AbstractIn this dissertation we present our work on representational languages for graph theory. We have shown that a knowledge representation can be structured to provide both expressive and procedural power.
Our major research contributions are three. First we have defined representations of infinite sets and recommended that mathematical concepts be considered as sets of objects with relations among them. Second, we have demonstrated how a carefully controlled hierarchy of representations is available through formal languages Third, we have employed a recursive formulation of concepts which enables their application to many of the behaviors of a research mathematician.
Two major families of representations are described: edge-set languages and recursive languages. The edge-set languages have finite expressive power and an interesting potential for hashing digraphs, characterizing classes of graphs and detecting differences among them. The recursive languages have extensible expressive power and impressive procedural power. Recursive languages appear to be an excellent implementation technique for artificial intelligence programs in mathematical research.
Our results enable us to compare the complexity of mathematical concepts (via floors). Concepts represented in our languages can be inverted (to test for the presence of a property) and merged (to combine properties). Conjectures are available through simple search, and most theorems easily proved under the representation.
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