Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 141-155). Available also in a digital version from Dissertation Abstracts.

Thesis (M.S.)--Mathematics, Georgia Institute of Technology, 2005. / Belinfante, Johan, Committee Chair ; Green, William, Committee Member ; Manolios, Panagiotis, Committee Member. Includes bibliographical references.

Thesi supervisor: Cesare Tinelli. Includes bibliographic references (p. 214-220).

Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 129-140).

Our work focuses on bringing the first-order reasoning closer to practicalapplications, particularly in software and hardware verification. The aim is to develop techniques that make first-order reasoners more scalablefor large problems and suitable for the applications. In pursuit of this goal the work focuses in three main directions. First, wedevelop an algorithm for an efficient pre-selection of axioms. This algorithmis already being widely used by the community and enables off-the-shelf theoremprovers to work with problems having millions of axioms that would otherwisebe overwhelming for them. Secondly, we focus on the saturation algorithm itself, and develop anew calculus for separate handling of propositional predicates. We also do anextensive research on various ways of clause splitting within the saturationalgorithm. The third main block of our work is focused on the use of saturation basedfirst-order theorem provers for software verification, particularly forgenerating invariants and computing interpolants. We base our work on theoretical results of Kovacs and Voronkov published in2009 on the CADE and FASE conferences. We develop a practical implementationwhich embraces all the extensions of the basic resolution and superposition calculus that are contained in the theorem prover Vampire. We have also developed a unique proof transforming algorithm which optimizes the computed interpolantswith respect to a user specified cost function.

<p>The findings of the study indicate that knowledge of logic does help to improve the ability of students to make logical connections (deductions) between and from<br /> statements. The results of the study, however, do not indicate that knowledge and understanding of logic translates into improved proving ability of mathematical<br /> statements by students.</p>

<p>The findings of the study indicate that knowledge of logic does help to improve the ability of students to make logical connections (deductions) between and from<br /> statements. The results of the study, however, do not indicate that knowledge and understanding of logic translates into improved proving ability of mathematical<br /> statements by students.</p>

Magister Scientiae - MSc / The findings of the study indicate that knowledge of logic does help to improve the ability of students to make logical connections (deductions) between and from statements.The results of the study, however, do not indicate that knowledge and understanding of logic translates into improved proving ability of mathematical statements by students. / South Africa

The discovery of unknown lemmas, case-splits and other so called eureka steps are challenging problems for automated theorem proving and have generally been assumed to require user intervention. This thesis is mainly concerned with the automated discovery of inductive lemmas. We have explored two approaches based on failure recovery and theory formation, with the aim of improving automation of firstand higher-order inductive proofs in the IsaPlanner system. We have implemented a lemma speculation critic which attempts to find a missing lemma using information from a failed proof-attempt. However, we found few proofs for which this critic was applicable and successful. We have also developed a program for inductive theory formation, which we call IsaCoSy. IsaCoSy was evaluated on different inductive theories about natural numbers, lists and binary trees, and found to successfully produce many relevant theorems and lemmas. Using a background theory produced by IsaCoSy, it was possible for IsaPlanner to automatically prove more new theorems than with lemma speculation. In addition to the lemma discovery techniques, we also implemented an automated technique for case-analysis. This allows IsaPlanner to deal with proofs involving conditionals, expressed as if- or case-statements.

In this thesis we describe an approach to automatically invent/explore new mathematical theories, with the goal of producing results comparable to those produced by humans, as represented, for example, in the libraries of the Isabelle proof assistant. Our approach is based on ‘schemes’, which are formulae in higher-order logic. We show that it is possible to automate the instantiation process of schemes to generate conjectures and definitions. We also show how the new definitions and the lemmata discovered during the exploration of a theory can be used, not only to help with the proof obligations during the exploration, but also to reduce redundancies inherent in most theory-formation systems. We exploit associative-commutative (AC) operators using ordered rewriting to avoid AC variations of the same instantiation. We implemented our ideas in an automated tool, called IsaScheme, which employs Knuth-Bendix completion and recent automatic inductive proof tools. We have evaluated our system in a theory of natural numbers and a theory of lists.