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.

Herbivores equalize the seedling height growth of three dominant tree species in an African tropical rain forest

Determining the impact of insect herbivores on forest tree seedlings and saplings is difficult without experimentation in the field. Moreover, this impact may be heterogeneous in time and space because of seasonal rainfall and canopy disturbances, or ‘gaps’, which can influence both insect abundance and plant performance. In this study we used fine netting to individually protect seedlings of Microberlinia bisulcata, Tetraberlinia bifoliolata and Tetraberlinia korupensis trees (Fabaceae = Leguminosae) from insects in 41 paired gap-understorey locations across 80 ha of primary rain forest (Korup, Cameroon). For all species, growth in height and leaf numbers was negligible in the understorey, where M. bisulcata had the lowest survival after c. 2 years. In gaps, however, all species responded positively with pronounced above-ground growth across seasons. When exposed to herbivores their seedling height growth was similar, but in the absence of herbivores, M. bisulcata significantly outgrew both Tetraberlinia species and matched their leaf numbers. This result suggests that insect herbivores might play an important role in maintaining species coexistence by mitigating sapling abundance of the more palatable M. bisulcata, which in gaps was eaten the most severely. The higher ratio in static leaf damage of control-to-caged M. bisulcata seedlings in gaps than understorey locations was consistent with the Plant Vigour Hypothesis. This result, however, did not apply to either Tetraberlinia species. For M. bisulcata and T. korupensis, but not T. bifoliolata (the most shade-tolerant species), caging improved relative seedling survival in the understory locations compared to gaps, providing restricted support for the Limiting Resource Model. Approximately 2.25 years after treatments were removed, the caged seedlings were taller and had more leaves than controls in all three species, and the effect remained strongest for M. bisulcata. We conclude that in this community the impact of leaf herbivory on seedling growth in gaps is strong for the dominant M. bisulcata, which coupled to a very low shade-tolerance contributes to limiting its regeneration. However, because gaps are common to most forests, insect herbivores may be having impacts upon functionally similar tree species that are also characterized by low sapling recruitment much more widely than currently appreciated. An implication for the restoration and management of M. bisulcata populations in forests outside of Korup is that physical protection from herbivores of new seedlings where the canopy is opened by gaps, or by harvesting, should substantially increase its subcanopy regeneration, and thus, too, its opportunities for adult recruitment.

Canonical proof nets for classical logic

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In Richard McKinley (2010) [22], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK∗ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of Richard McKinley (2010) [22]), we give a full proof of (c) for expansion nets with respect to LK∗, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.