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Formal verification is required for systems that require high assurance. Formal verification can require large and complex proofs that can drastically affect the development life cycle. Through the use of a verification system, such proofs can be managed and completed in an efficient manner. A verification system consists of a specification language that can express formal logic, and an automated theorem tool that can be used to verify theorems and conjectures within the specifications. One example of a verification system is Specware. This thesis presents an analysis of Specware against a set of evaluation criteria in order to determine the level of usefulness Specware can have in the verification of high assurance systems. This analysis revealed that Specware contains a powerful specification language capable of representing higher order logic in a simple and expressive manner. Specware is able to represent multiple levels of abstraction and generate proof obligations regarding specification correctness and interlevel mapping. The theorem prover associated with Specware was found to be lacking in capability. Through this analysis we found that Specware has great potential to be an excellent verification system given improvement upon the theorem prover and strengthening of weaknesses regarding linguistic components. / Naval Postgraduate School author (civilian).

Automated reasoning, and in particular automated theorem proving, has become a very important research field within the world of mathematics. Besides being used to verify proofs of theorems, it has also been used to discover proofs of theorems which were previously open problems. In this thesis, an automated reasoning assistant based on Godel's class theory is used to deduce several theorems.

Thesis (M.S.)--Boise State University, 2009. / Includes abstract. Includes bibliographical references (leaves 52-53).

Thesis (M.S.)--Boise State University, 2009. / Title from t.p. of PDF file (viewed June 15, 2010). Includes abstract. Includes bibliographical references (leaves 52-53).

Thesis (Ph. D.)--University of Washington, 2005. / Vita. Includes bibliographical references (p. 155-165).

The Grundlagen der Geometrie brought Euclid’s ancient axioms up to the standards of modern logic, anticipating a completely mechanical verification of their theorems. There are five groups of axioms, each focused on a logical feature of Euclidean geometry. The first two groups give us ordered geometry, a highly limited setting where there is no talk of measure or angle. From these, we mechanically verify the Polygonal Jordan Curve Theorem, a result of much generality given the setting, and subtle enough to warrant a full verification. Along the way, we describe and implement a general-purpose algebraic language for proof search, which we use to automate arguments from the first axiom group. We then follow Hilbert through the preliminary definitions and theorems that lead up to his statement of the Polygonal Jordan Curve Theorem. These, once formalised and verified, give us a final piece of automation. Suitably armed, we can then tackle the main theorem.

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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