterizes parallelism in concurrency theory. This operator is a
principal cause of the so-called combinatorial explosion that makes very hard -
at least from the point of view of computational complexity - the analysis of
process behaviours e.g. by model-checking. The originality of our approach is
to study this combinatorial explosion phenomenon on average, relying on
advanced analytic combinatorics techniques. We study various measures that
contribute to a better understanding of the process behaviours represented as
plane rooted trees: the number of runs (corresponding to the width of the
trees), the expected total size of the trees as well as their overall shape.
Two practical outcomes of our quantitative study are also presented: (1) a
linear-time algorithm to compute the probability of a concurrent run prefix,
and (2) an efficient algorithm for uniform random sampling of concurrent runs.
These provide interesting responses to the combinatorial explosion problem
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