Compressed intramolecular dispersion interactions.

The feasibility of the compression of localized virtual orbitals is explored in the context of intramolecular long-range dispersion interactions. Singular value decomposition (SVD) of coupled cluster doubles amplitudes associated with the dispersion interactions is analyzed for a number of long-chain systems, including saturated and unsaturated hydrocarbons and a silane chain. Further decomposition of the most important amplitudes obtained from these SVDs allows for the analysis of the dispersion-specific virtual orbitals that are naturally localized. Consistent with previous work on intermolecular dispersion interactions in dimers, it is found that three important geminals arise and account for the majority of dispersion interactions at the long range, even in the many body intramolecular case. Furthermore, it is shown that as few as three localized virtual orbitals per occupied orbital can be enough to capture all pairwise long-range dispersion interactions within a molecule

Formal Verification of a Geometry Algorithm: A Quest for Abstract Views and Symmetry in Coq Proofs

This extended abstract is about an effort to build a formal description of a triangulation algorithm starting with a naive description of the algorithm where triangles, edges, and triangulations are simply given as sets and the most complex notions are those of boundary and separating edges. When performing proofs about this algorithm, questions of symmetry appear and this exposition attempts to give an account of how these symmetries can be handled. All this work relies on formal developments made with Coq and the mathematical components library

Mining State-Based Models from Proof Corpora

Interactive theorem provers have been used extensively to reason about various software/hardware systems and mathematical theorems. The key challenge when using an interactive prover is finding a suitable sequence of proof steps that will lead to a successful proof requires a significant amount of human intervention. This paper presents an automated technique that takes as input examples of successful proofs and infers an Extended Finite State Machine as output. This can in turn be used to generate proofs of new conjectures. Our preliminary experiments show that the inferred models are generally accurate (contain few false-positive sequences) and that representing existing proofs in such a way can be very useful when guiding new ones.Comment: To Appear at Conferences on Intelligent Computer Mathematics 201