<p>Effective definedness checking is crucial for an implementation of a logic with undefinedness. The objective of the MathScheme project is to develop a new approach to mechanized mathematics that seeks to combine the capabilities of computer algebra systems and computer theorem proving systems. Chiron, the underlying logic of MathScheme, is a logic with undefinedness. Therefore, it is important to automate, to the greatest extent possible, the process of checking the definedness of Chiron expressions for the MathScheme project. This thesis provides an overview of information useful for checking definedness of Chiron expressions and presents the design and implementation of an AND/OR tree-based approach for automated definedness checking based on ideas from artificial intelligence. The theorems for definedness checking are outlined first, and then a three-valued AND/OR tree is presented, and finally, the algorithm for reducing Chiron definedness problems using AND/OR trees is illustrated. An implementation of the definedness checking system is provided that is based on the theorems and algorithm. The ultimate goal of this system is to provide a powerful mechanism to automatically reduce a definedness problem to simpler definedness problems that can be easily, or perhaps automatically, checked.</p> / Master of Science (MSc)

<p>MathScheme is a long-range research project being conducted at McMaster University with the aim to develop a mechanized mathematics system in which formal deduction and symbolic computation are integrated from the lowest level. The novel notion of a biform theory that is a combination of an axiomatic theory and an algorithmic theory is used to integrate formal deduction and symbolic computation into a uniform theory. A major focus of the project has currently been on building a library of formalized mathematics called the MathScheme Library. The MathScheme Library is based on the little theories method in which a portion of mathematical knowledge is represented as a network of biform theories interconnected via theory morphisms. In this thesis, we describe a systematic explanation of the underlying techniques which have been used for the construction of the MathScheme Library. Then we describe several algebraic constructions that can derive new useful machinery by leveraging the information extracted from a theory. For instance, we show a construction that can reify the term algebra of a (possibly multi-sorted) theory as an inductive data type.</p> / Master of Science (MSc)