Limits of Diagonalization and the Polynomial Hierarchy

Determining the computational complexity of problems is a large area of study. It seeks to separate these problems into ones with efficient solutions, and those with inefficient solutions. Of course, the strata is much more fine-grain than this. Of special interest are two classes of problems: P and NP. These have been of much interest to complexity theorists for quite some time, because both contain many instances of important real-world problems, and finding efficient solutions for those in NP would be beneficial for computing applications. Yet with all this attention, there are still important unanswered questions about the two classes. It is known that P ⊆ NP, however it is still unknown whether P = NP or if P ⊂ NP. Before we discuss why this problem is so crucial to complexity theory, an overview of P, NP, and coNP is necessary. The class P is a model of the notion of efficiently solvable , and thus contains all languages (problems) that are decidable in deterministic polynomial time. This means that any language in P has a deterministic Turing Machine (algorithm) that will either accept or reject any input in n^k steps, where n is the length of the input string, and k is a constant. The class NP contains all languages that are decidable in nondeterministic polynomial time. A nondeterministic Turing Machine is one that is allowed to guess the correct path of computation, and seems to be able to reach an accept or reject state faster than if it was forced to run deterministically. It is unknown whether NP is closed under complementation because of this nondeterminism. It is quite easy to show a class of deterministically-solvable languages (such as P) is closed under complementation: we simply reverse the accept and reject states. This method is not viable for a nondeterministic machine, since switching the accept and reject states will result in machine that computes a completely different language. Thus the class coNP is defined as containing the complement of every language in NP. In the rest of this paper we will present structural definitions of P and NP as well as present example languages from each. These structural definitions will give insight into the arrangement of the polynomial hierarchy, which is discussed in section 3. A diagonalization proof is presented in section 4, and an explanation of the general usage of diagonalization follows. In section 5, universal languages are defined and an important result from Kozen is given. In the final section, the limits of diagonalization as they pertain to P and NP are outlined, as well as the same limits for relativized classes

The Interictal State In Epilepsy And Behavior

Epilepsy is one of the most common neurological diseases, affecting up to 1% of the world population. Epilepsy remains poorly understood and there are currently no medications to cure it. Patients with epilepsy have both seizures as well as another type of abnormal activity between seizures, known as interictal spikes. Interictal spikes have thus far been poorly researched, yet growing evidence supports an important role for them in epilepsy. In this project, we first show the high variability between reviewers in marking interictal spikes on intracranial EEG, and then develop and test an automated detection method to solve this problem. Next, we use this automated detection algorithm to identify spikes on intracranial EEG in both tumor and non-tumor patients in order to determine the best spiking parameters to identify the seizure onset zone in each group. We then develop and characterize an animal model of chronic, neocortical interictal spiking to test our observations previously made in human epilepsy and to have a molecularly-accurate model on which to test new therapeutics. Finally, we show that interictal spikes are associated with behavioral changes in this animal model and that a targeted inhibitor can both prevent the development of a spiking focus and normalize behavior

Is laryngeal hypersensitivity the basis of unexplained or refractory chronic cough?

Refractory chronic cough (RCC) and unexplained chronic cough (UCC) are common problems seen in primary care and subspecialty clinics. The role of cough hypersensitivity and laryngeal dysfunction in contributing to the persistence of cough in RCC/UCC is not well recognised. Data of patients with RCC and UCC evaluated in 2019 by an interdisciplinary cough clinic led by a pulmonologist and speech–language pathology team was reviewed. Patients completed validated questionnaires including the Leicester cough questionnaire (LCQ), voice handicap index (VHI) and dyspnoea index (DI) questionnaire at initial encounter. Presence of cough hypersensitivity was based upon a history of allotussia and hypertussia. Laryngeal dysfunction was diagnosed in those with a history of laryngeal paresthesias, throat clearing, voice abnormalities, upper airway dyspnoea and documentation of functional or anatomic laryngeal abnormalities on nasoendoscopy. Of the 60 UCC/RCC patients analysed, 75% of patients were female and 85% were over 40 years of age. Cough hypersensitivity was documented in all patients and multiple cough triggers occurred in 75% of patients. 95%, 50% and 25% of patients reported laryngeal paresthesias, voice abnormalities and upper airway dyspnoea, respectively. Significant associations between LCQ and VHI and DI scores occurred when adjusting for age, sex, ethnicity and body mass index. Laryngeal functional abnormalities were documented on 44 out of 60 patients on nasoendoscopy. Hypertussia, allotussia and laryngeal dysfunction are common in patients with RCC and UCC. Evaluation of UCC and RCC can delineate laryngeal hypersensitivity and allows appropriate treatment to be directed at this phenotype

Influence of selective caries excavation on marginal penetration of class II composite restorations in vitro

Selective caries excavation may support pulp preservation. This in vitro study investigated the influence of selective removal of demineralized dentin on marginal integrity of composite restorations as determined by dye penetration. Dentinal caries-like lesions were produced in the approximal surfaces of 40 extracted human molars (ethylenediaminetetraacetate, 0.5 M, 96 h). The following test procedures were established: complete excavation, selective excavation, and caries-free control. Two class II cavities with enamel at the cervical margins were prepared per tooth and demineralization volume was determined by micro-computed tomography for the purpose of a stratified distribution to receive complete excavation or selective excavation. After complete or selective excavation (30 cavities each), adhesive composite restorations were placed. Cavities without demineralized dentin (20 cavities) served as control. The marginal integrity of restorations was evaluated by dye penetration with and without thermocycling or mechanical loading. Results were analyzed by non-parametrical statistical tests (Mann-WhitneyUTest) with an alpha = 0.05 level of significance. Dye penetration did not differ significantly among completely excavated, selectively excavated, or undemineralized teeth, but was increased by thermocycling and mechanical loading in all experimental groups. Selective caries removal did not increase marginal penetration in class II restorations. The presence of remaining demineralized dentin surrounded by sound dentin did not impair marginal integrity of restorations with margins placed in sound enamel