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.

Non-canonical actions of Nogo-A and its receptors

Nogo-A is a myelin associated protein and one of the most potent neurite growth inhibitors in the central nervous system. Interference with Nogo-A signaling has thus been investigated as therapeutic target to promote functional recovery in CNS injuries. Still, the finding that Nogo-A presents a fairly ubiquitous expression in many types of neurons in different brain regions, in the eye and even in the inner ear suggests for further functions besides the neurite growth repression. Indeed, a growing number of studies identified a variety of functions including regulation of neuronal stem cells, modulation of microglial activity, inhibition of angiogenesis and interference with memory formation. Aim of the present commentary is to draw attention on these less well-known and sometimes controversial roles of Nogo-A. Furthermore, we are addressing the role of Nogo-A in neuropathological conditions such as ischemic stroke, schizophrenia and neurodegenerative diseases.