Learning to Prove Trigonometric Identities

Automatic theorem proving with deep learning methods has attracted attentions recently. In this paper, we construct an automatic proof system for trigonometric identities. We define the normalized form of trigonometric identities, design a set of rules for the proof and put forward a method which can generate theoretically infinite trigonometric identities. Our goal is not only to complete the proof, but to complete the proof in as few steps as possible. For this reason, we design a model to learn proof data generated by random BFS (rBFS), and it is proved theoretically and experimentally that the model can outperform rBFS after a simple imitation learning. After further improvement through reinforcement learning, we get AutoTrig, which can give proof steps for identities in almost as short steps as BFS (theoretically shortest method), with a time cost of only one-thousandth. In addition, AutoTrig also beats Sympy, Matlab and human in the synthetic dataset, and performs well in many generalization tasks

Phylogenetic detection of numerous gene duplications shared by animals, fungi and plants

BACKGROUND: Gene duplication is considered a major driving force for evolution of genetic novelty, thereby facilitating functional divergence and organismal diversity, including the process of speciation. Animals, fungi and plants are major eukaryotic kingdoms and the divergences between them are some of the most significant evolutionary events. Although gene duplications in each lineage have been studied extensively in various contexts, the extent of gene duplication prior to the split of plants and animals/fungi is not clear. RESULTS: Here, we have studied gene duplications in early eukaryotes by phylogenetic relative dating. We have reconstructed gene families (with one or more orthogroups) with members from both animals/fungi and plants by using two different clustering strategies. Extensive phylogenetic analyses of the gene families show that, among nearly 2,600 orthogroups identified, at least 300 of them still retain duplication that occurred before the divergence of the three kingdoms. We further found evidence that such duplications were also detected in some highly divergent protists, suggesting that these duplication events occurred in the ancestors of most major extant eukaryotic groups. CONCLUSIONS: Our phylogenetic analyses show that numerous gene duplications happened at the early stage of eukaryotic evolution, probably before the separation of known major eukaryotic lineages. We discuss the implication of our results in the contexts of different models of eukaryotic phylogeny. One possible explanation for the large number of gene duplication events is one or more large-scale duplications, possibly whole genome or segmental duplication(s), which provides a genomic basis for the successful radiation of early eukaryotes

Divergence of exonic splicing elements after gene duplication and the impact on gene structures

An analysis of human exonic splicing elements in duplicated genes reveals their important role in the generation of new gene structures