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.

Clinical aspects, diagnostic challenges and management of patients with neuroendocrine tumors (NETs)

Neuroendocrine tumor (NET) entities are rare malignancies. Higher awareness and improved diagnostic methods have led to an increasing incidence of these diseases, and most oncologists deal with such patients in their daily practice. The symposium on NETs that was held in Merano (Italy) in October 2009 was organized by the German-speaking European School of Oncology (dESO) and gathered specialists from different disciplines of transalpine countries to bring together experiences and observations regarding these tumors. The goal of the meeting and of this review was to illustrate both well- and poorly differentiated NETs and to encourage interdisciplinary approaches.