Generalized Satisfiability Problems via Operator Assignments

Schaefer introduced a framework for generalized satisfiability problems on the Boolean domain and characterized the computational complexity of such problems. We investigate an algebraization of Schaefer's framework in which the Fourier transform is used to represent constraints by multilinear polynomials in a unique way. The polynomial representation of constraints gives rise to a relaxation of the notion of satisfiability in which the values to variables are linear operators on some Hilbert space. For the case of constraints given by a system of linear equations over the two-element field, this relaxation has received considerable attention in the foundations of quantum mechanics, where such constructions as the Mermin-Peres magic square show that there are systems that have no solutions in the Boolean domain, but have solutions via operator assignments on some finite-dimensional Hilbert space. We obtain a complete characterization of the classes of Boolean relations for which there is a gap between satisfiability in the Boolean domain and the relaxation of satisfiability via operator assignments. To establish our main result, we adapt the notion of primitive-positive definability (pp-definability) to our setting, a notion that has been used extensively in the study of constraint satisfaction problems. Here, we show that pp-definability gives rise to gadget reductions that preserve satisfiability gaps. We also present several additional applications of this method. In particular and perhaps surprisingly, we show that the relaxed notion of pp-definability in which the quantified variables are allowed to range over operator assignments gives no additional expressive power in defining Boolean relations

Narrow band imaging is a new technique in visualization of recurrent respiratory papillomatos

Objectives/Hypothesis: Recurrent respiratory papillomatosis (RRP) is a rare, benign, wart-like disease for which no curative treatment exists. The goal of treatment is total surgical removal of the epithelial lesions to keep the airway open and the voice sufficient. Therefore, it is essential to visualize all papillomatous lesions. The present study aims to evaluate the sensitivity of additional use of narrow band imaging (NBI) in detecting RRP during microlaryngoscopy. Study Design: Prospective study. Methods: Between January 2011 and July 2011, patients with RRP underwent systematic inspection during microlaryngoscopy using conventional white light (WL) immediately followed by inspection with NBI. Consensus was achieved about the number of lesions and number of RRP suspect lesions. All lesions were subsequently excised and sent for histopathological examination. Results: Eighty-six excisional biopsies were taken in 24 microlaryngoscopies performed in 14 RRP patients. Eleven out of the 13 additional biopsies taken, induced by the second inspection with NBI, proved to be papillomata after histopathological examination. The sensitivity increased from 80% with WL up to 97% with WL + NBI (P <.01), whereas the specificity remained poor (32% and 28%, respectively). Conclusions: NBI is an additional diagnostic tool in increasing the sensitivity of visualizing papillomata during microlaryngoscopy. Keywords: Endoscopy; microlaryngoscopy; imaging; narrow band imaging; laryngeal papillomatosis; papilloma; recurrent respiratory papillomatosis; Level of Evidence: