589 research outputs found
Intuition in formal proof : a novel framework for combining mathematical tools
This doctoral thesis addresses one major difficulty in formal proof: removing obstructions
to intuition which hamper the proof endeavour. We investigate this in the context
of formally verifying geometric algorithms using the theorem prover Isabelle, by first
proving the Graham’s Scan algorithm for finding convex hulls, then using the challenges
we encountered as motivations for the design of a general, modular framework
for combining mathematical tools.
We introduce our integration framework — the Prover’s Palette, describing in detail
the guiding principles from software engineering and the key differentiator of our
approach — emphasising the role of the user. Two integrations are described, using
the framework to extend Eclipse Proof General so that the computer algebra systems
QEPCAD and Maple are directly available in an Isabelle proof context, capable of running
either fully automated or with user customisation. The versatility of the approach
is illustrated by showing a variety of ways that these tools can be used to streamline the
theorem proving process, enriching the user’s intuition rather than disrupting it. The
usefulness of our approach is then demonstrated through the formal verification of an
algorithm for computing Delaunay triangulations in the Prover’s Palette
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Computer algebra techniques in object-oriented mathematical modelling.
This thesis proposes a rigorous object-oriented methodology, supported by computer algebra software, to generate and relate features in a mathematical model. Evidence shows that there is little heuristic or theoretical research into this problem from any of the three principal modelling methodologies: 'case study’, ‘scenario’ and ‘generic’. This thesis comprises two other major strands: applications of computer algebra software and the efficacy of symbolic computation in teaching and learning. Developing the principal algorithms in computer algebra has sometimes been done at the expense of ease of use. Developers have also not concentrated on integrating an algebra engine into other software. A thorough review of quantitative studies in teaching and learning mathematics highlights a serious difficulty in measuring the effect of using computer algebra. This arises because of the disparate nature of learning with and without a computer.
This research tackles relationship formulation by casting the problem domain into object-oriented terms and building an appropriate class hierarchy. This capitalises on the fact that specific problems are instances of generic problems involving prototype physical objects. The computer algebra facilitates calculus operations and algebraic manipulation. In conjunction, I develop an object-oriented design methodology applicable to small-scale mathematical modelling. An object model modifies the generic modelling cycle. This allows relationships between features in the mathematical model to be generated automatically. The software is validated by quantifying the benefits of using the object-oriented techniques, and the results are statistically significant.
The principal problem domain considered is Newtonian particle mechanics. Although the Newtonian axioms form a firm basis for a mathematical description of interactions between physical objects, applying them within a particular modelling context can cause problems. The goal is to produce an equation of motion. Applications to other contexts are also demonstrated.
This research is significant because it formalises feature and equation-generation in a novel way. A new modelling methodology ensures that this crucial stage in the modelling cycle is given priority and automated
Making Presentation Math Computable
This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book
An investigation into dynamic and functional properties of prokaryotic signalling networks
In this thesis, I investigate dynamic and computational properties of prokaryotic signalling architectures commonly known as the Two Component Signalling networks and phosphorelays. The aim of this study is to understand the information processing capabilities of different prokaryotic signalling architectures by examining the dynamics they exhibit.
I present original investigations into the dynamics of different phosphorelay architectures and identify network architectures that include a commonly found four step phosphorelay architecture with a capacity for tuning its steady state output to implement different signal-response behaviours viz. sigmoidal and hyperbolic response. Biologically, this tuning can be implemented through physiological processes like regulating total protein concentrations (e.g. via transcriptional regulation or feedback), altering reaction rate constants through binding of auxiliary proteins on relay components, or by regulating bi-functional activity in relays which are mediated by bifunctional histidine kinases. This study explores the importance of different biochemical arrangements of signalling networks and their corresponding response dynamics.
Following investigations into the significance of various biochemical reactions and topological variants of a four step relay architecture, I explore the effects of having different types of proteins in signalling networks. I show how multi-domain proteins in a phosphorelay architecture with multiple phosphotransfer steps occurring on the same protein can exhibit multistability through a combination of double negative and positive feedback loops. I derive a minimal multistable (core) architecture and show how component sharing amongst networks containing this multistable core can implement computational logic (like AND, OR and ADDER functions) that allows cells to integrate multiple inputs and compute an appropriate response.
I examine the genomic distribution of single and multi domain kinases and annotate their partner response regulator proteins across prokaryotic genomes to find the biological significance of dynamics that these networks embed and the processes they regulate in a cell. I extract data from a prokaryotic two component protein database and take a sequence based functional annotation approach to identify the process, function and localisation of different response regulators as signalling partners in these networks.
In summary, work presented in this thesis explores the dynamic and computational properties of different prokaryotic signalling networks and uses them to draw an insight into the biological significance of multidomain sensor kinases in living cells. The thesis concludes with a discussion on how this understanding of the dynamic and computational properties of prokaryotic signalling networks can be used to design synthetic circuits involving different proteins comprising two component and phosphorelay architectures.Dorothy Hodgkin Studentship funded by EPSRC and Microsoft Research
Making Presentation Math Computable
This Open-Access-book addresses the issue of translating mathematical expressions from LaTeX to the syntax of Computer Algebra Systems (CAS). Over the past decades, especially in the domain of Sciences, Technology, Engineering, and Mathematics (STEM), LaTeX has become the de-facto standard to typeset mathematical formulae in publications. Since scientists are generally required to publish their work, LaTeX has become an integral part of today's publishing workflow. On the other hand, modern research increasingly relies on CAS to simplify, manipulate, compute, and visualize mathematics. However, existing LaTeX import functions in CAS are limited to simple arithmetic expressions and are, therefore, insufficient for most use cases. Consequently, the workflow of experimenting and publishing in the Sciences often includes time-consuming and error-prone manual conversions between presentational LaTeX and computational CAS formats. To address the lack of a reliable and comprehensive translation tool between LaTeX and CAS, this thesis makes the following three contributions. First, it provides an approach to semantically enhance LaTeX expressions with sufficient semantic information for translations into CAS syntaxes. Second, it demonstrates the first context-aware LaTeX to CAS translation framework LaCASt. Third, the thesis provides a novel approach to evaluate the performance for LaTeX to CAS translations on large-scaled datasets with an automatic verification of equations in digital mathematical libraries. This is an open access book
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4
Using features for automated problem solving
We motivate and present an architecture for problem solving where an abstraction
layer of "features" plays the key role in determining methods to apply. The system
is presented in the context of theorem proving with Isabelle, and we demonstrate
how this approach to encoding control knowledge is expressively different to
other common techniques. We look closely at two areas where the feature
layer may offer benefits to theorem proving — semi-automation and learning
—
and find strong evidence that in these particular domains, the approach shows
compelling promise. The system includes a graphical theorem-proving user
interface for Eclipse ProofGeneral and is available from the project web page,
http://feasch.heneveld.org
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