41 research outputs found
The Quantum Query Complexity of Algebraic Properties
We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.Comment: 13 pages, 0 figure
HR: A System for Machine Discovery in Finite Algebras
We describe the HR concept formation program which invents mathematical definitions and conjectures in finite algebras such as group theory and ring theory. We give the methods behind and the reasons for the concept formation in HR, an evaluation of its performance in its training domain, group theory, and a look at HR in domains other than group theory
Automated theory formation in pure mathematics
The automation of specific mathematical tasks such as theorem proving and algebraic
manipulation have been much researched. However, there have only been a few isolated
attempts to automate the whole theory formation process. Such a process involves
forming new concepts, performing calculations, making conjectures, proving theorems
and finding counterexamples. Previous programs which perform theory formation are
limited in their functionality and their generality. We introduce the HR program
which implements a new model for theory formation. This model involves a cycle of
mathematical activity, whereby concepts are formed, conjectures about the concepts
are made and attempts to settle the conjectures are undertaken.HR has seven general production rules for producing a new concept from old ones and
employs a best first search by building new concepts from the most interesting old
ones. To enable this, HR has various measures which estimate the interestingness of a
concept. During concept formation, HR uses empirical evidence to suggest conjectures
and employs the Otter theorem prover to attempt to prove a given conjecture. If this
fails, HR will invoke the MACE model generator to attempt to disprove the conjecture
by finding a counterexample. Information and new knowledge arising from the attempt
to settle a conjecture is used to assess the concepts involved in the conjecture, which
fuels the heuristic search and closes the cycle.The main aim of the project has been to develop our model of theory formation and
to implement this in HR. To describe the project in the thesis, we first motivate
the problem of automated theory formation and survey the literature in this area.
We then discuss how HR invents concepts, makes and settles conjectures and how
it assesses the concepts and conjectures to facilitate a heuristic search. We present
results to evaluate HR in terms of the quality of the theories it produces and the
effectiveness of its techniques. A secondary aim of the project has been to apply HR to
mathematical discovery and we discuss how HR has successfully invented new concepts
and conjectures in number theory
Automorphism groups of some designs of steiner triple systems and the atomorphism groups of their block intersection graphs
>Magister Scientiae - MScA Steiner triple system of order v is a collection of subsets of size three
from a set of v-elements such that every pair of the elements of the set is
contained in exactly one 3-subset. In this study, we discuss some known
Steiner triple systems and their automorphism groups. We also construct
block intersection graphs of the Steiner triple systems of our consideration
and compare their automorphism groups to the automorphism groups of the
Steiner triple systems