330 research outputs found
Birnbaum Importance Patterns and Their Applications in the Component Assignment Problem
The Birnbaum importance (BI) is a well-known measure that evaluates the relative contribution of components to system reliability. It has been successfully applied to tackling some reliability problems. This dissertation investigates two topics related to the BI in the reliability field: the patterns of component BIs and the BI-based heuristics and meta-heuristics for solving the component assignment problem (CAP).There exist certain patterns of component BIs (i.e., the relative order of the BI values to the individual components) for linear consecutive-k-out-of-n (Lin/Con/k/n) systems when all components have the same reliability p. This study summarizes and annotates the existing BI patterns for Lin/Con/k/n systems, proves new BI patterns conditioned on the value of p, disproves some patterns that were conjectured or claimed in the literature, and makes new conjectures based on comprehensive computational tests and analysis. More importantly, this study defines a concept of segment in Lin/Con/k/n systems for analyzing the BI patterns, and investigates the relationship between the BI and the common component reliability p and the relationship between the BI and the system size n. One can then use these relationships to further understand the proved, disproved, and conjectured BI patterns.The CAP is to find the optimal assignment of n available components to n positions in a system such that the system reliability is maximized. The ordering of component BIs has been successfully used to design heuristics for the CAP. This study proposes five new BI-based heuristics and discusses their corresponding properties. Based on comprehensive numerical experiments, a BI-based two-stage approach (BITA) is proposed for solving the CAP with each stage using different BI-based heuristics. The two-stage approach is much more efficient and capable to generate solutions of higher quality than the GAMS/CoinBonmin solver and a randomization method.This dissertation then presents a meta-heuristic, i.e., a BI-based genetic local search (BIGLS) algorithm, for the CAP in which a BI-based local search is embedded into the genetic algorithm. Comprehensive numerical experiments show the robustness and effectiveness of the BIGLS algorithm and especially its advantages over the BITA in terms of solution quality
On the Notion of Interestingness in Automated Mathematical Discovery
Deciding whether something is interesting or not is of central importance in automated mathematical discovery, as it helps determine both the search space and search strategy for finding and evaluating concepts and conjectures
Sound and Automated Synthesis of Digital Stabilizing Controllers for Continuous Plants
Modern control is implemented with digital microcontrollers, embedded within
a dynamical plant that represents physical components. We present a new
algorithm based on counter-example guided inductive synthesis that automates
the design of digital controllers that are correct by construction. The
synthesis result is sound with respect to the complete range of approximations,
including time discretization, quantization effects, and finite-precision
arithmetic and its rounding errors. We have implemented our new algorithm in a
tool called DSSynth, and are able to automatically generate stable controllers
for a set of intricate plant models taken from the literature within minutes.Comment: 10 page
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
Computer-Assisted Proving of Combinatorial Conjectures Over Finite Domains: A Case Study of a Chess Conjecture
There are several approaches for using computers in deriving mathematical
proofs. For their illustration, we provide an in-depth study of using computer
support for proving one complex combinatorial conjecture -- correctness of a
strategy for the chess KRK endgame. The final, machine verifiable, result
presented in this paper is that there is a winning strategy for white in the
KRK endgame generalized to board (for natural greater than
). We demonstrate that different approaches for computer-based theorem
proving work best together and in synergy and that the technology currently
available is powerful enough for providing significant help to humans deriving
complex proofs
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