Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large corpora of proofs. This work develops learning-based premise selection in two ways. First, a newly available minimal dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed,extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50% improvement on the benchmark over the Vampire/SInE state-of-the-art system for automated reasoning in large theories.Comment: 26 page

Connectionist Theory Refinement: Genetically Searching the Space of Network Topologies

An algorithm that learns from a set of examples should ideally be able to exploit the available resources of (a) abundant computing power and (b) domain-specific knowledge to improve its ability to generalize. Connectionist theory-refinement systems, which use background knowledge to select a neural network's topology and initial weights, have proven to be effective at exploiting domain-specific knowledge; however, most do not exploit available computing power. This weakness occurs because they lack the ability to refine the topology of the neural networks they produce, thereby limiting generalization, especially when given impoverished domain theories. We present the REGENT algorithm which uses (a) domain-specific knowledge to help create an initial population of knowledge-based neural networks and (b) genetic operators of crossover and mutation (specifically designed for knowledge-based networks) to continually search for better network topologies. Experiments on three real-world domains indicate that our new algorithm is able to significantly increase generalization compared to a standard connectionist theory-refinement system, as well as our previous algorithm for growing knowledge-based networks.Comment: See http://www.jair.org/ for any accompanying file

Simulation of a Texas Hold'Em poker player

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