Discriminating Lambda-Terms Using Clocked Boehm Trees

As observed by Intrigila, there are hardly techniques available in the lambda-calculus to prove that two lambda-terms are not beta-convertible. Techniques employing the usual Boehm Trees are inadequate when we deal with terms having the same Boehm Tree (BT). This is the case in particular for fixed point combinators, as they all have the same BT. Another interesting equation, whose consideration was suggested by Scott, is BY = BYS, an equation valid in the classical model P-omega of lambda-calculus, and hence valid with respect to BT-equality but nevertheless the terms are beta-inconvertible. To prove such beta-inconvertibilities, we employ `clocked' BT's, with annotations that convey information of the tempo in which the data in the BT are produced. Boehm Trees are thus enriched with an intrinsic clock behaviour, leading to a refined discrimination method for lambda-terms. The corresponding equality is strictly intermediate between beta-convertibility and Boehm Tree equality, the equality in the model P-omega. An analogous approach pertains to Levy-Longo and Berarducci Trees. Our refined Boehm Trees find in particular an application in beta-discriminating fixed point combinators (fpc's). It turns out that Scott's equation BY = BYS is the key to unlocking a plethora of fpc's, generated by a variety of production schemes of which the simplest was found by Boehm, stating that new fpc's are obtained by postfixing the term SI, also known as Smullyan's Owl. We prove that all these newly generated fpc's are indeed new, by considering their clocked BT's. Even so, not all pairs of new fpc's can be discriminated this way. For that purpose we increase the discrimination power by a precision of the clock notion that we call `atomic clock'.Comment: arXiv admin note: substantial text overlap with arXiv:1002.257

Data mining an EEG dataset with an emphasis on dimensionality reduction

The human brain is obviously a complex system, and exhibits rich spatiotemporal dynamics. Among the non-invasive techniques for probing human brain dynamics, electroencephalography (EEG) provides a direct measure of cortical activity with millisecond temporal resolution. Early attempts to analyse EEG data relied on visual inspection of EEG records. Since the introduction of EEG recordings, the volume of data generated from a study involving a single patient has increased exponentially. Therefore, automation based on pattern classification techniques have been applied with considerable success. In this study, a multi-step approach for the classification of EEG signal has been adopted. We have analysed sets of EEG time series recording from healthy volunteers with open eyes and intracranial EEG recordings from patients with epilepsy during ictal (seizure) periods. In the present work, we have employed a discrete wavelet transform to the EEG data in order to extract temporal information in the form of changes in the frequency domain over time - that is they are able to extract non-stationary signals embedded in the noisy background of the human brain. Principal components analysis (PCA) and rough sets have been used to reduce the data dimensionality. A multi-classifier scheme consists of LVQ2.1 neural networks have been developed for the classification task. The experimental results validated the proposed methodology