A Note on the Use of Sum in the Logic of Proofs

The Logic of Proofs~LP, introduced by Artemov, encodes the same reasoning as the modal logic~S4 using proofs explicitly present in the language. In particular, Artemov showed that three operations on proofs (application~$\cdot$, positive introspection~!, and sum~+) are sufficient to mimic provability concealed in S4~modality. While the first two operations go back to G{\"o}del, the exact role of~+ remained somewhat unclear. In particular, it was not known whether the other two operations are sufficient by themselves. We provide a positive answer to this question under a very weak restriction on the axiomatization of LP.

How do subordinate and dominant species in semi-natural mountain grasslands relate to productivity and land-use change?

Changes in agricultural practices of semi-natural mountain grasslands are expected to modify plant community structure and shift dominance patterns. Using vegetation surveys of 11 sites in semi-natural grasslands of the Swiss Jura and Swiss and French Alps, we determined the relative contribution of dominant, subordinate and transient plant species in grazed and abandoned communities and observed their changes along a gradient of productivity and in response to abandonment of pasturing. The results confirm the humpbacked diversity–productivity relationship in semi-natural grassland, which is due to the increase of subordinate species number at intermediate productivity levels. Grazed communities, at the lower or higher end of the species diversity gradient, suffered higher species loss after grazing abandonment. Species loss after abandonment of pasturing was mainly due to a higher reduction in the number of subordinate species, as a consequence of the increasing proportion of dominant species. When plant biodiversity maintenance is the aim, our results have direct implications for the way grasslands should be managed. Indeed, while intensification and abandonment have been accelerated since few decades, our findings in this multi-site analysis confirm the importance of maintaining intermediate levels of pasturing to preserve biodiversity.

Genomewide association for a dominant pigmentation gene in sheep

Most published genomewide association studies (GWAS) in sheep have investigated recessively inherited monogenic traits. The objective here was to assess the feasibility of performing GWAS for a dominant trait for which the genetic basis was already known. A total of 42 Manchega and Rasa Aragonesa sheep that segregate solid black or white coat pigmentation were genotyped using the SNP50 BeadChip. Previous analysis in Manchegas demonstrated a complete association between the pigmentation trait and alleles of the MC1R gene, setting an a priori expectation for GWAS. Multiple methods were used to identify and quantify the strength of population substructure between black and white animals, before allelic association testing was performed for 49 034 SNPs. Following correction for substructure, GWAS identified the most strongly associated SNP (s26449) was also the closest to the MC1R gene. The finding was strongly supported by the permutation tree-based random forest (RF) analysis. Importantly, GWAS identified unlinked SNP with only slightly lower p-values than for s26449. Random forest analysis indicated these were false positives, suggesting interpretation based on both approaches was beneficial. The results indicate that a combined analytical approach can be successful in studies where a modest number of animals are available and substantial population stratification exists.