6 research outputs found
Implementing an Automatic Differentiator in ACL2
The foundational theory of differentiation was developed as part of the
original release of ACL2(r). In work reported at the last ACL2 Workshop, we
presented theorems justifying the usual differentiation rules, including the
chain rule and the derivative of inverse functions. However, the process of
applying these theorems to formalize the derivative of a particular function is
completely manual. More recently, we developed a macro and supporting functions
that can automate this process. This macro uses the ACL2 table facility to keep
track of functions and their derivatives, and it also interacts with the macro
that introduces inverse functions in ACL2(r), so that their derivatives can
also be automated. In this paper, we present the implementation of this macro
and related functions.Comment: In Proceedings ACL2 2011, arXiv:1110.447
Fourier Series Formalization in ACL2(r)
We formalize some basic properties of Fourier series in the logic of ACL2(r),
which is a variant of ACL2 that supports reasoning about the real and complex
numbers by way of non-standard analysis. More specifically, we extend a
framework for formally evaluating definite integrals of real-valued, continuous
functions using the Second Fundamental Theorem of Calculus. Our extended
framework is also applied to functions containing free arguments. Using this
framework, we are able to prove the orthogonality relationships between
trigonometric functions, which are the essential properties in Fourier series
analysis. The sum rule for definite integrals of indexed sums is also
formalized by applying the extended framework along with the First Fundamental
Theorem of Calculus and the sum rule for differentiation. The Fourier
coefficient formulas of periodic functions are then formalized from the
orthogonality relations and the sum rule for integration. Consequently, the
uniqueness of Fourier sums is a straightforward corollary.
We also present our formalization of the sum rule for definite integrals of
infinite series in ACL2(r). Part of this task is to prove the Dini Uniform
Convergence Theorem and the continuity of a limit function under certain
conditions. A key technique in our proofs of these theorems is to apply the
overspill principle from non-standard analysis.Comment: In Proceedings ACL2 2015, arXiv:1509.0552
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
Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
Proceedings of the 22nd Conference on Formal Methods in Computer-Aided Design – FMCAD 2022
The Conference on Formal Methods in Computer-Aided Design (FMCAD) is an annual conference on the theory and applications of formal methods in hardware and system verification. FMCAD provides a leading forum to researchers in academia and industry for presenting and discussing groundbreaking methods, technologies, theoretical results, and tools for reasoning formally about computing systems. FMCAD covers formal aspects of computer-aided system design including verification, specification, synthesis, and testing
An investigation into the merits of fuzzy logic control versus classical control.
A project report submitted to the Faculty of
Engineering, University of the
Witwatersrand, Johannesburg, in partial
fulfilment of the requirements for the degree
of Master of Science in Engineering.Up to now the benefits and problems with fuzzy control have not been fully
identified and its role in the control domain needs investigation. The past trend has
been to show that a fuzzy controller can provide better control than classical
control, without examining what is actually being achieved. The aim in this project
report is to give a fair comparison between classical and fuzzy control. Robustness,
disturbance rejection, noise suppression" nonminimurn phase and dead time are
examined for both controllers. The comparison is performed through computer
simulation of classical and fuzzy controlled plant models. Fuzzy control has the
advantage of non-linear performance and the ability to capture linguistic
information. Translating quantitative information into the fuzzy domain is difficult;
therefore when the system is easily mathematically modelled and linear, classical
control is usually better. Which controller should be used depends on the
application, control designer and information available.Andrew Chakane 201